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%I #68 Jun 12 2022 09:32:32
%S 1,459818240,51001180160,13188979363639752997731839211623940096,
%T 5157152737616023231698245840143799191339008,
%U 54530444405217553992377326508106948362108928,133821156044600922812153118065015159487725568,4989680372093758991515359988337845750507257510078971904
%N Multiperfect numbers m such that sigma(m) is also multiperfect.
%C Multiperfect numbers m such that sigma(m) divides sigma(sigma(m)).
%C Also k-multiperfect numbers m such that k*m is also multiperfect.
%C Corresponding values of numbers k(n) = sigma(a(n)) / a(n): 1, 3, 3, 5, 5, 5, 5, 5, ...
%C Corresponding values of numbers h(n) = sigma(k(n) * a(n)) / (k(n) * a(n)): 1, 4, 4, 6, 6, 6, 6, 6, ...
%C Number of k-multiperfect numbers m such that sigma(n) is also multiperfect for k = 3..6: 2, 0, 20, 0.
%C From _Antti Karttunen_, Mar 20 2021, Feb 18 2022: (Start)
%C 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.
%C Conjecture 2: This sequence is finite.
%C Conjecture 3: This sequence is the intersection of A007691 and A351458.
%C Conjecture 4: This is a subsequence of A349745, thus also of A351551 and of A351554.
%C 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.
%C (End)
%C From _Antti Karttunen_, May 16 2022: (Start)
%C Apparently for all n > 1, A336546(a(n)) = 0. [At least for n=2..23], while A353633(a(n)) = 1, for n=1..23.
%C 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.
%C 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.
%C (End)
%H Antti Karttunen, <a href="/A323653/b323653.txt">Table of n, a(n) for n = 1..23</a> (prepared from 1600 term b-file given in A007691).
%H Achim Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/mpn.html">The Multiply Perfect Numbers Page</a>
%H <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e 3-multiperfect number 459818240 is a term because number 3*459818240 = 1379454720 is a 4-multiperfect number.
%o (Magma) [n: n in [1..10^6] | SumOfDivisors(n) mod n eq 0 and SumOfDivisors(SumOfDivisors(n)) mod SumOfDivisors(n) eq 0]
%o (PARI) ismulti(n) = (sigma(n) % n) == 0;
%o isok(n) = ismulti(n) && ismulti(sigma(n)); \\ _Michel Marcus_, Jan 26 2019
%Y Cf. A000396, A005820, A027687, A046060, A046061, A053669, A054030, A145551, A323652, A336546, A342659, A347392, A349745, A351458, A351551, A351554, A353365.
%Y Intersection of any two of these three sequences: A007691, A019278, A066961.
%K nonn
%O 1,2
%A _Jaroslav Krizek_, Jan 21 2019
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