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A378980 Numbers k such that (A003961(k)-2*k) divides (A003961(k)-sigma(k)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

%I A378980 #23 Dec 19 2024 07:30:27
%S A378980 1,2,3,4,6,7,10,25,26,28,33,46,55,57,69,91,93,496,1034,1054,1558,2211,
%T A378980 2626,4825,8128,11222,12046,12639,28225,32043,68727,89575,970225,
%U A378980 1392386,2245557,8550146,12371554,16322559,22799825,33550336,48980427,51326726,55037217,60406599,68258725,142901438,325422273,342534446
%N A378980 Numbers k such that (A003961(k)-2*k) divides (A003961(k)-sigma(k)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.
%C A378980 Numbers k such that A252748(k) divides A286385(k).
%C A378980 Conjecture: Apart from a(5)=6, this is a subsequence of A319630, i.e., for all terms k<>6, gcd(k, A003961(k)) = 1. See also A372562, A372566.
%H A378980 Amiram Eldar, <a href="/A378980/b378980.txt">Table of n, a(n) for n = 1..69</a> (terms below 10^11; terms 1..60 from Antti Karttunen)
%H A378980 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%H A378980 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%t A378980 f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^e; q[k_] := Module[{fct = FactorInteger[k], m, s}, s = Times @@ f1 @@@ fct; m = Times @@ f2 @@@ fct; Divisible[m - s, m - 2*k]]; q[1] = True; Select[Range[10^5], q] (* _Amiram Eldar_, Dec 19 2024 *)
%o A378980 (PARI)
%o A378980 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A378980 A378981(n) = { my(u=A003961(n)); ((u-sigma(n))%((2*n)-u)); };
%o A378980 isA378980(n) = !A378981(n);
%Y A378980 Cf. A000203, A003961, A252748, A286385, A319630, A372562, A372566.
%Y A378980 Positions of 0's in A378981.
%Y A378980 Subsequence of A263837.
%Y A378980 Subsequences: A000396, A048674, A348514, A326134, A349753 (odd terms of this sequence).
%Y A378980 Cf. also A378983.
%K A378980 nonn
%O A378980 1,2
%A A378980 _Antti Karttunen_, Dec 12 2024