A336702 -id:A336702 - cp's OEIS frontend

A325637 Numbers k for which gcd(2k, sigma(k)) = 2k.

6, 28, 496, 8128, 30240, 32760, 2178540, 23569920, 33550336, 45532800, 142990848, 1379454720, 8589869056, 43861478400, 66433720320, 137438691328, 153003540480, 403031236608, 704575228896, 181742883469056, 6088728021160320, 14942123276641920, 20158185857531904, 275502900594021408, 622286506811515392, 2305843008139952128

Offset: 1

Antti Karttunen, May 21 2019

nonn

Multiply-perfect numbers (A007691) k with an even abundancy index sigma(k)/k. - Amiram Eldar, Jun 26 2024

Antti Karttunen, Table of n, a(n) for n = 1..1013 (computed from the b-file of A007691 prepared by T. D. Noe, using Flammenkamp's data)
Index entries for sequences related to sigma(n).
Index entries for sequences where any odd perfect numbers must occur.

Cf. A000203, A033879, A224832, A325636, A325638, A325654.
Subsequences: A000396, A336702 (after its initial 1).
Subsequence of A007691.

isA325637(n) = ((n+n)==gcd(n+n,sigma(n)));

a(n) = A224832(n)/2. - Amiram Eldar, Jun 26 2024

A347879 The nearest common ancestor of n and sigma(n) in the Doudna tree (A005940).

Original entry on oeis.org

1, 2, 2, 2, 3, 6, 2, 2, 2, 2, 3, 3, 7, 3, 6, 2, 2, 2, 5, 10, 2, 2, 3, 6, 2, 5, 2, 28, 3, 2, 2, 2, 3, 2, 6, 2, 19, 3, 7, 3, 5, 3, 11, 5, 3, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 3, 5, 3, 3, 3, 31, 3, 5, 2, 5, 2, 17, 5, 3, 2, 2, 2, 37, 17, 2, 3, 6, 5, 5, 5, 4, 5, 5, 5, 2, 7, 3, 3, 3, 3, 5, 5, 2, 2, 3, 3, 2, 2, 7, 2, 13, 2

Offset: 1

Antti Karttunen, Oct 13 2021

nonn

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.

Cf. A000203, A005940, A252463, A332221, A336702, A347381, A347391, A347392, A348040, A348041.

A347879(n) = A348041sq(n,sigma(n)); \\ Needs also code from A348041.

a(n) = A348041(n, A000203(n)).

A348743 Odd nonsquares k for which A161942(k) >= k, where A161942 is the odd part of sigma.

Original entry on oeis.org

2205, 19845, 108045, 143325, 178605, 187425, 236925, 266805, 319725, 353925, 372645, 407925, 452025, 462825, 584325, 637245, 646425, 658125, 672525, 789525, 796005, 804825, 845325, 920205, 972405, 981225, 1007325, 1055925, 1069425, 1102725, 1113525, 1116225, 1166445, 1201725, 1245825, 1289925, 1378125, 1380825, 1442925

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

The first non-multiples of 5 are a(103) = 6243237 and a(125) = 8164233.
From Antti Karttunen, Nov 28 2024: (Start)
This is not a subsequence of A228058. At least k = A000040(28)*(A002110(27)/2)^2 = 15388519572341080054329140040512468358441210638435506649120749687401476705908239675 is a number of the form 4m+3 such that A161942(k) >= k.
Another such number is A000040(28)*81*(A002110(25)/6)^2 = 1279741205456530915782536871495922949062895982530933679752838870798129159675.
Question: What is the smallest term of this sequence that is of the form 4m+3, and thus not in A386427 (in A191218 and in A228058)?
(End)

Intersection of A088828 and A348742.
Cf. A386427 (a subsequence, which agrees for a very long time).
Cf. A033630, A065205, A083206, A161942, A191218, A228058, A336702, A348741.
Cf. also A065235, A162284.

A000265(n) = (n >> valuation(n, 2));
isA348743(n) = ((n%2)&&!issquare(n)&&A000265(sigma(n))>=n); \\ Edited Nov 28 2024

Definition changed (from > to >=) to formally include also any hypothetical odd perfect numbers - Antti Karttunen, Nov 28 2024

Comment removed, because it was more related to sequence A386427. - Antti Karttunen, Aug 21 2025

A348941 a(n) = n / gcd(n, A326042(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 10, 11, 6, 13, 7, 15, 16, 17, 18, 19, 20, 21, 22, 23, 4, 25, 13, 27, 14, 29, 15, 31, 32, 33, 34, 35, 36, 37, 19, 39, 40, 41, 21, 43, 4, 45, 23, 47, 24, 49, 25, 17, 13, 53, 27, 11, 28, 57, 58, 59, 30, 61, 62, 63, 64, 65, 33, 67, 68, 23, 35, 71, 24, 73, 37, 75, 38, 77, 39, 79, 80, 81, 82

Offset: 1

Antti Karttunen, Nov 04 2021

Denominator of ratio A326042(n) / n.

If there are no more 1's in this sequence after the initial one, then there are no odd terms of A336702 (numbers whose abundancy index is a power of 2) larger than one, and neither there are odd terms in A005820 or in A046060. Compare to similar conditions given in A336848, A336849 and A337339.

Cf. A005820, A046060, A326042, A336702, A336848, A336849, A337339, A348736, A348940, A348942 (numerators).

f1[2, e_] := 1; f1[p_, e_] := NextPrime[p, -1]^e; s[n_] := Times @@ f1 @@@ FactorInteger[n]; f[p_, e_] := s[((q = NextPrime[p])^(e + 1) - 1)/(q - 1)]; s2[1] = 1; s2[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := n/GCD[n, s2[n]]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
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)};
A326042(n) = A064989(sigma(A003961(n)));
A348941(n) = (n / gcd(n, A326042(n)));

a(n) = n / A348940(n) = n / gcd(n, A326042(n)).

A353365 Numbers k such that the odd part of sigma(sigma(k)) is equal to the odd part of sigma(k).

Original entry on oeis.org

1, 5, 12, 427, 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, 12640, 12924, 12980, 13000, 13216, 13340, 13554, 13804, 13806, 13962, 13984, 14022, 14056, 14094, 14178, 14212, 14336, 14380

Offset: 1

Antti Karttunen, Apr 17 2022

nonn

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.

Index entries for sequences related to sigma(n)

Cf. A000203, A000265, A000396, A005820, A051027, A161942, A336702.
Subsequence of A066961.
Cf. also A336700, A336701, A347383, A347392, A349745, A351458 and also A353363.

A000265(n) = (n>>valuation(n,2));
isA353365(n) = (A000265(sigma(sigma(n)))==A000265(sigma(n)));

A043305 Numbers k such that the numerator of the sum of the reciprocals of the divisors of k (=A017665(k)) is a power of 2.

Original entry on oeis.org

1, 3, 6, 7, 15, 21, 28, 31, 33, 42, 69, 84, 91, 93, 105, 127, 135, 141, 186, 217, 231, 270, 273, 285, 381, 420, 465, 483, 496, 546, 573, 651, 762, 775, 819, 861, 868, 889, 924, 945, 987, 1023, 1149, 1185, 1302, 1365, 1419, 1485, 1488, 1561, 1638, 1743, 1890

Offset: 1

Benoit Cloitre, Apr 04 2002

After 1, a subsequence of A216782. If both x and y are terms and gcd(x, y) = 1, then x*y is also present. - Antti Karttunen, Mar 20 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..910 from Harvey P. Dale)

Cf. A017665, A216782, A361465 (characteristic function).

Subsequences: A000396, A336702, A348943 (odd terms).

Select[Range[2000],IntegerQ[Log[2,Numerator[Total[1/Divisors[#]]]]]&] (* Harvey P. Dale, Nov 29 2014 *)

isok(n) = (ispower(num = numerator(sigma(n)/n), , &s) && (s == 2)) || (num == 2) || (num == 1); \\ Michel Marcus, Nov 21 2013

isA043305(n) = { n=sigma(n)/gcd(sigma(n),n); !bitand(n,n-1); }; \\ Antti Karttunen, Mar 20 2023

A347393 Positions of 3's in A347381.

Original entry on oeis.org

7, 10, 11, 14, 16, 18, 25, 27, 39, 45, 63, 77, 81, 99, 105, 135, 182, 270, 819, 1365, 1392, 1638, 4250, 15631, 21275, 63767, 122944, 161257, 203203, 446369, 936100, 1128799, 1773827, 2808300

Offset: 1

Antti Karttunen, Aug 31 2021

nonn

Cf. A347381.

Cf. also A046060, A336702, A347391, A347392, A347394.

A347394 Positions of 4's in A347381.

Original entry on oeis.org

21, 26, 30, 32, 33, 36, 37, 40, 42, 44, 48, 49, 50, 54, 60, 65, 75, 80, 84, 90, 91, 112, 120, 125, 126, 153, 162, 175, 176, 198, 208, 220, 231, 252, 272, 275, 304, 325, 343, 368, 400, 425, 475, 546, 575, 725, 765, 775, 11132, 12750, 13167, 31262, 46893, 55660, 63825, 78155, 93500, 171171, 191301, 406406, 483771, 609609

Offset: 1

Antti Karttunen, Aug 31 2021

nonn

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

Cf. A347381.

Cf. also A336702, A347391, A347392, A347393.

A351440 Numbers k for which A003958(sigma(k)) + A064989(sigma(k)) is equal to A003958(k) + A064989(k).

Original entry on oeis.org

1, 6, 28, 496, 8128, 30240, 32760, 240408, 2178540, 6828720, 13042080, 23569920, 33550336, 42402048, 45532800, 142990848, 1379454720

Offset: 1

Antti Karttunen, Feb 12 2022

Cf. A000203, A003958, A064989, A351442.
Subsequence of A351446.
Subsequences: A000396, A336702.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 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); };
isA351440(n) = { my(s=sigma(n)); ((A003958(s)+A064989(s)) == (A003958(n)+A064989(n))); };

A351447 Numbers k for which A003958(sigma(k)) = 2*A003958(k), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

2, 98, 120, 136, 312, 520, 672, 888, 1080, 1120, 1464, 1480, 1752, 2440, 2520, 2808, 2912, 2920, 3420, 3768, 3848, 4632, 5880, 6048, 6280, 6344, 6552, 6648, 6664, 7512, 7592, 7720, 7992, 8181, 8288, 8892, 9528, 10104, 10968, 11080, 12464, 12520, 12984, 13176, 13664, 14712, 15288

Offset: 1

Antti Karttunen, Feb 12 2022

nonn

Numbers k such that A351442(k) = 2*A003958(k).

In contrast, numbers x for which A064989(sigma(x)) = 2*A064989(x) seem to consist just of {2} followed by A005820: 2, 120, 672, 523776, ..., etc, which (also) contains as its subsequence all the odd terms of A336702 multiplied by 2.

Index entries for sequences related to sigma(n)

Cf. A000203, A003958, A064989, A351442.

Subsequences: A005820 (3-perfect numbers), odd terms of A336702 doubled, the terms of A351443 doubled (2, 98, 81810, ...), A351448 (odd terms in this sequence).

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
isA351447(n) = (A003958(sigma(n))==2*A003958(n));

cp's OEIS Frontend

A325637 Numbers k for which gcd(2k, sigma(k)) = 2k.

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A347879 The nearest common ancestor of n and sigma(n) in the Doudna tree (A005940).

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A348743 Odd nonsquares k for which A161942(k) >= k, where A161942 is the odd part of sigma.

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A348941 a(n) = n / gcd(n, A326042(n)).

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A353365 Numbers k such that the odd part of sigma(sigma(k)) is equal to the odd part of sigma(k).

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A043305 Numbers k such that the numerator of the sum of the reciprocals of the divisors of k (=A017665(k)) is a power of 2.

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A347393 Positions of 3's in A347381.

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A347394 Positions of 4's in A347381.

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A351440 Numbers k for which A003958(sigma(k)) + A064989(sigma(k)) is equal to A003958(k) + A064989(k).

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