A349746 - OEIS

%I #19 Mar 12 2024 09:54:01

%S 1,11466,114660,411264,804384,871416,4999680,46332000,176417280,

%T 378069120,396168192,485188704,709430400,2004912000,3921372000,

%U 5600534400,6128179200,6956471808,7556976000,7746979968,9904204800,14092001280,14182439040,23423662080,31998395520

%N Numbers k for which k * gcd(sigma(k), u) is equal to sigma(k) * gcd(k, u), where u is obtained by shifting the prime factorization of k two steps toward larger primes [with u = A003961(A003961(k))].

%C Sigma preserves both the 2-adic and 3-adic valuation of the terms of this sequence.

%C All 65 known 5-multiperfect numbers (A046060) are included in this sequence, as well as the smallest 7-multiperfect number, 141310897947438348259849402738485523264343544818565120000 = A007539(7), and probably the majority of other p-multiperfect numbers as well, where p is a prime > 3. However, any term that is in A349747 is not included in this sequence.

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F For all n >= 1, A007814(A000203(a(n))) = A007814(a(n)) and A007949(A000203(a(n))) = A007949(a(n)). [See comment]

%t f[p_, e_] := NextPrime[p, 2]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := n * GCD[(sigma = DivisorSigma[1, n]), (u = s[n])] == sigma * GCD[n, u]; Select[Range[10^6], q] (* _Amiram Eldar_, Dec 01 2021 *)

%o (PARI)

%o A003961twice(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(1+nextprime(1+f[i, 1]))); factorback(f); };

%o isA349746(n) = { my(s=sigma(n),u=A003961twice(n)); (n*gcd(s,u) == (s*gcd(n,u))); };

%Y Cf. A000203, A003961, A007539, A007814, A007949, A046060, A349747.

%Y Cf. also A349169, A349745.

%K nonn

%O 1,2

%A _Antti Karttunen_, Nov 30 2021

%E a(15)-a(25) from _Martin Ehrenstein_, Dec 17 2021