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).

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, 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, 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

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

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

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

Maple

g:= proc(n) option remember; local t, x;
  min(map(t -> rhs(op(t)), {msolve(x*(x+1), n)}) minus {0})
end proc:
g(1):= 1: g(2):= 1:
f:= proc(n) local d, v;
  v:= g(n);
  nops(select(t -> g(t) < v, numtheory:-divisors(n)))
end proc:
map(f, [$1..100]); # Robert Israel, Jun 17 2022

Mathematica

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 *)

Prog

(PARI)
A344005(n) = for(m=1, oo, if((m*(m+1))%n==0, return(m))); \\ From A344005
A354992(n) = { my(x=A344005(n)); sumdiv(n, d, A344005(d)<x); };

Crossrefs

Cf. A000005, A344005, A354991.

Cf. also A344589, A345936.

Sequence in context: A337093 A320266 A277692 * A129138 A353378 A112970

Adjacent sequences: A354989 A354990 A354991 * A354993 A354994 A354995

Keyword

nonn

Author

Antti Karttunen, Jun 17 2022

Status

approved