A352997 -id:A352997 - 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).

A353001 Numbers k such that the k-th triangular number mod the sum (with multiplicity) of prime factors of k, and the k-th triangular number mod the sum of divisors of k, are both prime.

Original entry on oeis.org

4, 57, 70, 93, 129, 217, 322, 381, 417, 453, 513, 565, 597, 646, 682, 781, 813, 921, 925, 1057, 1081, 1102, 1137, 1165, 1197, 1261, 1317, 1393, 1405, 1558, 1582, 1641, 1750, 1798, 1846, 1857, 1918, 1929, 2073, 2101, 2110, 2173, 2181, 2305, 2329, 2361, 2482, 2506, 2569, 2577, 2626, 2649, 2653

Offset: 1

Robert Israel, Apr 14 2022

nonn

Numbers k such that A232324(k) and A352996(k) are prime.

			a(3) = 70 is a term because 70*71/2 = 2485, A000217(70) = 144, A001414(70) = 14, and both 2485 mod 144 = 37 and 2485 mod 14 = 7 are prime.

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

Intersection of A352908 and A352997.

Cf. A000217, A001414, A232324, A352996.

filter:= proc(n) local a,b,c,t;
  a:= n*(n+1)/2;
  b:= add(t[1]*t[2],t=ifactors(n)[2]);
  if not isprime(a mod b) then return false fi;
  c:= numtheory:-sigma(n);
  isprime(a mod c)
end proc:
select(filter, [$2..3000]);

Select[Range[3000], And @@ PrimeQ[Mod[#*(# + 1)/2, {DivisorSigma[1, #], Plus @@ Times @@@ FactorInteger[#]}]] &] (* Amiram Eldar, Apr 15 2022 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula