A275494 Number of primitive weird numbers (A002975) between 2^n and 2^(n+1).

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 3, 2, 1, 1, 4, 2, 4, 4, 5, 4, 13, 9, 6, 18, 8, 16, 25, 24, 21, 61, 32, 47, 90, 80, 78, 195, 94, 90

Offset

0,13

Comments

It is not known unconditionally whether there are infinitely many primitive weird numbers (PWN, A002975), although numerical data provides strong evidence: even the number of weird numbers of the form 2^k*p*q (A258882, A258333) seems to increase rapidly as k increases. Melfi has shown that Cramer's conjecture implies the infiniteness of PWN.

Links

Table of n, a(n) for n=0..39.

Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Volume 147, February 2015, Pages 508-514.

Formula

a(n) = A275493(n+1) - A275493(n).

Example

The first primitive weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, 10792, ..., so there is one between 2^6 and 2^7 = 128, one between 2^9 and 2^10 = 1024, one between 2^11 and 2^12 = 4096, three between 2^12 and 2^13, etc.

Prog

(PARI) a(n)=sum(n=2^n\2+1, 2^n, is_A002975(n*2))

Crossrefs

Cf. A002975, A006037, A258333, A258882, A275493, A275491, A275492.

Sequence in context: A057160 A239938 A097794 * A137683 A329231 A259341

Adjacent sequences: A275491 A275492 A275493 * A275495 A275496 A275497

Keyword

nonn,more

Author

M. F. Hasler, Jul 30 2016

Extensions

a(39) from Amiram Eldar, Sep 02 2023

Status

approved