cp's OEIS Frontend

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.

%I A353001 #14 Apr 19 2022 09:24:43
%S A353001 4,57,70,93,129,217,322,381,417,453,513,565,597,646,682,781,813,921,
%T A353001 925,1057,1081,1102,1137,1165,1197,1261,1317,1393,1405,1558,1582,1641,
%U A353001 1750,1798,1846,1857,1918,1929,2073,2101,2110,2173,2181,2305,2329,2361,2482,2506,2569,2577,2626,2649,2653
%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.
%C A353001 Numbers k such that A232324(k) and A352996(k) are prime.
%H A353001 Robert Israel, <a href="/A353001/b353001.txt">Table of n, a(n) for n = 1..10000</a>
%e A353001 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.
%p A353001 filter:= proc(n) local a,b,c,t;
%p A353001   a:= n*(n+1)/2;
%p A353001   b:= add(t[1]*t[2],t=ifactors(n)[2]);
%p A353001   if not isprime(a mod b) then return false fi;
%p A353001   c:= numtheory:-sigma(n);
%p A353001   isprime(a mod c)
%p A353001 end proc:
%p A353001 select(filter, [$2..3000]);
%t A353001 Select[Range[3000], And @@ PrimeQ[Mod[#*(# + 1)/2, {DivisorSigma[1, #], Plus @@ Times @@@ FactorInteger[#]}]] &] (* _Amiram Eldar_, Apr 15 2022 *)
%Y A353001 Intersection of A352908 and A352997.
%Y A353001 Cf. A000217, A001414, A232324, A352996.
%K A353001 nonn
%O A353001 1,1
%A A353001 _Robert Israel_, Apr 14 2022