Calendar for Sequence of the Day in July

Calendar for Sequence of the Day in June * Calendar for Sequence of the Day in August

		Template:Sequence of the Day for July 1 A070308: Numbers such that the sum of each base 10 digit squared adds up to the sum of the nontrivial divisors. { 125, 581, 8549, 16999 } If we take the base 10 digits of these "Canada perfect numbers," square each of them, and add them up, we find that they equal the sum of their nontrivial divisors (excluding 1 and the number itself). For example, with 125, we have $1^{2}+2^{2}+5^{2}=30$. The divisors of 125 are 1, 5, 25, and 125, and we see that 5 and 25 ad up to 30. Prime numbers are excluded from consideration since their nontrivial divisors add up to 0. These numbers were defined by mathematicians at the University of Manitoba to celebrate Canada's 125th anniversary.	Template:Sequence of the Day for July 2 A024702: $\scriptstyle a(n)\,=\,{\frac {({p_{n}}^{2}-1)}{24}}\,$, where $\scriptstyle p_{n},\ n\,\geq \,3,\,$ is a prime congruent to –1 or +1 (mod 6). { 1, 2, 5, 7, 12, 15, 22, 35, 40, 57, 70, ... } For all numbers $\scriptstyle p\,$ and $\scriptstyle q\,$ congruent to –1 or +1 (mod 6) we have $\scriptstyle p^{2}-q^{2}\,$ is divisible by 24, so this applies for the unit 1 or any primes $\scriptstyle p\,$ and $\scriptstyle q\,$ other than 2 or 3. Iterate $\scriptstyle p\,$ through all primes other than 2 and 3 and let $\scriptstyle q\,$ be 1. This is a subsequence of the generalized pentagonal numbers, A001318.	Template:Sequence of the Day for July 3 A020639 lpf(n): least prime dividing n (a(1) = 1) { 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 19, 2, 3, 2, ... } I'm listing the elements without the initial 1. I'm a purist, and given that 1 has no prime factors, I feel a(1) is undefined and should not appear in the sequence. I'm sure I (David W. Wilson) did not include it when I authored the sequence. Why do I find this sequence SOTD-worthy? It is because of a question I posed back in 1997. I noticed that in this sequence, between every pair of 2's, there was an element > 2, easy enough to prove. Similarly, I could show that between every pair of 3's there is an element > 3. I found the same to be true of 5, 7, 11, and 13. I began to think it might true for every prime number, but I could not find a general proof. So I posed my question the seqfan list. Apparently it was interesting, because it sent John Conway, Johan de Jong, Derek Smith, and Manjul Bhargava to the blackboard find a counterexample. After two hours, they found one: n = 126972592296404970720882679404584182254788131, with a(n) = a(n+226) = 113 and all intervening values < 113. Later, Fred Helenius found that the smallest prime that generates a counterexample is 71, for which n = 7310131732015251470110369, with a(n) = a(n+142) = 71, and all intervening values < 71. He also found the earliest known counterexample, n = 2061519317176132799110061, with a(n) = a(n+146) = 73, and all intervening values < 73. I was rather amazed that you had to look so far into the sequence.	Template:Sequence of the Day for July 4 A123456: Sequence name { –20, 56, 55, 56, 55, 56, 51, ... } Paragraph or two of info.	Template:Sequence of the Day for July 5 A177854: Smallest prime of rank $\scriptstyle n\,$. { 2, 3, 11, 131, 1571, 43717, 5032843, 1047774137, ... } This is a re-imagination of the Erdős-Selfridge classification of primes. 2 is rank 0 by definition, and for other primes $\scriptstyle p\,$ the rank is the lesser of the maximum rank of the primes dividing $\scriptstyle p-1\,$ and the maximum rank of the primes dividing $\scriptstyle p+1\,$. There are many interesting questions that could be asked about this sequence. What are $\scriptstyle a(8)\,$, $\scriptstyle a(9)\,$, and $\scriptstyle a(10)\,$? The sequence is infinite: how closely can its growth be bounded? (Look at the logarithmic graph, it seems to follow quite close to a regular pattern) How does the closely-related sequence A169818 behave around random large integers? (This last question is related to the performance of modern $\scriptstyle n-1\,/\,n+1\,/\,n^{2}-1\,$ algorithms!) Trivial bounds: If $\scriptstyle p\,$ and $\scriptstyle q\,$ are the smallest primes of rank $\scriptstyle r-1\,$, then the smallest prime of rank $\scriptstyle r\,$ is at least $\scriptstyle 2p+1\,$ and at most $\scriptstyle c(pq)^{L}\,$ where $\scriptstyle L\,$ is Linnik's constant. These give rise to bounds like $\scriptstyle 2^{n}\,$ and $\scriptstyle c^{L^{n}}\,$, respectively.
Template:Sequence of the Day for July 6 A036236: Least inverse of A015910: smallest integer $\scriptstyle k\,>\,0\,$ such that $\scriptstyle 2^{k}{\bmod {k}}\,=\,n,\,n\,\geq \,0,\,$ or 0 if no such $\scriptstyle k\,$ exists. { 1, 0, 3, 4700063497, 6, 19147, 10669, 25, ... } Fermat's little theorem says that $\scriptstyle {2^{p}}\,\equiv \,2{\pmod {p}}\,$ for odd prime $\scriptstyle p\,$, and on some random day long ago I got to wondering what values $\scriptstyle {2^{n}}{\bmod {n}}\,$ might take for other $\scriptstyle n\,$. I did some experimenting and found that that for small $\scriptstyle n\,$, $\scriptstyle {2^{n}}{\bmod {n}}\,$ took on many small values, but 1 and 3 remained elusive. I asked about them on the seqfan list, and found that $\scriptstyle {2^{n}}{\bmod {n}}\,=\,1\,$ is provably insoluble, while D. H. Lehmer had found the impressively large minimal solution $\scriptstyle n\,$ = 4700063497 for $\scriptstyle {2^{n}}{\bmod {n}}\,=\,3\,$, an impressively large solution for such a simple identity. Joe Crump has done some amazing work with this sequence, see 2^n mod n = c.	Template:Sequence of the Day for July 7 A000032: Lucas numbers (beginning with 2): $\scriptstyle L(n)\,=\,L(n-1)+L(n-2)\,$ { 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ... } John Blythe Dobson connected some dots regarding Lucas and Fibonacci numbers: The comments submitted by Miklos Kristof from Mar 19, 2007 for the Fibonacci numbers (A000045) contain four important identities which have close analogues in the Lucas numbers (A000204): For $\scriptstyle a\,\geq \,b\,$ and odd $\scriptstyle b\,$, $\scriptstyle L(a+b)+L(a-b)\,=\,5F(a)F(b)\,$. For $\scriptstyle a\,\geq \,b\,$ and even $\scriptstyle b\,$, $\scriptstyle L(a+b)+L(a-b)\,=\,L(a)L(b)\,$. For $\scriptstyle a\,\geq \,b\,$ and odd $\scriptstyle b\,$, $\scriptstyle L(a+b)-L(a-b)\,=\,L(a)L(b)\,$. For $\scriptstyle a\,\geq \,b\,$ and even $\scriptstyle b\,$, $\scriptstyle L(a+b)-L(a-b)\,=\,5F(a)F(b)\,$.	Template:Sequence of the Day for July 8 A181780: Numbers $\scriptstyle n\,$ which are Fermat pseudoprimes to some base $\scriptstyle b\,$, with $\scriptstyle 2\,\leq \,b\,\leq \,n-2\,$. { 15, 21, 25, 28, 33, 35, 39, ... } For example 15 is Fermat pseudoprime to every base of the form $\scriptstyle 15k+4\,$ and $\scriptstyle 15k+11\,$ with $\scriptstyle k\,\geq \,0\,$. The sequence shows that the distribution of the Fermat pseudoprimes is the opposite of rare, if it is not determined on a fixed base $\scriptstyle b\,$. There are 429 Fermat pseudoprimes below 1000, but only 168 prime numbers. Fermat pseudoprimes to all bases $\scriptstyle b\,$, with $\scriptstyle 2\,\leq \,b\,\leq \,n-2\,$ are called Carmichael numbers.	Template:Sequence of the Day for July 9 A156695: Odd numbers which are not of the form $\scriptstyle p+2^{a}+2^{b}\,$, with $\scriptstyle a\,\geq \,b\,>\,0\,$, and $\scriptstyle p\,$ prime. { 1, 3, 5, 6495105, 848629545, 1117175145, 2544265305, 3147056235, 3366991695, ... } Can you find a member (> 5) which is not divisible by 255? There should be infinitely many, but none have been found!	Template:Sequence of the Day for July 10 A060851: $\scriptstyle (2n-1)~3^{(2n-1)}\,$ { 3, 81, 1215, 15309, 177147, 1948617, ... } Simon Plouffe published a helpful list of odd zeta values from ζ(9) up to ζ(2051) with a precision of 1000 decimal digits — an older version of this list also covered ζ(3), ζ(5) and ζ(7) and is now available in separate files with more digits. You could hardwire these values in projects (e.g. RxShell (ZETA and RX.MATH)) where computing small odd ζ(2n+1) values on the fly is no option, because it would take far too long. The odd ζ(2n+1) values can be used to determine Euler's constant C0 among other things.  A060851 can then help to verify computed values for Euler's C0 (usually referred to as $\scriptstyle \gamma \,$ (gamma) or Euler-Mascheroni constant,) Apéry's constant ζ(3), or ln(2) after dynamic changes of the chosen precision, e.g., Open Object Rexx NUMERIC DIGITS 500.	Template:Sequence of the Day for July 11 A004022: Repunit primes in base 10 { 11, 1111111111111111111, 11111111111111111111111, ... } These primes are of the form $\scriptstyle {\frac {10^{n}-1}{9}}\,$. The exponent $\scriptstyle n\,$ must itself also be prime. Those three repunit primes have 2, 19 and 23 digits, respectively. The next repunit prime has 317 digits!	Template:Sequence of the Day for July 12 A097478: Atomic numbers of halogens in the periodic table { 9, 17, 35, 53, 85 } This sequence demonstrates that there is a mathematical structure to physical matter. We can compute this sequence mathematically by taking the formula for magic numbers, which happen to be the atomic numbers of the noble gases and then subtracting 1. The outermost shell of a noble gas atom has as many electrons as it can possibly have. Halogen atoms, on the other hand, can admit precisely one more electron to their outer shell (specifically in their outer p subshell.) Note that as you go down the periodic table the elements tend to acquire more metallic properties, which happens to be the case of astatine (85), which has somewhat more metallic properties than iodine (53), although astatine is still considered a halogen.
Template:Sequence of the Day for July 13 A118372: $\scriptstyle {\mathcal {S}}\,$-perfect numbers { 6, 24, 28, 96, 126, 224, 384, 496, ... } Also called Granville numbers after the mathematician Andrew Granville. In October 2010, William Rex Marshall offered a much clearer explanation† of these numbers as well as a few more terms to list in the sequence (which includes the familiar 2-perfect numbers of A000396). † William Marshall, Who understands Granville numbers?, posting to SeqFan on Oct 28 2010	Template:Sequence of the Day for July 14 A106108: Rowland's sequence: $\scriptstyle a(1)\,=\,7\,$, then $\scriptstyle a(n)\,=\,a(n-1)+\gcd(n,a(n-1)),\,n\,>\,1\,$. { 7, 8, 9, 10, 15, 18, 19, 20, 21, 22, 33, 36, 37, 38, ... } The difference between two consecutive terms is a noncomposite number (either 1 or a prime number.)	Template:Sequence of the Day for July 15 A001783: $\scriptstyle n\,$-phi-torial, or phi-torial of $\scriptstyle n\,$ is the product of all positive integers up to and coprime to $\scriptstyle n$. $\varphi _{_{_{!}}}(n)=\prod _{\stackrel {i=1}{i\bot n}}^{n}i=\prod _{\stackrel {i=1}{(i,n)=1}}^{n}i,\quad n\,\geq \,1.\,$ Here $\scriptstyle i\bot n\,$ means that $\scriptstyle i$ and $\scriptstyle n$ are relatively prime and $\scriptstyle (i,n)\,$ is the GCD of $\scriptstyle i\,$ and $n.$ { 1, 1, 2, 3, 24, 5, 720, 105, 2240, ... } The phi-torial of $\scriptstyle n\,$ is a divisor of the factorial of $\scriptstyle n\,$. $\varphi _{_{_{!}}}(n)=n!\ /\ {\prod _{\stackrel {i=1}{i\not \bot n}}^{n}i}.$ We take the positive integers below $\scriptstyle n\,$, cull out those $\scriptstyle i\,$ that have prime factors in common with $\scriptstyle n\,$ and then multiply the residual together. For example, with $\scriptstyle n\,$ = 8, we cull out 2, 4, 6, 8, and multiply 1, 3, 5, 7, giving 105. Of course when $\scriptstyle n\,$ is prime this works out to $\scriptstyle \varphi _{_{_{!}}}(n)\,=n!/n=\,(n-1)!\,$.	Template:Sequence of the Day for July 16 A004215: Numbers that are the sum of 4 but no fewer nonzero squares. { 7, 15, 23, 28, 31, 39, 47, ... } From Waring's problem we know that all positive integers can be expressed as a sum of at most four squares. In fact, most numbers can be expressed as the sum of fewer than four squares, even though the perfect squares thin out quadratically.	Template:Sequence of the Day for July 17 A083216: Fibonacci-like primefree sequence { 20615674205555510, 3794765361567513, 24410439567123023, 28205204928690536, ... } Like the sequence of Fibonacci numbers, this sequence has two initial terms set, and the rest are given by the familiar recurrence relation $\scriptstyle a_{n}\,=\,a_{n-1}+a_{n-2}\,$. The amazing thing about this sequence is that there are no prime numbers among any of its terms even though the initial terms are coprime. And whereas the search for prime numbers seeks larger and larger primes, the search for primefree sequences seeks smaller initial terms. When in 1964, Ronald Graham (of Graham's number fame) proved that this kind of sequence is possible, his example had initial terms with more than thirty digits each (in base 10). Donald Knuth found a 17-digit pair in 1990, and later that same year Herbert Wilf found this sequence with slightly smaller initial terms. The current record is a 12-digit pair found by John W. Nicol in 1999.	Template:Sequence of the Day for July 18 A002372: Number of decompositions of $\scriptstyle 2n,\,n\,\geq \,1,\,$ into ordered sums of two odd primes { 0, 0, 1, 2, 3, 2, 3, 4, 4, ... } If the Goldbach conjecture is true, then $\scriptstyle a(n)\,>\,0\,$ for all $\scriptstyle n\,>\,2\,$. Note that this sequence does not require the primes to be distinct to be counted, and counts solutions of different primes twice (for example, for 8, both 3 + 5 and 5 + 3 are counted). Also note in the b-file that above about 3000, all values of this sequence are over a hundred.	Template:Sequence of the Day for July 19 A059999: ${\tfrac {1}{6}}n^{5}-{\tfrac {19}{8}}n^{4}+{\tfrac {51}{4}}n^{3}-{\tfrac {253}{8}}n^{2}+{\tfrac {445}{12}}n-14\,$ { 2, 3, 5, 7, 11, 42, 168, ... } If instead of the first seven terms I had given the first five, you might have guessed that the next five terms are 13, 17, 19, 23, 29. That is, unless you know by heart that the sequence of prime numbers has a much lower A-number in the OEIS, namely A000040. Rainer Rosenthal "deliberately contrived" this sequence "to begin with [the] first five primes" in order to illustrate the "absurdity of many 'guess the next term' puzzles." I think this sequence also makes two other important points. First, it shows the futility of searching for an arithmetic formula that will give the prime numbers in order, since such a level of complication is required for a formula that gives just the first five primes in order. Second, it shows the usefulness of the OEIS in finding formulae when one only knows a few terms of the sequence.
Template:Sequence of the Day for July 20 A007955: Product of divisors of $\scriptstyle n,\,n\,\geq \,1,\,$ $\prod _{\stackrel {i=1}{i\mid n}}^{n}i=\prod _{i=1}^{n}i^{[i\mid n]},\quad n\geq 1.\,$ where $\scriptstyle [\cdot ]\,$ is the Iverson bracket. { 1, 2, 3, 8, 5, 36, 7, ... } In Only Problems, Not Solutions!, Florentin Smarandache lists this sequence (page 24), describes it, and urges the reader to study it, saying nothing else. It is easy enough to see that each prime number $\scriptstyle p\,$ will occur once and only once in this sequence, namely, at $\scriptstyle a(p)\,$. T. D. Noe goes much further than that and actually proves that each term of this sequence occurs just once. Of course not every positive integer occurs in this sequence (e.g., squares of primes). The product of divisors of $\scriptstyle n\,$ is sometimes called the divisorial of $\scriptstyle n\,$.	Template:Sequence of the Day for July 21 A175607 Largest number $\scriptstyle x\,$ such that the greatest prime factor of $\scriptstyle x^{2}-1\,$ is $\scriptstyle p_{n},\,n\,\geq \,1.\,$ { 3, 17, 161, 8749, 19601, 246401, 672281, ... } It surprised me, but after a little reflection, it makes a lot of sense: for any prime $\scriptstyle p\,$, there is a short (that is to say, finite) list of numbers $\scriptstyle x\,$ such that $\scriptstyle x^{2}-1\,$ has $\scriptstyle p\,$ as its largest prime factor (Cf. Largest prime dividing n^2 - 1, n ≥ 2.) For example, for $\scriptstyle p_{1}\,=\,2\,$, we find that $\scriptstyle 3^{2}-1\,=\,8\,=\,2^{3}\,$. No larger number satisfies $\scriptstyle k^{2}\,\equiv \,1{\pmod {2^{m}}}\,$. Even-indexed powers of two are squares of a smaller power of two, and therefore $\scriptstyle {\sqrt {2^{m}+1}}\,$ is not an integer if $\scriptstyle m\,$ is even. The reason why higher odd-indexed powers of two don't work for this purpose is, as so many math textbooks say, left as an exercise for the Reader. According to Artur Jasinski, every prime $\scriptstyle p\,$ has a corresponding $\scriptstyle x\,$ with the property described here.	Template:Sequence of the Day for July 22 A038770 Numbers divisible by at least one of their base 10 digits. { ..., 20, 21, 22, 24, 25, 26, 28, ... } I choose to start the listing at 20 because 1-digit numbers are trivially members of this sequence, while 2-digit numbers with a 1 among their digits are also trivially in this sequence. The choice of base 10 here encourages multiples of 5 but discourages odd multiples of 3 or 7. Regardless of the base, most prime numbers are excluded if in that base they are more than one digit and none of their digits in that base are 1.	Template:Sequence of the Day for July 23 A000945: The Euclid-Mullin sequence: $\scriptstyle a(1)\,=\,2\,$, $\scriptstyle a(n)\,$ is the smallest prime factor of $\scriptstyle 1+\prod _{k=1}^{n-1}a(k)\,$ { 2, 3, 7, 43, 13, 53, 5, 6221671, ... } The first four terms coincide with the Sylvester sequence (where each term is the product of all preceding terms plus 1) (A000058), an (thus strictly increasing) infinite coprime sequence whose terms may be prime or composite.	Template:Sequence of the Day for July 24 A115368: Decimal expansion of first zero of the Bessel function $\scriptstyle J_{0}(z)\,$ 2.4048255576957727686... The most famous zeros of a function are of course the nontrivial zeros of the Riemann zeta function. But other functions also have interesting zeros which mathematicians have studied, the Bessel function $\scriptstyle J_{0}(z)\,$ being one of them. (Mathematicians have also studied the zeros of the Bessel functions $\scriptstyle J_{n}(z)\,$ with other integer values of $\scriptstyle n\,$.) See Weisstein, Eric W., Bessel Function of the First Kind, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html]	Template:Sequence of the Day for July 25 A000058: Sylvester's sequence, which gives the denominators $\scriptstyle a(n)\,$ for the greedy Egyptian representation of 1: $1={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{43}}+{\frac {1}{1807}}+\ldots \,$ (Of course the numerators are given by A000012.) The denominators are given by the quadratic recurrence $a(1)=2,\,$ $a(n)=a(n-1)(a(n-1)-1)+1$ $=a(n)^{2}-a(n)+1,\quad n\geq 2.\,$ and by the formula (which shows that it is an infinite coprime sequence) $a(n)=1+\prod _{i=1}^{n-1}a(i),\,$ where for $\scriptstyle n\,=\,1\,$ we get 1 + (empty product, i.e. 1) = 2.	Template:Sequence of the Day for July 26 A034872: Central column of Lozanić's triangle { 1, 1, 1, 2, 4, 6, 10, 19, 38, 66, 126, 236, 472, 868, ... } Note that although the values of odd-numbered columns are only listed once, they appear twice in the triangle. Lozanić's triangle A005418 Row sums $\sum _{i=0}^{n}T(n,i)$ $n\,$ = 0 1 1 1 1 1 2 2 1 1 1 3 3 1 2 2 1 6 4 1 2 4 2 1 10 5 1 3 6 6 3 1 20 6 1 3 9 10 9 3 1 36 7 1 4 12 19 19 12 4 1 72
Template:Sequence of the Day for July 27 A008892: Aliquot sequence starting at 276. { 276, 396, 696, 1104, 1872, 3770, 3790, ... } When The Encyclopedia of Integer Sequences was published in 1995, it was believed that there exist numbers such that their aliquot sequences neither terminate nor eventually cycle (the aliquot sequences for perfect numbers, amicable pairs and sociable numbers of order t [with t = 1 , t = 2 and t > 2 respectively] very obviously cycle, while others may take a dozen or so terms to reach 1 [after hitting a prime number] which begets the empty sum, i.e. 0, and the sequence terminates) but no example was known. D. H. Lehmer succeeded in ruling out the first thousand positive integers except for 276, 552, 564, 660 and 966. Today, 276 still looks like a good candidate for a number with an aliquot sequence that neither terminates nor eventually cycles!? According to FactorDB.com, the last term in this sequence to be factored and checked is a (1663), a number which has among its proper divisors 2, 32, 7, 9551, 26441307811, and 72269874661669001519651. An non-terminating aliquot sequence which never cycle would have to manage to never hit neither a previous term nor any prime number while growing (monotonically or not) without bounds!	Template:Sequence of the Day for July 28 A006127: $2^{n}+n$ { 1, 3, 6, 11, 20, 37, 70, 135, 264, 521, 1034, 2059, 4108, 8205, ... } In The Book of Numbers, John Conway and Richard Guy coyly describe this sequence as "not very different from the powers of 2."	Template:Sequence of the Day for July 29 A029578: An obvious mixture of two sequences { 0, 0, 1, 2, 2, 4, 3, 6, 4, 8, 5, 10, 6, 12, 7, 14, 8, ... } Can you tell what the two sequences being mixed are?	Template:Sequence of the Day for July 30 A105417: Numbers that are pandigital in Roman numerals, using each of the symbols I, V, X, L, C, D and M at least once. { MCDXLIV, MCDXLVI, MCDXLVII, MCDXLVIII, MCDLXIV, MCDLXVI, MCDLXVII, ... } We consider the largest member of this sequence to be MMMDCCCLXXXVIII, since none of the many systems to cope with greater numbers became standard prior to Roman numerals being displaced by modern numerals. There are 192 = 3 * 43 pandigital Roman numerals 3 ways to start: M*, MM*, MMM* (since no Roman numeral stood for 5000) 4 ways for the hundreds: *CD*, *DC*, *DCC*, *DCCC* 4 ways for the decades: *XL*, *LX*, *LXX*, *LXXX* 4 ways for the units: *IV, *VI, *VII, *VIII Note that the Roman numeral representation of numbers is not admissible in the OEIS, thus the decimal representation of those numbers is used in A105417.	Template:Sequence of the Day for July 31 A173419: Length of shortest computation yielding $\scriptstyle n,\,n\,\geq \,1,\,$ using addition, subtraction and multiplication. { 0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 4, 4, 5, ... } So you start with $\scriptstyle x_{0}\,=\,1,\,x_{m}\,=\,n\,$ and then through successive $\scriptstyle x_{k}\,=\,x_{i}+x_{j}\,$, $\scriptstyle x_{k}\,=\,x_{i}x_{j}\,$, or $\scriptstyle x_{k}\,=\,x_{i}-x_{j}\,$ operations for some $\scriptstyle 0\,\leq \,i,\,j\,<\,k\,$, you get to $\scriptstyle n\,$. The fun is of course getting there in as few steps as possible. Prime $\scriptstyle n\,$ are not necessarily more difficult to get to than their neightbors: 29, for example, requires five operations, just as 28 and 30 do.