A353365 -id:A353365 - cp's OEIS frontend

A323653 Multiperfect numbers m such that sigma(m) is also multiperfect.

1, 459818240, 51001180160, 13188979363639752997731839211623940096, 5157152737616023231698245840143799191339008, 54530444405217553992377326508106948362108928, 133821156044600922812153118065015159487725568, 4989680372093758991515359988337845750507257510078971904

Offset: 1

Jaroslav Krizek, Jan 21 2019

nonn

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)

			3-multiperfect number 459818240 is a term because number 3*459818240 = 1379454720 is a 4-multiperfect number.

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
Index entries for sequences where any odd perfect numbers must occur
Index entries for sequences related to sigma(n)

Cf. A000396, A005820, A027687, A046060, A046061, A053669, A054030, A145551, A323652, A336546, A342659, A347392, A349745, A351458, A351551, A351554, A353365.

Intersection of any two of these three sequences: A007691, A019278, A066961.

[n: n in [1..10^6] | SumOfDivisors(n) mod n eq 0 and SumOfDivisors(SumOfDivisors(n)) mod SumOfDivisors(n) eq 0];

ismulti(n) = (sigma(n) % n) == 0;
isok(n) = ismulti(n) && ismulti(sigma(n)); \\ Michel Marcus, Jan 26 2019

A347392 Numbers k such that nearest common ancestor of k and sigma(k) in Doudna tree (A347879) is the grandparent of k.

Original entry on oeis.org

8, 9, 12, 13, 24, 35, 160, 455, 42550, 127650, 8041950, 22469750, 58506250, 67409250, 175518750, 394055550, 4246782750

Offset: 1

Antti Karttunen, Aug 30 2021

Note how 13 * 35 = 455.
If there exists any odd perfect numbers x, with sigma(x) = 2x, then 2*x would be a term of this sequence, as then sigma(2*x) = 6*x would be situated as a descendant under the other branch of the grandparent of 2*x (a parent of x), which is m = A064989(x), with m in A005101. Opn x itself would be a term of A336702. Furthermore, if such x is not a multiple of 3 (in which case m is odd and in A005231), then also 3x would be a term of this sequence as sigma(3*x) = 4*sigma(x) = 8*x would be situated as a grandchild of 2x, with 2x being a first cousin of 3x. Also, in that case, 6*x would be located in A336702 (particularly, in A027687) because then sigma(6*x) = 12*sigma(x) = 24*x = 4*(6*x).
.
                <--A003961--  m  ---(*2)--->
              .............../ \...............
             /                                 \
            /                                   \
           /                                     \
          x                                       2m
  etc..../ \......2x = sigma(x)          3x....../ \......4m
                  / \                     / \            / \
               etc.  \                 etc.  \        etc.  etc.
                      \                       \
                       4x          sigma(2x) = 6x
                      / \                     / \
                  etc    \                 etc.  \
                          \                       \
                          8x = sigma(3x)          12x
                             if m odd               \
                                                     \
                                                    24x = sigma(6x) if m odd.
.
Furthermore, if there were any hypothetical odd terms y in A005820 (triperfect numbers), then 2y would be a term of this sequence. See the diagram in A347391.
If it exists, a(18) > 2^33.

			455 is included in the sequence as sigma(455) = 672, and the nearest common ancestor of 455 and 672 in Doudna tree is 42, which is the grandparent of 455 [as 455 = A003961(A003961(42))] and the grand-grand-grand-parent of 672 [as 672 = (2^4)*42].

Positions of 2's in A347381.
Cf. A000203, A005101, A005231, A005820, A005940, A027687, A336702, A347391, A347393, A347394, A347879, A348041.
Cf. also A336834, A353365.

PARI
```
isA347392(n) = (2==A347381(n));
```

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
isA347391(n) = if(1==n,0,my(m=A252463(n), s=sigma(n)); while(s>m, if(s==n, return(0)); s = A252463(s)); (s==m));
isA347391_or_A347392(n) = if(1==n,0,my(m=A252463(A252463(n)), s=sigma(n)); while(s>m, if(s==n, return(0)); s = A252463(s)); (s==m));
isA347392(n) = (isA347391_or_A347392(n) && !isA347391(n));

A066961 Numbers k such that sigma(k) divides sigma(sigma(k)).

Original entry on oeis.org

1, 5, 12, 54, 56, 87, 95, 276, 308, 427, 429, 446, 455, 501, 581, 611, 9120, 9180, 9504, 9720, 9960, 10296, 10620, 10740, 10824, 11070, 11310, 11480, 11484, 11556, 11628, 11748, 11934, 11960, 12024, 12036, 12072, 12084, 12376, 12460, 12510, 12570

Offset: 1

Benoit Cloitre, Jan 26 2002

nonn

Is this sequence finite?

These are numbers k such that sigma(k) is a multiply-perfect number (A007691). - Ivan N. Ianakiev, Sep 13 2016

			12 is in the sequence since sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28 divides sigma(28) = 1 + 2 + 4 + 7 + 14 + 28 = 56. - _Michael B. Porter_, Sep 22 2016

Antti Karttunen, Table of n, a(n) for n = 1..3718 (first 1000 terms from Harry J. Smith)
Index entries for sequences where odd perfect numbers must occur, if they exist at all
Index entries for sequences related to sigma(n)

Cf. A000203, A051027.

Subsequences: A323653 (intersection with A007691, or equally, with A019278), A353365 (where the quotient is a power of 2).

[n: n in [1..13000] | (SumOfDivisors(SumOfDivisors(n)) mod SumOfDivisors(n) eq 0)]; // Vincenzo Librandi, Sep 13 2016

with(numtheory): A066961:=n->`if`(sigma(sigma(n)) mod sigma(n) = 0, n, NULL): seq(A066961(n), n=1..2*10^4); # Wesley Ivan Hurt, Sep 22 2016

Select[Range[30000], Divisible[DivisorSigma[1, DivisorSigma[1, #]], DivisorSigma[1, #]] &] (* Ivan N. Ianakiev, Sep 13 2016 *)

isok(n) = my(s=sigma(n)); s && ((sigma(s) % s) == 0); \\ Michel Marcus, Sep 17 2016

More terms from Lior Manor, Feb 06 2002

A353363 Numbers k such that A046523(A332223(A332223(k))) is equal to A046523(A332223(k)).

Original entry on oeis.org

1, 2, 6, 25, 38, 155, 235, 294, 322, 365, 390, 465, 528, 638, 646, 744, 765, 768, 867, 884, 924, 946, 1152, 1172, 1290, 1421, 1600, 1653, 1675, 1677, 1729, 1785, 1980, 1989, 2002, 2028

Offset: 1

Antti Karttunen, Apr 17 2022

nonn

Numbers k such that A332223(k) is in A353308.

If there existed any 3-perfect number (A005820) of the form x = 4u+2 and not divisible by 3, then x would be also included in A353365 and A005940(1+x) would be included in this sequence. See comments in A353365.

Cf. A000203, A005940, A046523, A066961, A156552, A332223, A353365.

Cf. also A353308.

A354198 a(n) = A064989(A064989(sigma(sigma(sigma(A003961(A003961(n))))))), where A003961 shifts the prime factorization of n one step towards larger primes, and A064989 shifts it back towards smaller primes.

Original entry on oeis.org

1, 3, 1, 3, 3, 3, 2, 26, 23, 3, 3, 3, 1, 3, 21, 6, 3, 9, 14, 22, 2, 2, 7, 182, 3, 14, 313, 201, 3, 3, 3, 603, 3, 3, 3, 115, 3, 3, 2, 3, 3, 21, 2, 9, 9, 3, 2, 75, 2, 22, 3, 109, 3, 21, 46, 109, 2, 23, 7, 154, 3, 6, 22, 222, 2, 14, 2, 22, 29, 6, 1, 78, 3, 161, 69, 1407, 6, 2, 21, 44, 7, 21, 14, 201, 21, 39, 3, 529

Offset: 1

Antti Karttunen, May 24 2022

nonn

For any hypothetical odd perfect number opn that is not a multiple of 3, it holds that a(n) = A354196(n) = A348750(n) = n, where n = A064989(A064989(opn)). See also comments in A353365.

Cf. A000203, A003961, A064989, A066971, A326042, A353365, A354197.

Cf. also A348750, A354196.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f=factor(n>>valuation(n, 2))); for(i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
A354198(n) = A064989(A064989(sigma(sigma(sigma(A003961(A003961(n)))))));

a(n) = A064989(A354197(A003961(n))) = A064989(A064989(A066971(A003961(A003961(n))))).

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A323653 Multiperfect numbers m such that sigma(m) is also multiperfect.

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A347392 Numbers k such that nearest common ancestor of k and sigma(k) in Doudna tree (A347879) is the grandparent of k.

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A066961 Numbers k such that sigma(k) divides sigma(sigma(k)).

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A353363 Numbers k such that A046523(A332223(A332223(k))) is equal to A046523(A332223(k)).

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A354198 a(n) = A064989(A064989(sigma(sigma(sigma(A003961(A003961(n))))))), where A003961 shifts the prime factorization of n one step towards larger primes, and A064989 shifts it back towards smaller primes.

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