A344763 - OEIS

%I #23 Jun 17 2022 16:03:12

%S 0,-1,1,-3,1,3,1,-7,1,6,1,4,1,7,10,-15,1,10,1,5,15,11,1,9,1,14,1,21,1,

%T 15,1,-31,22,18,21,28,1,19,27,25,1,22,1,12,36,23,1,16,1,26,34,13,1,27,

%U 45,8,39,30,1,45,1,31,36,-63,40,55,1,52,46,50,1,9,1,38,51,20,56,66,1,16,1,42,1,36,51,43,58,56,1,55

%N a(n) = n - A011772(n).

%H Antti Karttunen, <a href="/A344763/b344763.txt">Table of n, a(n) for n = 1..16383</a>

%H Antti Karttunen, <a href="/A344763/a344763.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%F a(n) = n - A011772(n).

%F a(n) = A344765(n) - A001065(n).

%F a(2^k) = 1-2^k. - _Chai Wah Wu_, Jun 15 2022

%t A011772[n_] := Module[{m = 1}, While[!IntegerQ[(m(m+1))/(2n)], m++]; m];

%t a[n_] := n - A011772[n];

%t Array[a, 100] (* _Jean-François Alcover_, Jun 12 2021 *)

%o (PARI) A344763(n) = (n-A011772(n));

%o (Python)

%o from sympy.ntheory.modular import crt

%o from sympy import factorint

%o from math import prod

%o from itertools import combinations

%o def A344763(n):

%o     plist = tuple(p**q for p, q in factorint(2*n).items())

%o     return 1-n if len(plist) == 1 else n-int(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 15 2022

%Y Cf. A001065, A011772, A344764, A344765, A344769.

%K sign

%O 1,4

%A _Antti Karttunen_, May 30 2021