A353001 - OEIS

%I #14 Apr 19 2022 09:24:43

%S 4,57,70,93,129,217,322,381,417,453,513,565,597,646,682,781,813,921,

%T 925,1057,1081,1102,1137,1165,1197,1261,1317,1393,1405,1558,1582,1641,

%U 1750,1798,1846,1857,1918,1929,2073,2101,2110,2173,2181,2305,2329,2361,2482,2506,2569,2577,2626,2649,2653

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

%H Robert Israel, <a href="/A353001/b353001.txt">Table of n, a(n) for n = 1..10000</a>

%e 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 filter:= proc(n) local a,b,c,t;

%p   a:= n*(n+1)/2;

%p   b:= add(t[1]*t[2],t=ifactors(n)[2]);

%p   if not isprime(a mod b) then return false fi;

%p   c:= numtheory:-sigma(n);

%p   isprime(a mod c)

%p end proc:

%p select(filter, [$2..3000]);

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

%Y Intersection of A352908 and A352997.

%Y Cf. A000217, A001414, A232324, A352996.

%K nonn

%O 1,1

%A _Robert Israel_, Apr 14 2022