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%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
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