Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #21 May 12 2022 15:19:46
%S 93,2653,30433,1922113,15421122,28776673,240409057,611393953,
%T 2713190397,5413336381
%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 the same prime.
%e 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.
%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 t:= a mod b; if not isprime(t) then return false fi;
%p c:= numtheory:-sigma(n);
%p a mod c = t
%p end proc:
%p select(filter, [$2..2*10^7]);
%t Select[Range[2*10^6], (r = Mod[#*(# + 1)/2, DivisorSigma[1, #]]) == Mod[#*(# + 1)/2, Plus @@ Times @@@ FactorInteger[#]] && PrimeQ[r] &] (* _Amiram Eldar_, Apr 15 2022 *)
%Y Cf. A000217, A001414, A232324, A352996, A353001.
%K nonn,more
%O 1,1
%A _J. M. Bergot_ and _Robert Israel_, Apr 15 2022
%E a(8) from _Amiram Eldar_, Apr 15 2022
%E a(9)-a(10) from _Daniel Suteu_, May 12 2022