A025488 - OEIS

%I #25 May 08 2022 02:41:54

%S 1,2,3,5,7,10,14,18,25,32,40,51,63,80,98,119,145,173,207,248,292,346,

%T 404,473,552,639,742,855,984,1129,1289,1477,1681,1912,2170,2452,2771,

%U 3121,3514,3951,4426,4955,5536,6182,6898,7674,8535,9470,10500,11633,12869

%N Number of distinct prime signatures of the positive integers up to 2^n.

%C The distinct prime signatures, in the order in which they occur, are listed in A124832. - _M. F. Hasler_, Jul 16 2019

%C The subsequence a(n) = A085089(2^n) is strictly increasing since it counts at least the additional prime signature (n) which did not occur for the previously considered numbers. All other partitions of n are prime signatures of numbers larger than 2^n and therefore counted only as part of later terms. - _M. F. Hasler_, Jul 17 2019

%H Ray Chandler, <a href="/A025488/b025488.txt">Table of n, a(n) for n = 0..182</a> (first 151 terms from T. D. Noe)

%F a(n) = Sum_{k=0..n} A056099(k). - _M. F. Hasler_, Jul 16 2019

%F a(n) = A085089(2^n). - _M. F. Hasler_, Jul 17 2019

%e From _M. F. Hasler_, Jul 16 2019: (Start)

%e For n = 0, the only integer k to be considered is 1, so the only prime signature is the empty one, (), whence a(0) = 1.

%e For n = 1, the integers k to be considered are {1, 2}; the prime signatures are {(), (1)}, whence a(1) = 2.

%e For n = 2, the integers k to be considered are {1, 2, 3, 4}; the distinct prime signatures are {(), (1), (2)}, whence a(2) = 3.

%e For n = 3, the integers k to be considered are {1, 2, 3, 4, 5, 6, 7, 8}; the distinct prime signatures are {(), (1), (2), (1,1), (3)}, whence a(3) = 5. (End)

%o (PARI) A025488(n)=A085089(2^n) \\ For illustrative purpose, n not too large. - _M. F. Hasler_, Jul 16 2019

%Y A025487(a(n)) = 2^n.

%Y Partial sums of A056099.

%Y Cf. A085089, A124832.

%K nonn

%O 0,2

%A _David W. Wilson_

%E Name edited by _M. F. Hasler_, Jul 16 2019