A352989 -id:A352989 - cp's OEIS frontend

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

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

nonn

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

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

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

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

Cf. A352989, A352990, A352997.

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

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

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

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

A352990 Numbers k such that the k-th triangular number == 1 mod the integer log of k.

Original entry on oeis.org

2, 6, 12, 21, 33, 45, 52, 63, 136, 162, 201, 205, 208, 225, 245, 253, 301, 304, 344, 441, 494, 531, 550, 637, 697, 720, 742, 806, 901, 910, 918, 1078, 1233, 1242, 1274, 1333, 1376, 1540, 1566, 1573, 1625, 1680, 1695, 1792, 1863, 1909, 2025, 2041, 2107, 2295, 2466, 2497, 2774, 2896, 2926, 2965

Offset: 1

J. M. Bergot and Robert Israel, Apr 13 2022

nonn

Numbers k such that A000217(k) == 1 (mod A001414(k)).

			a(3) = 12 = 2*2*3 is a term because 12*13/2 = 78 == 1 (mod 2+2+3 = 7).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000217, A001414, A352989.

filter:= proc(n) local t; (n*(n+1)/2) mod add(t[1]*t[2],t=ifactors(n)[2]) = 1 end proc:
select(filter, [$2..3000]);

Select[Range[3000], Mod[#*(# + 1)/2, Plus @@ Times @@@ FactorInteger[#]] == 1 &] (* Amiram Eldar, Apr 14 2022 *)

cp's OEIS Frontend

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

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