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%I #15 Sep 02 2023 03:29:54
%S 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,
%T 61,32,47,90,80,78,195,94,90
%N Number of primitive weird numbers (A002975) between 2^n and 2^(n+1).
%C 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.
%H Giuseppe Melfi, <a href="http://dx.doi.org/10.1016/j.jnt.2014.07.024">On the conditional infiniteness of primitive weird numbers</a>, Journal of Number Theory, Volume 147, February 2015, Pages 508-514.
%F a(n) = A275493(n+1) - A275493(n).
%e 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.
%o (PARI) a(n)=sum(n=2^n\2+1,2^n,is_A002975(n*2))
%Y Cf. A002975, A006037, A258333, A258882, A275493, A275491, A275492.
%K nonn,more
%O 0,13
%A _M. F. Hasler_, Jul 30 2016
%E a(39) from _Amiram Eldar_, Sep 02 2023