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A354992 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 A354992 #11 Jun 17 2022 18:11:37
%S A354992 0,0,1,2,1,2,1,3,2,2,1,4,1,2,3,4,1,4,1,3,2,2,1,7,2,2,3,5,1,6,1,5,3,2,
%T A354992 3,6,1,2,2,7,1,4,1,5,5,2,1,8,2,4,3,3,1,6,2,5,2,2,1,11,1,2,5,6,3,6,1,3,
%U A354992 3,6,1,7,1,2,4,5,3,4,1,7,4,2,1,11,3,2,3,7,1,10,3,5,2,2,3,11,1,4,5,6,1,6,1,7,6
%N A354992 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 A354992 Antti Karttunen, <a href="/A354992/b354992.txt">Table of n, a(n) for n = 1..65537</a>
%F A354992 a(n) = Sum_{d|n} [A344005(d) < A344005(n)], where [ ] is the Iverson bracket.
%F A354992 a(n) = A000005(n) - A354991(n).
%p A354992 g:= proc(n) option remember; local t,x;
%p A354992   min(map(t -> rhs(op(t)), {msolve(x*(x+1),n)}) minus {0})
%p A354992 end proc:
%p A354992 g(1):= 1: g(2):= 1:
%p A354992 f:= proc(n) local d,v;
%p A354992   v:= g(n);
%p A354992   nops(select(t -> g(t) < v, numtheory:-divisors(n)))
%p A354992 end proc:
%p A354992 map(f, [$1..100]); # _Robert Israel_, Jun 17 2022
%t A354992 s[n_] := Module[{m = 1}, While[! Divisible[m*(m + 1), n], m++]; m]; a[n_] := Module[{sn = s[n]}, DivisorSum[n, 1 &, # < n && s[#] < sn &]]; Array[a, 100] (* _Amiram Eldar_, Jun 17 2022 *)
%o A354992 (PARI)
%o A354992 A344005(n) = for(m=1, oo, if((m*(m+1))%n==0, return(m))); \\ From A344005
%o A354992 A354992(n) = { my(x=A344005(n)); sumdiv(n, d, A344005(d)<x); };
%Y A354992 Cf. A000005, A344005, A354991.
%Y A354992 Cf. also A344589, A345936.
%K A354992 nonn
%O A354992 1,4
%A A354992 _Antti Karttunen_, Jun 17 2022