OFFSET
1,2
COMMENTS
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.
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)
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..23 (prepared from 1600 term b-file given in A007691).
Achim Flammenkamp, The Multiply Perfect Numbers Page
EXAMPLE
3-multiperfect number 459818240 is a term because number 3*459818240 = 1379454720 is a 4-multiperfect number.
PROG
(Magma) [n: n in [1..10^6] | SumOfDivisors(n) mod n eq 0 and SumOfDivisors(SumOfDivisors(n)) mod SumOfDivisors(n) eq 0]
(PARI) ismulti(n) = (sigma(n) % n) == 0;
isok(n) = ismulti(n) && ismulti(sigma(n)); \\ Michel Marcus, Jan 26 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Jaroslav Krizek, Jan 21 2019
STATUS
approved