This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A054272 Number of primes in the interval [prime(n), prime(n)^2]. 14
 2, 3, 7, 12, 26, 34, 55, 65, 91, 137, 152, 208, 251, 270, 315, 394, 471, 502, 591, 656, 685, 790, 864, 977, 1139, 1227, 1268, 1354, 1395, 1494, 1847, 1945, 2109, 2157, 2455, 2512, 2693, 2878, 3005, 3202, 3396, 3471, 3826, 3902, 4045, 4119, 4581, 5059 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS These primes are candidates for fortunate numbers (A005235). These are precisely the primes available for the solution of Aguilar's conjecture or Haga's conjecture in Carlos Rivera's The Prime Puzzles and Problems Connection, (conjecture 26). Aguilar's conjecture states that at least one prime will be available for placement on each row and column of a p X p square array. Haga's conjecture states that just p primes are required for such placement in any p X p array. - Enoch Haga, Jan 23 2002 Also number of times p_n (the n-th prime) occurs as the least prime factor (A020639) among numbers in range [(p_n)+1, ((p_n)^3)-1]. For n=1, p_1 = 2 and there are two even numbers in range [3, 7], namely 4 and 6, so a(1) = 2. See also A250474. - Antti Karttunen, Dec 05 2014 The number of consecutive primes after the leading 1 in the prime(n)-rough numbers. - Benedict W. J. Irwin, Mar 24 2016 LINKS Antti Karttunen, Table of n, a(n) for n = 1..5000 Carlos Rivera, Conjecture 26. The Calendar-like square Conjecture, The Prime Puzzles and Problems Connection. FORMULA a(n) = A000879(n) - n + 1. From Antti Karttunen, Dec 05-08 2014: (Start) a(n) = A250473(n) - 1 = A250474(n) - 2. a(n) = sum_{d | A002110(n)} moebius(d) * floor((p_n)^2 / d). [Where p_n is the n-th prime (A000040(n)) and A002110(n) gives the product of the first n primes. Because the latter is always squarefree, one could also use Liouville's lambda (A008836) instead of Moebius mu (A008683).] The ratio (a(n) * A002110(n)) / (A001248(n) * A005867(n)) stays near 1, which follows from the above summation formula. See also A249747. (End) EXAMPLE n=4, the zone in question is [7,49] and encloses a(4)=12 primes, as follows: {7,11,13,17,19,23,29,31,37,41,43,47}. MATHEMATICA a[n_] := PrimePi[Prime[n]^2] - n + 1; Array[a, 50] (* Jean-François Alcover, Dec 07 2015 *) PROG (PARI) \\ A fast version: default(primelimit, 2^31 + 2^30); A054272(n) = 1 + primepi(prime(n)^2) - n; for(n=1, 5000, write("b054272.txt", n, " ", A054272(n))); \\ The following mirrors the given new formula. It is far from an optimal way to compute this sequence: allocatemem(234567890); A002110(n) = prod(i=1, n, prime(i)); A054272(n) = { my(p2); p2 = prime(n)^2; sumdiv(A002110(n), d, moebius(d)*floor(p2/d)); }; for(n=1, 22, print1(A054272(n), ", ")); \\ Antti Karttunen, Dec 05 2014 CROSSREFS One less than A250473, two less than A250474. First differences: A251723. Cf. A000040, A000879, A001248, A002110, A005235, A005867, A008683, A008836, A020639, A078898, A249747. Sequence in context: A090596 A321838 A298897 * A259593 A129016 A099163 Adjacent sequences:  A054269 A054270 A054271 * A054273 A054274 A054275 KEYWORD nonn AUTHOR Labos Elemer, May 05 2000 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 22 17:02 EDT 2019. Contains 325225 sequences. (Running on oeis4.)