A156749 For all numbers k(n) congruent to -1 or +1 (mod 4) starting with k(n) = {3,5,7,9,11,...}, a(k(n)) is incremented by the congruence (mod 4) if k(n) is prime and by 0 if k(n) is composite.

-1, 0, -1, -1, -2, -1, -1, 0, -1, -1, -2, -2, -2, -1, -2, -2, -2, -1, -1, 0, -1, -1, -2, -2, -2, -1, -1, -1, -2, -1, -1, -1, -2, -2, -3, -2, -2, -2, -3, -3, -4, -4, -4, -3, -3, -3, -3, -2, -2, -1, -2, -2, -3, -2, -2, -1, -1, -1, -1, -1, -1, -1, -2, -2, -3, -3, -3, -2, -3, -3

Offset

1,5

Comments

The fact that a(k(n)) is predominantly negative exhibits the Chebyshev Bias (where the congruences that are not quadratic residues generally lead in the prime number races, at least for "small" integers, over the congruences that are quadratic residues).

This bias seems caused (among other causes?) by the presence of all those squares (even powers) coprime to 4 taking away opportunities for primes to appear in the quadratic residue class +1 (mod 4), while the non-quadratic residue class -1 (mod 4) is squarefree.

The density of squares congruent to +1 (mod 4) is 1/(4*sqrt(k(n))) since 1/2 of squares are congruent to +1 (mod 4), while the density of primes in either residue class -1 or +1 (mod 4) is 1/(phi(4)*log(k(n))), with phi(4) = 2.

Here 1 is quadratic residue mod 4, but 3 (or equivalently -1) is quadratic non-residue mod 4. All the even powers (included in the squares) map congruences {-1, +1} to {+1, +1} respectively and so contribute to the bias, whereas all the odd powers map {-1, +1} to {-1, +1} respectively and so do not contribute to the bias.

One would then expect the ratio of this bias, if caused exclusively by the even powers, relative to the number of primes in either congruences to asymptotically tend towards to 0 as k(n) increases (since 1/(4*sqrt(k(n))) is o(1/(phi(4)*log(k(n)))).

The persistence or not of such bias in absolute value then does not contradict The Prime Number Theorem for Arithmetic Progressions (Dirichlet) which states that the asymptotic (relative) ratio of the count of prime numbers in each congruence class coprime to m tends to 1 in the limit towards infinity. (Cf. 'Prime Number Races' link below.)

Also, even if this bias grows in absolute value, it is expected to be drowned out (albeit very slowly) by the increasing fluctuations in the number of primes in each congruence class coprime to 4 since, assuming the truth of the Riemann Hypothesis, their maximum amplitude would be, with x standing for k(n) in our case, h(x) = O(sqrt(x)*log(x)) <= C*sqrt(x)*log(x) in absolute value which gives relative fluctuations of order h(x)/x = O(log(x)/sqrt(x)) <= C*log(x)/sqrt(x) in the densities of primes pi(x, {4, 1})/x and pi(x, {4, 3})/x in either congruence class.

Since 1/(4*sqrt(x)) is o(log(x)/sqrt(x)) the bias will eventually be overwhelmed by the "pink noise or nearly 1/f noise" corresponding to the fluctuations in the prime densities in either congruence class. The falsehood of the Riemann Hypothesis would imply even greater fluctuations since the RH corresponds to the minimal h(x).

We get pink noise or nearly 1/f noise if we consider the prime density fluctuations of pi(x, {4, k})/x as an amplitude spectrum over x (with a power density spectrum of (C*log(x)/sqrt(x))^2 = ((C*log(x))^2)/x and see x as the frequency f. This power density spectrum is then nearly 1/x and would have nearly equal energy (although slowly increasing as (C*log(x))^2) for each octave of x. (Cf. 'Prime Numbers: A Computational Perspective' link below.)

Among the positive integers k(n) up to 100000 that are congruent to -1 or +1 (mod 4) [indexed from n = 1 to 49999, with k(n) = 4*ceiling(n/2) + (-1)^n], a tie is attained or maintained, with a(k(n)) = 0, for only 34 integers and that bias favoring the non-quadratic residue class -1 (mod 4) gets violated only once, i.e., a(k(n)) = +1, for index n = 13430 (corresponding to the prime k(n) = 26861 congruent to +1 (mod 4) since n is even) where the congruence +1 leads once!

References

Richard E. Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective

Links

Daniel Forgues, Table of n, a(n) for n = 1..49999

A. Granville and G. Martin, Prime Number Races, arXiv:math/0408319 [math.NT], 2004.

Eric Weisstein, Chebyshev Bias

Wikipedia, Pink noise

Formula

a(n) = -A066520(2*n+1) = A066339(2*n+1) - A066490(2*n+1). - Jonathan Sondow, May 17 2013

Mathematica

Table[Which[!PrimeQ[2*n+1], 0, Mod[2*n+1, 4] == 1, 1, True, -1], {n, 1, 100}] // Accumulate (* Jean-François Alcover, Dec 09 2014 *)

Crossrefs

Cf. A066339, A066490, A066520, A007350, A007351, A038691, A096628, A156707, A156709, A156706, A101264, A075743.

Keyword

sign

Author

_Daniel Forgues_, Feb 14 2009

Extensions

Edited by _Daniel Forgues_, Mar 01 2009, Mar 29 2009

Status

approved

A174658 Balanced ternary numbers with equal count of negative trits and positive trits.

0, 2, 6, 8, 16, 18, 20, 24, 26, 32, 46, 48, 52, 54, 56, 60, 62, 70, 72, 74, 78, 80, 86, 96, 98, 104, 130, 136, 138, 142, 144, 146, 154, 156, 160, 162, 164, 168, 170, 178, 180, 182, 186, 188, 194, 208, 210, 214, 216, 218, 222, 224, 232, 234, 236, 240, 242, 248, 258

Offset

1,2

Comments

Numbers for which the sum of trits is zero.

Links

Daniel Forgues, Table of n, a(n) for n=1..14309

Mathematica

(* First run the program for A065363 to define balTernDigits *) A174658 = Select[Range[0, 299], Plus@@balTernDigits[#] == 0 &] (* Alonso del Arte, Feb 24 2011 *)

Prog

(Python)
def a(n):
    s=0
    x=0
    while n>0:
        x=n%3
        n //= 3
        if x==2:
            x=-1
            n+=1
        s+=x
    return s
print([n for n in range(301) if a(n)==0]) # Indranil Ghosh, Jun 07 2017

Crossrefs

Cf. A174657, A174659, A065363, A140267, A257869.

Keyword

base,nonn

Author

_Daniel Forgues_, Mar 26 2010

Status

approved

A131605 Perfect powers of nonprimes (m^k where m is a nonprime positive integer and k >= 2).

1, 36, 100, 144, 196, 216, 225, 324, 400, 441, 484, 576, 676, 784, 900, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 1600, 1728, 1764, 1936, 2025, 2116, 2304, 2500, 2601, 2704, 2744, 2916, 3025, 3136, 3249, 3364, 3375, 3600, 3844, 3969, 4225, 4356, 4624

Offset

1,2

Comments

Although 1 is a square, is a cube, and so on..., 1 is sometimes excluded from perfect powers since it is not a well-defined power of 1 (1 = 1^k for any k in [2, 3, 4, 5, ...])

Links

Daniel Forgues, Table of n, a(n) for n=1..8649

Prog

(PARI) isok(n) = if (n == 1, return (1), return (ispower(n, , &np) && (! isprime(np)))); \\ Michel Marcus, Jun 12 2013

Crossrefs

Cf. A000961, A001597, A024619, A025475.

Keyword

nonn

Author

_Daniel Forgues_, May 27 2008

Extensions

Klaus Brockhaus previously provided a table of n, a(n) for n=1..1323, May 28 2008

Status

approved

A004231 Ackermann's sequence: n^^n := n^n^n^...^n (with n n's).

1, 1, 4, 7625597484987

Offset

0,3

Comments

Using Knuth's arrow notation, this is n^^^2 (n-penta-2) or n^^n (n-tetra-n). - Andrew Robbins, Apr 16 2009

Comment from Trevor Green: The fourth term in this sequence has about as many digits - 8.07 * 10^153 - as the *square* of the number of protons in the universe.

We could prepend a(0) = 1 (since 0^^0 = 1, that is, the "empty power tower" gives the "empty product"). - _Daniel Forgues_, May 17 2013

The last 60 decimal digits of a(4) are ...67586985427238232605843019607448189676936860456095261392896. - _Daniel Forgues_, Jun 25 2016

From _Daniel Forgues_, Jul 06 2016: (Start)

a(4) has (the following number having 154 decimal digits)

80723047260282253793826303970853990300713679217387 \

43031867082828418414481568309149198911814701229483 \

451981557574771156496457238535299087481244990261351117 decimal digits.

a(4) = 4^4^4^4 = 4^

13407807929942597099574024998205846127479365820592 \

39337772356144372176403007354697680187429816690342 \

7690031858186486050853753882811946569946433649006084096,

the exponent of 4 having 155 decimal digits. (End)

The fractional part of 4^4^4*log[10](4) starts .373100157363599870..., so the first few digits of a(4) are 23610226714597313.... - Robert Israel, Jul 06 2016

Links

Table of n, a(n) for n=0..3.

W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Ann. 99 (1928), 118-133, DOI:10.1007/BF01459088.

Eric Weisstein's World of Mathematics, Ackermann Number

Wikipedia, Knuth's up-arrow notation

Maple

b:= (n, i)-> `if`(i=0, 1, n^b(n, i-1)):
a:= n-> b(n, n):
seq(a(n), n=0..3);  # Alois P. Heinz, Aug 22 2017

Mathematica

a[n_] := If[n == 0, 1, Nest[n^#&, n, n-1]];
Table[a[n], {n, 0, 3}] (* Jean-François Alcover, Mar 19 2019 *)

Crossrefs

Cf. A027747, A008868 (which mentions an older estimate for the same number), A266200.

Main diagonal of A321312.

Keyword

nonn

Author

Daniel Wild (wild(AT)edumath.u-strasbg.fr)

Status

approved

A164572 Numbers k such that k and k+4 are both prime powers.

1, 3, 4, 5, 7, 9, 13, 19, 23, 25, 27, 37, 43, 49, 67, 79, 97, 103, 109, 121, 127, 163, 169, 193, 223, 229, 239, 277, 289, 307, 313, 343, 349, 379, 397, 439, 457, 463, 487, 499, 613, 643, 673, 729, 739, 757, 769, 823, 853, 859, 877, 883, 907, 937, 967, 1009, 1087

Offset

1,2

Comments

Numbers n such that n + (0, 4) is a prime power pair.

A generalization of the cousin primes. The cousin primes are a subsequence.

n + (0, 2m), m >= 1, being an admissible pattern for prime pairs, since (0, 2m) = (0, 0) (mod 2), has high density.

n + (0, 2m-1), m >= 1, being a non-admissible pattern for prime pairs, since (0, 2m-1) = (0, 1) (mod 2), has low density [the only possible pairs are (2^a - 2m-1, 2^a) or (2^a, 2^a + 2m-1), a >= 0.]

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1256 from Daniel Forgues)

Mathematica

Select[Range[1000], PrimeNu[#] < 2 && PrimeNu[# + 4] < 2 &] (* Amiram Eldar, Oct 01 2020 *)

Prog

(PARI) is(n)=if(n==1, return(1)); isprimepower(n) && isprimepower(n+4) \\ Charles R Greathouse IV, Apr 24 2015

Crossrefs

Cf. A023200, A000961.

k and (x) are prime powers: A006549 (k+1), A120431 (k+2), A164571 (k+3), this sequence (k+4), A164573 (k+5), A164574 (k+6).

Keyword

nonn

Author

_Daniel Forgues_, Aug 16 2009, Aug 17 2009

Status

approved

A168066 If n = Product p(k)^e(k) then a(n) = (Product (p(k)+1)^e(k) - Product (p(k)-1)^e(k))/2, a(1) = 0.

0, 1, 1, 4, 1, 5, 1, 13, 6, 7, 1, 17, 1, 9, 8, 40, 1, 22, 1, 25, 10, 13, 1, 53, 10, 15, 28, 33, 1, 32, 1, 121, 14, 19, 12, 70, 1, 21, 16, 79, 1, 42, 1, 49, 40, 25, 1, 161, 14, 46, 20, 57, 1, 92, 16, 105, 22, 31, 1, 104, 1, 33, 52, 364, 18, 62, 1, 73, 26, 60, 1, 214, 1, 39, 56, 81, 18, 72

Offset

1,4

Comments

a(n) = 0 iff n is 1;

a(n) = 1 iff n is a prime;

a(n) = p+q iff n is a biprime, i.e., n = p*q, p <= q primes;

a(n) = (p*q + p*r + q*r)+1 iff n is a triprime, i.e., n = p*q*r, p <= q <= r primes;

a(n) = (p*q*r + p*q*s + p*r*s + q*r*s) + (p+q+r+s) iff n is a quadprime, i.e., n = p*q*r*s, p <= q <= r <= s primes;

...

Links

Daniel Forgues, Table of n, a(n) for n = 1..100000

Formula

a(n) = (A003959(n) - A003958(n))/2.

Prog

(PARI) a(n) = {f = factor(n); return ((prod(k=1, #f~, (f[k, 1]+1)^f[k, 2]) - prod(k=1, #f~, (f[k, 1]-1)^f[k, 2]))/2); } \\ Michel Marcus, Jun 13 2013

Crossrefs

Cf. A003958, A003959, A168065.

Keyword

nonn

Author

_Daniel Forgues_, Nov 18 2009

Status

approved

A182979 Fermi-Dirac representation of n. Let n have factorization p1^(2^e1) * p2^(2^e2) * ... * pr^(2^er), where each factor is in A050376. The number n is represented by a binary string that indicates which terms of A050376 appear in the factorization of n.

0, 1, 10, 100, 1000, 11, 10000, 101, 100000, 1001, 1000000, 110, 10000000, 10001, 1010, 100000000, 1000000000, 100001, 10000000000, 1100, 10010, 1000001, 100000000000, 111, 1000000000000, 10000001, 100010, 10100, 10000000000000, 1011, 100000000000000, 100000001, 1000010, 1000000001, 11000, 100100

Offset

1,3

Comments

Every number has a unique representation as a product of terms from A050376. - N. J. A. Sloane, Feb 11 2011

The "Fermi-Dirac factorization" of n, i.e., the factorization of n into prime powers of the form p_k^(2^e_k), e_k >= 0, (A050376) allows each of those prime powers to be used at most once, since this corresponds to the binary representation of the exponents of the prime powers p^a of the "Bose-Einstein factorization" of n, i.e., the classic prime factorization of n. (Cf. A050376 comments.)

The prime powers of the form p_k^(2^e_k), e_k >= 0 (A050376) might be called "Fermi-Dirac primes" since they may appear at most once (thus raised to powers 0 or 1) in the "Fermi-Dirac factorization" of n. Compare with the classic prime factorization of n, which might be called the "Bose-Einstein factorization" of n, where the primes (which might be called "Bose-Einstein primes") may appear any number of times >= 0.

In the "Fermi-Dirac representation" of n, if a given prime power with powers of two as exponents does not appear in the factorization of n into prime powers with powers of two as exponents, we use 0 as a placeholder; otherwise, we use 1 to indicate that the given prime power with powers of two as exponents does appear in the "Fermi-Dirac factorization" of n.

In the base-b representation of n, we do not show the leading 0's, except for 0 where it is more convenient to show it than to show nothing. Similarly, for the "Fermi-Dirac representation" of n, we do not show the leading 0's, except for 0, which is the representation of 1, where it is more convenient to show it than to show nothing.

The limit of the supremum of the number of "binary digits" of the representation of n is asymptotic to the number of primes up to n, i.e., n/log(n), making this representation absolutely impractical!

See A052330 for the numbers having representation as 0, 1, 10, 11, 100, 101, 110, 111, ... which is an ordering of the positive integers. (Cf. OEIS Wiki page.)

Let n have factorization (f_r)^g_r * ... * (f_2)^g_2 * (f_1)^g_1, where f_i is the i-th prime power of the form p_k^(2^e_k), e_k >= 0 (A050376, A302778); then a(n) = Sum_{i=1..r} g_i * 2^(i-1).

The number of 1's in a(n) is the number of terms of A050376 dividing n with odd maximal exponent. For example, if n=96, then the maximal exponent of 2 that divides 96 is 5, for 3 it is 1, for 4 it is 2, for 16 it is 1. Thus only 2, 3 and 16 divide n with odd maximal exponents. Therefore, the number of 1's in a(96) is 3. Moreover, since 2=A050376(1), 3=A050376(2) and 16=A050376(9), then 1's appear in positions 1,2,9 from the right. - Vladimir Shevelev, Nov 02 2013

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

OEIS Wiki, "Fermi-Dirac representation" of n

OEIS Wiki, Ordering of positive integers by increasing "Fermi-Dirac representation"

Formula

Let q_1,q_2,q_3,... be consecutive terms of A050376 and n = q_1^a_1 * q_2^a_2 *...* q_r^a_r, where a_i = 0 or 1. Then a(n) = a_1 + 10*a_2 + ... +10^(r-1)*a_r. For example, since 30 = 2^1 * 3^1 * 4^0 * 5^1, then a(30)= 1 + 10 + 1000 = 1011. - Vladimir Shevelev, Nov 02 2013

a(n) = A007088(A052331(n)). - Antti Karttunen, Apr 17 2018

Example

"Fermi-Dirac factorizations" (cf. A050376 examples, here with exponents of "Fermi-Dirac primes" being either 0 or 1):
6 = 3*2 = 3^1*2^1, so a(6) = 11;
8 = 4*2 = 4^1*3^0*2^1, so a(8) = 101;
20 = 5*4 = 5^1*4^1*3^0*2^0, so a(20) = 1100;
24 = 4*3*2 = 4^1*3^1*2^1, so a(24) = 111;
27 = 9*3 = 9^1*7^0*5^0*4^0*3^1*2^0, so a(27) = 100010;
32 = 16*2 = 16^1*13^0*11^0*9^0*7^0*5^0*4^0*3^0*2^1, so a(32) = 100000001;
64 = 16*4 = 16^1*13^0*11^0*9^0*7^0*5^0*4^1*3^0*2^0, so a(64) = 100000100;
108 = 9*4*3 = 9^1*7^0*5^0*4^1*3^1*2^0, so a(108) = 100110;
120 = 5*4*3*2 = 5^1*4^1*3^1*2^1, so a(120) = 1111;
...

Mathematica

nn=24; p=Select[Range[nn], PrimeQ]; Do[p=Select[Union[p, p^2], #<=nn&], {Floor[Log[2, Log[2, nn]]]}]; Table[m=n; FromDigits[Table[If[Mod[m, i]==0, m=m/i; 1, 0], {i, Reverse[p]}]], {n, nn}]

Crossrefs

Cf. A050376, A052330, A052331, A302778.

Keyword

nonn

Author

_Daniel Forgues_, Feb 10 2011, Feb 13 2011

Extensions

Clearer definition from T. D. Noe, Feb 11 2011

Edited by N. J. A. Sloane, Jul 21 2018

Status

approved

A164571 Numbers n such that n and n+3 are prime powers.

1, 2, 4, 5, 8, 13, 16, 29, 61, 64, 125, 128, 509, 1021, 4093, 4096, 16381, 32768, 65536, 262144, 1048573, 4194301, 16777213, 268435456, 536870909, 1073741824, 36028797018963968

Offset

1,2

Comments

Numbers n such that n + (0, 3) is a prime power pair.

n + (0, 2m), m >= 1, being an admissible pattern for prime pairs, since (0, 2m) = (0, 0) (mod 2), has high density.

n + (0, 2m-1), m >= 1, being a non-admissible pattern for prime pairs, since (0, 2m-1) = (0, 1) (mod 2), has low density [the only possible pairs are (2^a - 2m-1, 2^a) or (2^a, 2^a + 2m-1), a >= 0.]

n + (0, 3) being a non-admissible pattern for prime pairs, has only prime power pairs (2^a - 3, 2^a) or (2^a, 2^a + 3), a >= 0.

Numbers n such that n and n+3 are primes would give only 2, for the prime pair (2, 5).

10^18 < a(28) <= 19807040628566084398385987581. - Donovan Johnson, Aug 17 2009

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..50

Prog

(PARI) ispp(n) = (n==1) || isprime(n) || (ispower(n, , &p) && isprime(p));
isok(n) = ispp(n) && ispp(n+3); \\ Michel Marcus, Aug 31 2013
(PARI) v=List(); for(n=0, 1e3, if(isprimepower(2^n-3), listput(v, 2^n-3)); if(isprimepower(2^n+3), listput(v, 2^n))); Set(v) \\ Charles R Greathouse IV, Apr 24 2015

Crossrefs

Cf. A000961.

Cf. A006549 Numbers n such that n and n+1 are prime powers.

Cf. A120431 Numbers n such that n and n+2 are prime powers.

Cf. A164571 Numbers n such that n and n+3 are prime powers.

Cf. A164572 Numbers n such that n and n+4 are prime powers.

Cf. A164573 Numbers n such that n and n+5 are prime powers.

Cf. A164574 Numbers n such that n and n+6 are prime powers.

Keyword

nonn

Author

_Daniel Forgues_, Aug 16 2009

Extensions

Edited by _Daniel Forgues_, Aug 17 2009

a(20)-a(27) from Donovan Johnson, Aug 17 2009

Status

approved

A164573 Numbers n such that n and n+5 are prime powers.

2, 3, 4, 8, 11, 27, 32, 59, 251, 1019, 2048, 4091, 262139, 1048571, 67108859, 4294967291, 68719476731, 140737488355328, 9007199254740992, 72057594037927931, 73786976294838206459, 332306998946228968225951765070086139

Offset

1,1

Comments

Numbers n such that n + (0, 5) is a prime power pair.

n + (0, 2m), m >= 1, being an admissible pattern for prime pairs, since (0, 2m) = (0, 0) (mod 2), has high density.

n + (0, 2m-1), m >= 1, being a non-admissible pattern for prime pairs, since (0, 2m-1) = (0, 1) (mod 2), has low density [the only possible pairs are (2^a - 2m-1, 2^a) or (2^a, 2^a + 2m-1), a >= 0.]

Numbers n such that n and n+5 are primes would give only 2, for the prime pair (2, 7).

10^18 < a(21) <= 73786976294838206459. - Donovan Johnson, Aug 17 2009

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..37

Prog

(PARI) is(n)=if(n<5, return(n>1)); isprimepower(n) && isprimepower(n+5) \\ Charles R Greathouse IV, Apr 24 2015
(PARI) v=List(); for(n=0, 1e3, if(isprimepower(2^n-5), listput(v, 2^n-5)); if(isprimepower(2^n+5), listput(v, 2^n))); Set(v) \\ Charles R Greathouse IV, Apr 24 2015

Crossrefs

Cf. A006549 Numbers n such that n and n+1 are prime powers.

Cf. A120431 Numbers n such that n and n+2 are prime powers.

Cf. A164571 Numbers n such that n and n+3 are prime powers.

Cf. A164572 Numbers n such that n and n+4 are prime powers.

Cf. A164573 Numbers n such that n and n+5 are prime powers.

Cf. A164574 Numbers n such that n and n+6 are prime powers.

Keyword

nonn

Author

_Daniel Forgues_, Aug 16 2009

Extensions

Edited by _Daniel Forgues_, Aug 17 2009

a(13)-a(20) from Donovan Johnson, Aug 17 2009

a(21)-a(22) from Charles R Greathouse IV, Apr 24 2015

Status

approved

A164574 Numbers k such that k and k+6 are both prime powers.

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 31, 37, 41, 43, 47, 53, 61, 67, 73, 83, 97, 101, 103, 107, 121, 125, 131, 151, 157, 163, 167, 173, 191, 193, 223, 227, 233, 251, 257, 263, 271, 277, 283, 307, 311, 331, 337, 343, 347, 353, 361, 367, 373, 383, 433, 443, 457

Offset

1,2

Comments

Numbers n such that n + (0, 6) is a prime power pair.

n + (0, 2m), m >= 1, being an admissible pattern for prime pairs, since (0, 2m) = (0, 0) (mod 2), has high density.

n + (0, 2m-1), m >= 1, being a non-admissible pattern for prime pairs, since (0, 2m-1) = (0, 1) (mod 2), has low density [the only possible pairs are (2^a - 2m-1, 2^a) or (2^a, 2^a + 2m-1), a >= 0.]

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2492 from Daniel Forgues)

Mathematica

Join[{1}, Select[Range[500], AllTrue[{#, #+6}, PrimePowerQ]&]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 30 2018 *)

Prog

(PARI) is(n)=if(n<4, return(n>0)); isprimepower(n) && isprimepower(n+6) \\ Charles R Greathouse IV, Apr 24 2015

Crossrefs

Cf. A023201, A000961.

k and (x) are prime powers: A006549 (k+1) A120431 (k+2), A164571 (k+3), A164572 (k+4), A164573 (k+5), this sequence (k+6).

Keyword

nonn

Author

_Daniel Forgues_, Aug 16 2009

Extensions

Edited by _Daniel Forgues_, Aug 17 2009

Status

approved