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
Examples
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.
Programs
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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]);
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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 *)
Extensions
a(8) from Amiram Eldar, Apr 15 2022
a(9)-a(10) from Daniel Suteu, May 12 2022