A025488 Number of distinct prime signatures of the positive integers up to 2^n.

1, 2, 3, 5, 7, 10, 14, 18, 25, 32, 40, 51, 63, 80, 98, 119, 145, 173, 207, 248, 292, 346, 404, 473, 552, 639, 742, 855, 984, 1129, 1289, 1477, 1681, 1912, 2170, 2452, 2771, 3121, 3514, 3951, 4426, 4955, 5536, 6182, 6898, 7674, 8535, 9470, 10500, 11633, 12869

Offset

0,2

Comments

The distinct prime signatures, in the order in which they occur, are listed in A124832. - M. F. Hasler, Jul 16 2019

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

Links

Ray Chandler, Table of n, a(n) for n = 0..182 (first 151 terms from T. D. Noe)

Formula

a(n) = Sum_{k=0..n} A056099(k). - M. F. Hasler, Jul 16 2019

a(n) = A085089(2^n). - M. F. Hasler, Jul 17 2019

Example

From M. F. Hasler, Jul 16 2019: (Start)
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.
For n = 1, the integers k to be considered are {1, 2}; the prime signatures are {(), (1)}, whence a(1) = 2.
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.
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)

Prog

(PARI) A025488(n)=A085089(2^n) \\ For illustrative purpose, n not too large. - M. F. Hasler, Jul 16 2019

Crossrefs

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

Partial sums of A056099.

Cf. A085089, A124832.

Sequence in context: A173538 A130053 A177277 * A306473 A175846 A088585

Adjacent sequences: A025485 A025486 A025487 * A025489 A025490 A025491

Keyword

nonn

Author

David W. Wilson

Extensions

Name edited by M. F. Hasler, Jul 16 2019

Status

approved