A354992 - OEIS

%I #11 Jun 17 2022 18:11:37

%S 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 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 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 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 Antti Karttunen, <a href="/A354992/b354992.txt">Table of n, a(n) for n = 1..65537</a>

%F a(n) = Sum_{d|n} [A344005(d) < A344005(n)], where [ ] is the Iverson bracket.

%F a(n) = A000005(n) - A354991(n).

%p g:= proc(n) option remember; local t,x;

%p   min(map(t -> rhs(op(t)), {msolve(x*(x+1),n)}) minus {0})

%p end proc:

%p g(1):= 1: g(2):= 1:

%p f:= proc(n) local d,v;

%p   v:= g(n);

%p   nops(select(t -> g(t) < v, numtheory:-divisors(n)))

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Jun 17 2022

%t 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 (PARI)

%o A344005(n) = for(m=1, oo, if((m*(m+1))%n==0, return(m))); \\ From A344005

%o A354992(n) = { my(x=A344005(n)); sumdiv(n, d, A344005(d)<x); };

%Y Cf. A000005, A344005, A354991.

%Y Cf. also A344589, A345936.

%K nonn

%O 1,4

%A _Antti Karttunen_, Jun 17 2022