A352996 a(n) = n*(n+1)/2 mod (sum (with multiplicity) of prime factors of n).

1, 0, 2, 0, 1, 0, 0, 3, 6, 0, 1, 0, 6, 0, 0, 0, 3, 0, 3, 1, 6, 0, 3, 5, 6, 0, 10, 0, 5, 0, 8, 1, 6, 6, 6, 0, 6, 12, 6, 0, 3, 0, 0, 1, 6, 0, 10, 7, 3, 6, 1, 0, 0, 4, 10, 3, 6, 0, 6, 0, 6, 1, 4, 3, 3, 0, 15, 23, 7, 0, 0, 0, 6, 3, 5, 15, 3, 0, 3, 9, 6, 0, 0, 3, 6, 20, 6, 0, 0, 6, 12, 19, 6, 0, 2, 0

Offset: 2

Author

J. M. Bergot and Robert Israel, Apr 14 2022

Keywords

• nonn

Comments

If n is an odd prime, a(n) = 0.

If n is prime, a(n^2) = n.

Examples

			For n = 6, A000217(6) = 6*7/2 = 21 and A001414(6) = 2+3 = 5, so a(6) = 21 mod 5 = 1.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A352989, A352990, A352997.

Programs

Maple

seq((n*(n+1)/2) mod add(t[1]*t[2],t=ifactors(n)[2]), n=2..100);

Mathematica

a[n_] := Mod[n*(n + 1)/2, Plus @@ Times @@@ FactorInteger[n]]; Array[a, 100, 2] (* Amiram Eldar, Apr 15 2022 *)

Python

from sympy import factorint
def a(n): return (n*(n+1)//2)%sum(p*e for p, e in factorint(n).items())
print([a(n) for n in range(2, 98)]) # Michael S. Branicky, Apr 24 2022

Formula

a(n) = A000217(n) mod A001414(n).