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A344769 a(n) = A005187(n) - A011772(n).

%I A344769 #21 Jan 26 2025 20:28:31
%S A344769 0,0,2,0,4,7,5,0,8,14,9,14,11,18,21,0,16,26,17,23,33,30,20,31,23,37,
%T A344769 24,46,26,41,27,0,53,50,53,62,35,54,62,63,39,61,40,53,77,65,43,62,47,
%U A344769 73,81,62,50,77,95,61,92,84,55,101,57,88,93,0,103,119,65,118,112,117,68,79,71,109,122,93,129,140,75,94,79,121
%N A344769 a(n) = A005187(n) - A011772(n).
%H A344769 Antti Karttunen, <a href="/A344769/b344769.txt">Table of n, a(n) for n = 1..16384</a>
%H A344769 Antti Karttunen, <a href="/A344769/a344769.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A344769 a(n) = A005187(n) - A011772(n).
%F A344769 a(n) = A344765(n) + A294898(n).
%F A344769 a(2^e) = 0, for e >= 0.
%F A344769 If n is an odd prime power, then a(n) = n+1-A000120(n). - _Chai Wah Wu_, Jun 03 2021
%o A344769 (PARI)
%o A344769 A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
%o A344769 A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
%o A344769 A344769(n) = (A005187(n) - A011772(n));
%o A344769 (Python)
%o A344769 from itertools import combinations
%o A344769 from math import prod
%o A344769 from sympy import factorint, divisors
%o A344769 from sympy.ntheory.modular import crt
%o A344769 def A344769(n):
%o A344769     c = 2*n-bin(n).count('1')
%o A344769     plist = [p**q for p, q in factorint(2*n).items()]
%o A344769     if len(plist) == 1:
%o A344769         return int(c+1+(plist[0] % 2 - 2)*n)
%o A344769     return int(c-min(min(crt([m,2*n//m],[0,-1])[0],crt([2*n//m,m],[0,-1])[0]) for m in (prod(d) for l in range(1,len(plist)//2+1) for d in combinations(plist,l)))) # _Chai Wah Wu_, Jun 03 2021
%Y A344769 Cf. A000120, A005187, A011772, A294898, A344763, A344765.
%K A344769 nonn
%O A344769 1,3
%A A344769 _Antti Karttunen_, May 30 2021