%I #60 Feb 20 2022 23:04:10

%S 1,120,216,672,2464,22176,228480,523776,640640,837760,5581440,5765760,

%T 7539840,12999168,19603584,33860736,38342304,71344000,95472000,

%U 102136320,197308800,220093440,345080736,459818240,807009280,975576960,1476304896,1510831360,1773584640

%N Numbers k for which k * gcd(sigma(k), A003961(k)) is equal to sigma(k) * gcd(k, A003961(k)), where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

%C Numbers k for which k * A342671(k) = A000203(k) * A322361(k).

%C Numbers k such that gcd(A064987(k), A191002(k)) = gcd(A064987(k), A341529(k)).

%C Obviously, all odd terms in this sequence must be squares.

%C All the terms k of A005820 that satisfy A007949(k) < A007814(k) [i.e., whose 3-adic valuation is strictly less than their 2-adic valuation] are also terms of this sequence. Incidentally, the first six known terms of A005820 satisfy this condition, while on the other hand, any hypothetical odd 3-perfect number would be excluded from this sequence. Also, as a corollary, any hypothetical 3-perfect numbers of the form 4u+2 must not be multiples of 3 if they are to appear here. Similarly for any k which occurs in A349169, for 2*k to occur in this sequence, it shouldn't be a multiple of 3 and k should also be a term of A191218. See question 2 and its partial answer in A349169.

%C From _Antti Karttunen_, Feb 13-20 2022: (Start)

%C Question: Are all terms/2 (A351548) abundant, from n > 1 onward?

%C Note that of the 65 known 5-multiperfect numbers, all others except these three 1245087725796543283200, 1940351499647188992000, 4010059765937523916800 are also included in this sequence. The three exceptions are distinguished by the fact that their 3 and 5-adic valuations are equal. In 62 others the former is larger.

%C If k satisfying the condition were of the form 4u+2, then it should be one of the terms of A191218 doubled as only then both k and sigma(k) are of the form 4u+2, with equal 2-adic valuations for both. More precisely, one of the terms of A351538.

%C (End)

%H Amiram Eldar, <a href="/A349745/b349745.txt">Table of n, a(n) for n = 1..52</a> (terms below 10^11)

%H Antti Karttunen, <a href="https://oeis.org/plot2a?name1=A342671&amp;name2=A322361&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawlines=true">Ratio A342671(n)/A322361(n) plotted with OEIS Plot2 tool</a>

%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)). [sigma preserves the 2-adic valuation of the terms of this sequence]

%t f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^e; q[1] = True; q[n_] := n * GCD[(s = Times @@ f1 @@@ (f = FactorInteger[n])), (r = Times @@ f2 @@@ f)] == s*GCD[n, r]; Select[Range[10^6], q] (* _Amiram Eldar_, Nov 29 2021 *)

%o (PARI)

%o A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o isA349745(n) = { my(s=sigma(n),u=A003961(n)); (n*gcd(s,u) == (s*gcd(n,u))); };

%Y Cf. A000203, A003961, A005101, A005820, A007814, A007949, A046060, A064987, A191002, A191218, A248150, A322361, A341529, A342671, A347870, A351534, A351536, A351538.

%Y Cf. also A349169, A349746, A351458, A351549 for other variants.

%Y Subsequence of A351554 and also of its subsequence A351551.

%Y Cf. A351459 (subsequence, intersection with A351458), A351548 (terms halved).

%K nonn

%O 1,2

%A _Antti Karttunen_, Nov 29 2021