cp's OEIS Frontend

Sum_{ d divides n } 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Denominators of coefficients in expansion of Sum_{n >= 1} x^n/(n*(1-x^n)) = Sum_{n >= 1} log(1/(1-x^n)).

Also n/gcd(n, sigma(n)) = n/A009194(n); also n/lcm(all common divisors of n and sigma(n)). Equals 1 if 6,28,120,496,672,8128,..., i.e., if n is from A007691. - Labos Elemer, Aug 14 2002

a(A007691(n)) = 1. - Reinhard Zumkeller, Apr 06 2012

Denominator of sigma(n)/n = A000203(n)/n. a(n) = 1 for numbers n in A007691 (multiply-perfect numbers), a(n) = 2 for numbers n in A159907 (numbers n with half-integral abundancy index), a(n) = 3 for numbers n in A245775, a(n) = n for numbers n in A014567 (numbers n such that n and sigma(n) are relatively prime). See A162657 (n) - the smallest number k such that a(k) = n. - Jaroslav Krizek, Sep 23 2014

For all n, a(n) <= n, and thus records are obtained for terms of A014567. - Michel Marcus, Sep 25 2014

Conjecture: If a(n) is in A005153, then n is in A005153. In particular, if n has dyadic rational abundancy index, i.e., a(n) is in A000079 (such as A007691 and A159907), then n is in A005153. Since every term of A005153 greater than 1 is even, any odd n such that a(n) in A005153 must be in A007691. It is natural to ask if there exists a generalization of the indicator function for A005153, call it m(n), such that m(n) = 1 for n in A005153, 0 < m(n) < 1 otherwise, and m(a(n)) <= m(n) for all n. See also A050972. - Jaycob Coleman, Sep 27 2014

If a(n) = 0, then n is a multiply-perfect number (A007691). - Alonso del Arte, Mar 30 2014

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.

Numbers n such that a(n)=0 are: 1, 2, 24, 4320, 4680, ... (see A159907, conjecture by Jaroslav Krizek and further comments). - Michel Marcus, Sep 23 2013

Numbers n such that a(n)=n/2 are: 6, 28, 120, 496, 672, ... = A007691 \ {1}. - Michel Marcus, Sep 25 2013

200286975596707184640 belongs to this sequence. - Gerard P. Michon, May 10 2009

81703797123392614369698250752 is in this sequence. - Gerard P. Michon, May 11 2009

This sequence includes many terms but it is conjectured to be finite.

If m belongs to the sequence, then sigma(2*m)/m is an integer, so sigma(2*m)/(2*m) is either an integer or half of an integer, so 2*m is either perfect, multiperfect or hemiperfect. - Michel Marcus, Jul 09 2013

Numbers n such that antisigma(n) mod n = A229110(n) = 0.

If there are any odd multiply-perfect numbers, they are members of this sequence.

If there is no odd multiply-perfect number, then a(n) = A159907(n-1) for n >= 2.