A353002 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 the same prime.

93, 2653, 30433, 1922113, 15421122, 28776673, 240409057, 611393953, 2713190397, 5413336381

Offset

1,1

Links

Table of n, a(n) for n=1..10.

Example

a(1) = 93 is a term because 93*94/2 = 4371, A000217(93) = 128, A001414(93) = 34, and 4371 mod 128 = 4371 mod 34 = 19, which is prime.

Maple

filter:= proc(n) local a, b, c, t;
  a:= n*(n+1)/2;
  b:= add(t[1]*t[2], t=ifactors(n)[2]);
  t:= a mod b; if not isprime(t) then return false fi;
  c:= numtheory:-sigma(n);
  a mod c = t
end proc:
select(filter, [$2..2*10^7]);

Mathematica

Select[Range[2*10^6], (r = Mod[#*(# + 1)/2, DivisorSigma[1, #]]) == Mod[#*(# + 1)/2, Plus @@ Times @@@ FactorInteger[#]] && PrimeQ[r] &] (* Amiram Eldar, Apr 15 2022 *)

Crossrefs

Cf. A000217, A001414, A232324, A352996, A353001.

Sequence in context: A250373 A250374 A146125 * A017809 A017756 A246991

Adjacent sequences: A352999 A353000 A353001 * A353003 A353004 A353005

Keyword

nonn,more

Author

J. M. Bergot and Robert Israel, Apr 15 2022

Extensions

a(8) from Amiram Eldar, Apr 15 2022

a(9)-a(10) from Daniel Suteu, May 12 2022

Status

approved