cp's OEIS Frontend

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

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

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

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.

From Antti Karttunen, Feb 13-20 2022: (Start)

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

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.

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.

(End)

Numbers k such that k * A324644(k) = A000203(k) * A324198(k).

Numbers k such that gcd(A064987(k), A324580(k)) = gcd(A064987(k), A351252(k)).

Numbers k such that their abundancy index [sigma(k)/k] is equal to A324644(k)/A324198(k). See A364286.

A324644 gives odd values for even numbers and for the odd squares. A324198 is odd on all arguments, therefore on odd squares the above equation reduces to odd * odd = odd * odd, and on odd nonsquares as odd * even = even * odd. It is an open question whether there are any odd terms after the initial a(1)=1.

If k is even, but not a multiple of 3, then A276086(k) is a multiple of 3, but not even (i.e., is an odd multiple of 3). If for such k also sigma(k) = 3*k, then A007949(A324644(k)) = min(A007949(sigma(k)), A007949(A276086(k))) = 1, while A007949(A324198(k)) = min(A007949(k), A007949(A276086(k))) = 0, therefore all such k's do occur in this sequence, for example, the two known terms of A005820 (3-perfect numbers) that are not multiples of three: 459818240, 51001180160, but also any hypothetical term of A005820 of the form 4u+2, where 2u+1 is not multiple of 3, and which by necessity is then also an odd perfect number.

Similarly, of the 65 known 5-multiperfect numbers (A046060), those 20 that are not multiples of five are included in this sequence. Note that all 65 are multiples of six.

It is conjectured that the intersection of this sequence with the multiperfect numbers (A007691) gives A323653, see comments in the latter.

For all even terms k of this sequence, A007814(A000203(k)) = A007814(k), sigma preserves the 2-adic valuation, and A007949(A000203(k)) >= A007949(k), i.e., does not decrease the 3-adic valuation. The condition is equivalence (=) when k is a multiple of 6. With odd terms, any hypothetical odd perfect number x would yield a one greater 2-adic valuation for sigma(x) than for x, but would satisfy the main condition of this sequence. - Corrected Feb 17 2022

If k is a nonsquare positive odd number (in A088828), then it must be a term of A191218. - Antti Karttunen, Mar 10 2024

Conjecture 1: the initial 1 is the only square in this sequence, and a(2) = 2 is the only term that is twice a square.

Conjecture 2: A323653 is a subsequence (which would follow from conjecture 1 (c) given there).

Questions: Are all nonzero terms abundant (in A005101)? Are all terms even? Could either be proved? See also comments in A351538 and in A351549.

The terms a(2) .. a(52) are all also practical (A005153) and Zumkeller (A083207). - Antti Karttunen, Dec 05 2024