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A354991 Number of divisors d of n for which A344005(d) = A344005(n), where A344005(n) is the smallest positive integer m such that n divides m*(m+1).

%I A354991 #11 Jun 17 2022 18:11:32
%S A354991 1,2,1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,3,2,2,1,1,1,2,1,1,1,2,1,1,1,2,
%T A354991 1,3,1,2,2,1,1,4,1,1,1,2,1,2,1,2,1,3,1,2,2,3,2,2,1,1,1,2,1,1,1,2,1,3,
%U A354991 1,2,1,5,1,2,2,1,1,4,1,3,1,2,1,1,1,2,1,1,1,2,1,1,2,2,1,1,1,2,1,3,1,2,1,1,2
%N A354991 Number of divisors d of n for which A344005(d) = A344005(n), where A344005(n) is the smallest positive integer m such that n divides m*(m+1).
%H A354991 Antti Karttunen, <a href="/A354991/b354991.txt">Table of n, a(n) for n = 1..65537</a>
%F A354991 a(n) = Sum_{d|n} [A344005(d) = A344005(n)], where [ ] is the Iverson bracket.
%F A354991 a(n) = A000005(n) - A354992(n).
%F A354991 a(n) <= A354990(n).
%t A354991 s[n_] := Module[{m = 1}, While[! Divisible[m*(m + 1), n], m++]; m]; a[n_] := Module[{sn = s[n]}, 1 + DivisorSum[n, 1 &, # < n && s[#] == sn &]]; Array[a, 100] (* _Amiram Eldar_, Jun 17 2022 *)
%o A354991 (PARI)
%o A354991 A344005(n) = for(m=1, oo, if((m*(m+1))%n==0, return(m))); \\ From A344005
%o A354991 A354991(n) = { my(x=A344005(n)); sumdiv(n, d, A344005(d)==x); };
%Y A354991 Cf. A000005, A344005, A354990, A354992, A354994 (positions of 1's).
%Y A354991 Cf. also A344590, A345935.
%K A354991 nonn
%O A354991 1,2
%A A354991 _Antti Karttunen_, Jun 17 2022