cp's OEIS Frontend

Apart from missing 2, this sequence gives all numbers k such that the binary expansion of A156552(k) is a prefix of that of A156552(sigma(k)), that is, for k > 1, numbers k for which sigma(k) is a descendant of k in A005940-tree. This follows because of the two transitions x -> A005843(x) (doubling) and x -> A003961(x) (prime shift) used to generate descendants in A005940-tree, using A003961 at any step of the process will ruin the chances of encountering sigma(k) anywhere further down that subtree.

Proof: Any left child in A005940 (i.e., A003961(k) for k) is larger than sigma(k), for any k > 2 [see A286385 for a proof], and A003961(n) > n for all n > 1. Thus, apart from A003961(2) = 3 = sigma(2), A003961^t(k) > sigma(k), where A003961^t means t-fold application of prime shift, here with t >= 1. On the other hand, sigma(2n) > sigma(n) for all n, thus taking first some doubling steps before a run of one or more prime shift steps will not rescue us, as neither will taking further doubling steps after a bout of prime shifts.

The first terms of A325637 not included in this sequence are 154345556085770649600 and 9186050031556349952000, as they have abundancy index 6.

From Antti Karttunen, Nov 29 2021: (Start)

Odd terms of this sequence are given by the intersection of A349169 and A349174.

A064989 applied to the odd terms of this sequence gives the fixed points of A326042, i.e., the positions of zeros in A348736, and a subset of the positions of ones in A348941.

Odd terms of this sequence form a subsequence of A348943, but should occur neither in A348748 nor in A348749.

(End)

a(n) tells about the degree of relatedness between n and sigma(n) in Doudna tree (see the illustration in A005940). It is 0 for those n where sigma(n) is one of the descendants of n, 1 for those n where the nearest common ancestor of n and sigma(n) is the parent of n, 2 for those n where the nearest common ancestor of n and sigma(n) is the grandparent of n, and so on.

Numbers k > 1 such that nearest common ancestor of k and sigma(k) in Doudna tree is the parent of k, and sigma(k) is not a descendant of k.

Any hypothetical odd term x in A005820 (triperfect numbers) would also be a member of this sequence. This is illustrated in the following diagram which shows how the neighborhood of such x would look like in the Doudna tree (A005940). If m (the parent of x, x = A003961(m), m = A064989(x)) is even, then x is a multiple of 3, while if m is odd, then 3 does not divide x. Because the abundancy index decreases when traversing leftwards in the Doudna tree, m must be a term of A068403. Both x and m would also need to be squares, by necessity.

.

<--A003961-- m ---(*2)--->

.............../ \...............

/ \

/ \

x 2m

/ \ / \

etc.../ \.....2x sigma(x) = 3x..../ \.....4m

/ \ / \ / \

etc. etc. etc. \ / etc.

\ /

6x 9x = sigma(2x)

/ \ / \

etc. \ etc. etc.

\

12x = sigma(3x) if m odd.

.

From the diagram we also see that 2x would then need to be a term of A347392 (as well as that of A159907 and also in A074388, thus sqrt(x) should be a term of A097023), and furthermore, if x is not a multiple of 3 (i.e., when m is odd), then sigma(3*x) = 4*sigma(x) = 4*(3*x), thus 3*x = sigma(x) would be a term of A336702 (particularly, in A027687) and x would be a term of A323653.

Moreover, any odd square x in this sequence (for which sigma(x) would also be odd), would have an abundancy index of at least three (sigma(x)/x >= 3). See comments in A347383.

Note how 401625 = 6375 * 63 = 945 * 425, 46305 = 945 * 49, 9261 = 189 * 49, 6375 = 2125 * 3, 945 = 189 * 5 = 15 * 63 and 9261*2125 = 19679625. It seems that when the multiplicands are coprime, then they are both terms of this sequence, e.g. 2125 and 3, 189 and 5, 2125 and 9261.

From Antti Karttunen, Jul 10 2024: (Start)

Regarding the observation above, for two coprime odd numbers x, y, if both are included here because sigma(x) = 2^a * A064989(x) and sigma(y) = 2^b * A064989(y), then also their product x*y is included because in that case sigma(x*y) = 2^(a+b) * A064989(x*y).

Also, for two coprime odd numbers x, y, if both are included here because sigma(x) = A065119(i) * x and sigma(y) = A065119(j) * y, then also their product x*y is included because sigma(x*y) = A065119(k) * x*y, where A065119(k) = A065119(i)*A065119(j). The existence of such numbers (that would include odd triperfect and odd 6-perfect numbers, see A046061) is so far hypothetical, none is known.

It is not possible that the odd x is in this sequence if sigma(x) = k*A003961^e(x) and e = A061395(k)-2 >= 1.

Note that all odd terms < 2^33 here are some of the exponentially odd divisors of 19679625 (see A374199, also A374463 and A374464).

(End)

Question: from a(6) = 189 onward, are the rest of terms all in A347390?

Conjecture: sequence is finite.

If it exists, a(14) > 2^33.

Multiperfect numbers m such that sigma(m) divides sigma(sigma(m)).

Also k-multiperfect numbers m such that k*m is also multiperfect.

Corresponding values of numbers k(n) = sigma(a(n)) / a(n): 1, 3, 3, 5, 5, 5, 5, 5, ...

Corresponding values of numbers h(n) = sigma(k(n) * a(n)) / (k(n) * a(n)): 1, 4, 4, 6, 6, 6, 6, 6, ...

Number of k-multiperfect numbers m such that sigma(n) is also multiperfect for k = 3..6: 2, 0, 20, 0.

From Antti Karttunen, Mar 20 2021, Feb 18 2022: (Start)

Conjecture 1 (a): This sequence consists of those m for which sigma(m)/m is an integer (thus a term of A007691), and coprime with m. Or expressed in a slightly weaker form (b): {1} followed by those m for which sigma(m)/m is an integer, but not a divisor of m. In a slightly stronger form (c): For m > 1, sigma(m)/m is always the least prime not dividing m. This would imply both (a) and (b) forms.

Conjecture 2: This sequence is finite.

Conjecture 3: This sequence is the intersection of A007691 and A351458.

Conjecture 4: This is a subsequence of A349745, thus also of A351551 and of A351554.

Note that if there existed an odd perfect number x that were not a multiple of 3, then both x and 2*x would be terms in this sequence, as then we would have: sigma(x)/x = 2, sigma(2*x)/(2*x) = 3, sigma(6*x)/(6*x) = 4. See also the diagram in A347392 and A353365.

(End)

From Antti Karttunen, May 16 2022: (Start)

Apparently for all n > 1, A336546(a(n)) = 0. [At least for n=2..23], while A353633(a(n)) = 1, for n=1..23.

The terms a(1) .. a(23) are only cases present among the 5721 known and claimed multiperfect numbers with abundancy <> 2, as published 03 January 2022 under Flammenkamp's site, that satisfy the condition for inclusion in this sequence.

They are also the only 23 cases among that data such that gcd(n, sigma(n)/n) = 1, or in other words, for which the n and its abundancy are relatively prime, with abundancy in all cases being the least prime that does not divide n, A053669(n), which is a sufficient condition for inclusion in A351458.

(End)

Here nontrivial unitary divisor d of k means any divisor d|k, such that 1 < d < k and gcd(d, k/d) = 1.

Any hypothetical odd term x in A005820 (triperfect numbers) would also be a member of this sequence. Proof: such an odd number cannot be a prime power (although it must be a square), thus it must have at least two nontrivial unitary divisors (with A034444(x) >= 4). Because sigma(x) = 3*x, it must be a term of A347391. From the illustration given there, we see that any odd square y in that sequence (i.e. with A347381(y)=1) would have an abundancy index of at least three (sigma(y)/y >= 3). But because abundancy index is multiplicative and always > 1 for n > 1, any nontrivial unitary divisor d of an odd triperfect number x must have sigma(d)/d < 3, thus for all such d, A347381(d) <> 1. And neither such divisor d can be a term of A336702, because 3*x is odd, therefore we must have A347381(d) > 1 for all nontrivial unitary divisors d of such a hypothetical x.

Any odd term of A000396, i.e., an odd perfect number, if such a hypothetical number exists, must also be a term of this sequence, by reasoning similar to above. See also illustration in A347392.

The fixed points of this sequence is given by the union of {2} and A336702.

The positions x such that a(x) = A252463(x) is given by the union of {1} and A347391.

The positions x such that a(x) = A252463(A252463(x)) is given by the union of {1} and A347392.

Numbers k such that sigma(sigma(k)) = 2^e * sigma(k), for some e >= 0.

Numbers k such that sigma(k) is in A336702.

Numbers k for which A000265(A051027(k)) = A161942(k).

If there existed any hypothetical 3-perfect number (A005820) of the form x = 4u+2 and not divisible by 3, then x would be also included in this sequence, as then sigma(sigma(x)) = 12*x = 4*sigma(x). Such x would be also a term of A349745 and of A351458, and x/2 would be a rare odd term of A000396, and also in A336702. See also the diagram in A347392.

Question: Why the sudden rarification of terms after a(48) = 775?

This sequence tells how near sigma(x) is to each x in Doudna-tree, A005940, with x iterating over the vertices of the tree in the breadth-first fashion. Positions that correspond to perfect numbers or (hypothetical) odd triperfect numbers get values 0 and 1 respectively. 1's occur also elsewhere. (Clarified Jul 03 2023)

See the illustrations in A347391 and in A347392.