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

Author

M. F. Hasler, Jul 30 2016

Keywords

• nonn
• more

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

PARI

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

Formula

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

Extensions

a(39) from Amiram Eldar, Sep 02 2023