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A344590 Number of divisors d of n for which A011772(d) = A011772(n), where A011772(n) is the smallest number m such that m(m+1)/2 is divisible by n.

%I A344590 #20 Jun 12 2021 06:05:43
%S A344590 1,1,1,1,1,2,1,1,1,2,1,1,1,1,1,1,1,2,1,1,2,1,1,2,1,2,1,3,1,1,1,1,1,2,
%T A344590 1,4,1,1,2,3,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,1,2,2,1,3,1,1,1,1,1,3,1,3,
%U A344590 1,1,1,1,1,2,2,1,1,4,1,1,1,2,1,1,1,1,1,2,1,1,1,2,2,1,1,2,1,2,1,3,1,1,1,1,2
%N A344590 Number of divisors d of n for which A011772(d) = A011772(n), where A011772(n) is the smallest number m such that m(m+1)/2 is divisible by n.
%H A344590 Antti Karttunen, <a href="/A344590/b344590.txt">Table of n, a(n) for n = 1..65537</a>
%F A344590 a(n) = Sum_{d|n} [A011772(d) = A011772(n)], where [ ] is the Iverson bracket.
%F A344590 a(n) = A000005(n) - A344589(n).
%F A344590 a(n) <= A344770(n).
%e A344590 36 has 9 divisors: 1, 2, 3, 4, 6, 9, 12, 18, 36. When A011772 is applied to them, one obtains values [1, 3, 2, 7, 3, 8, 8, 8, 8], thus there are four divisors that obtain the maximal value 8 obtained at 36 itself, therefore a(36) = 4.
%t A344590 A011772[n_] := Module[{m = 1}, While[Not[IntegerQ[m(m+1)/(2n)]], m++]; m];
%t A344590 a[n_] := With[{m = A011772[n]}, Count[Divisors[n], d_ /; A011772[d] == m]];
%t A344590 Array[a, 100] (* _Jean-François Alcover_, Jun 12 2021 *)
%o A344590 (PARI)
%o A344590 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 A344590 A344590(n) = { my(x=A011772(n)); sumdiv(n,d,A011772(d)==x); };
%o A344590 (Python)
%o A344590 from itertools import combinations
%o A344590 from functools import reduce
%o A344590 from operator import mul
%o A344590 from sympy import factorint, divisors
%o A344590 from sympy.ntheory.modular import crt
%o A344590 def A011772(n):
%o A344590     plist = [p**q for p, q in factorint(2*n).items()]
%o A344590     if len(plist) == 1:
%o A344590         return n-1 if plist[0] % 2 else 2*n-1
%o A344590     return min(min(crt([m,2*n//m],[0,-1])[0],crt([2*n//m,m],[0,-1])[0]) for m in (reduce(mul, d) for l in range(1,len(plist)//2+1) for d in combinations(plist,l)))
%o A344590 def A344590(n):
%o A344590     m = A011772(n)
%o A344590     return sum(1 for d in divisors(n) if A011772(d) == m) # _Chai Wah Wu_, Jun 02 2021
%Y A344590 Cf. A000005, A011772, A344588 (positions of records), A344589, A344758, A344770, A344881 (positions of ones), A344882 (of terms > 1).
%K A344590 nonn
%O A344590 1,6
%A A344590 _Antti Karttunen_, May 31 2021