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.

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

1,1

Comments

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

Links

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

Example

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.

Maple

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]);

Mathematica

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

Crossrefs

Intersection of A352908 and A352997.

Cf. A000217, A001414, A232324, A352996.

Sequence in context: A092273 A193745 A181437 * A156873 A320977 A071540

Adjacent sequences: A352998 A352999 A353000 * A353002 A353003 A353004

Keyword

nonn

Author

Robert Israel, Apr 14 2022

Status

approved