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%I #19 Jul 28 2024 10:08:07
%S 1,2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,30,31,32,36,37,41,42,43,
%T 47,49,53,59,61,64,66,67,70,71,72,73,78,79,81,83,89,97,100,101,102,
%U 103,105,107,108,109,110,113,114,120,121,125,127,128,130,131,137
%N Numbers k such that sigma_0(k) != omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.
%H Amiram Eldar, <a href="/A328957/b328957.txt">Table of n, a(n) for n = 1..10000</a>
%F A000005(a(n)) != A001222(a(n)) * A001221(a(n)).
%e The sequence of terms together with their prime indices begins:
%e 1: {}
%e 2: {1}
%e 3: {2}
%e 4: {1,1}
%e 5: {3}
%e 7: {4}
%e 8: {1,1,1}
%e 9: {2,2}
%e 11: {5}
%e 13: {6}
%e 16: {1,1,1,1}
%e 17: {7}
%e 19: {8}
%e 23: {9}
%e 25: {3,3}
%e 27: {2,2,2}
%e 29: {10}
%e 30: {1,2,3}
%e 31: {11}
%e 32: {1,1,1,1,1}
%t Select[Range[100],DivisorSigma[0,#]!=PrimeOmega[#]*PrimeNu[#]&]
%o (PARI) is(k) = {my(f = factor(k)); numdiv(f) != omega(f) * bigomega(f);} \\ _Amiram Eldar_, Jul 28 2024
%Y Nonzeros of A328958.
%Y The complement is A328956.
%Y Prime signature is A124010.
%Y Omega-sequence is A323023.
%Y omega(n) * Omega(n) is A113901(n).
%Y (Omega(n) - 1) * omega(n) is A307409(n).
%Y sigma_0(n) - Omega(n) * omega(n) is A328958(n).
%Y sigma_0(n) - 2 - (Omega(n) - 1) * omega(n) is A328959(n).
%Y Cf. A060687, A070175, A090858, A112798, A303555, A320632, A328960, A328961, A328962, A328963, A328964, A328965.
%K nonn
%O 1,2
%A _Gus Wiseman_, Nov 01 2019
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