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

Author

Robert Israel, Apr 14 2022

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Intersection of A352908 and A352997.

Cf. A000217, A001414, A232324, A352996.

Programs

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 *)