A046060 -id:A046060 - cp's OEIS frontend

A351458 Numbers k for which k * gcd(sigma(k), A276086(k)) is equal to sigma(k) * gcd(k, A276086(k)), where A276086 is the primorial base exp-function, and sigma gives the sum of divisors of its argument.

1, 10, 56, 9196, 9504, 56160, 121176, 239096, 354892, 411264, 555520, 716040, 804384, 904704, 1063348, 1387386, 1444352, 1454112, 1884800, 2708640, 3317248, 3548920, 4009824, 4634784, 6179712, 6795360, 7285248, 14511744, 16328466, 28377216, 29855232, 31940280, 37444736, 42711552, 49762944, 52815744

Offset: 1

Antti Karttunen, Feb 13 2022

nonn

Numbers k such that k * A324644(k) = A000203(k) * A324198(k).
Numbers k such that gcd(A064987(k), A324580(k)) = gcd(A064987(k), A351252(k)).
Numbers k such that their abundancy index [sigma(k)/k] is equal to A324644(k)/A324198(k). See A364286.
A324644 gives odd values for even numbers and for the odd squares. A324198 is odd on all arguments, therefore on odd squares the above equation reduces to odd * odd = odd * odd, and on odd nonsquares as odd * even = even * odd. It is an open question whether there are any odd terms after the initial a(1)=1.
If k is even, but not a multiple of 3, then A276086(k) is a multiple of 3, but not even (i.e., is an odd multiple of 3). If for such k also sigma(k) = 3*k, then A007949(A324644(k)) = min(A007949(sigma(k)), A007949(A276086(k))) = 1, while A007949(A324198(k)) = min(A007949(k), A007949(A276086(k))) = 0, therefore all such k's do occur in this sequence, for example, the two known terms of A005820 (3-perfect numbers) that are not multiples of three: 459818240, 51001180160, but also any hypothetical term of A005820 of the form 4u+2, where 2u+1 is not multiple of 3, and which by necessity is then also an odd perfect number.
Similarly, of the 65 known 5-multiperfect numbers (A046060), those 20 that are not multiples of five are included in this sequence. Note that all 65 are multiples of six.
It is conjectured that the intersection of this sequence with the multiperfect numbers (A007691) gives A323653, see comments in the latter.
For all even terms k of this sequence, A007814(A000203(k)) = A007814(k), sigma preserves the 2-adic valuation, and A007949(A000203(k)) >= A007949(k), i.e., does not decrease the 3-adic valuation. The condition is equivalence (=) when k is a multiple of 6. With odd terms, any hypothetical odd perfect number x would yield a one greater 2-adic valuation for sigma(x) than for x, but would satisfy the main condition of this sequence. - Corrected Feb 17 2022
If k is a nonsquare positive odd number (in A088828), then it must be a term of A191218. - Antti Karttunen, Mar 10 2024

Cf. A000203, A005820, A007691, A007814, A007949, A046060, A088828, A191218, A276086, A323653, A324198, A324644, A327938, A364286, A351459 (intersection with a similar sequence, A349745).

Cf. also A351549.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA351458(n) = { my(s=sigma(n), z=A276086(n)); (n*gcd(s,z))==(s*gcd(n,z)); };

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]); \\ Works OK with rationals also!
isA351458(n) = { my(orgn=n, s=sigma(n), abi=s/n, p=2, q=A006530(abi), d, e1, e2); while((1!=abi)&&(p<=q), d = n%p; e1 = min(d, valuation(s, p)); e2 = min(d, valuation(orgn, p)); d = e1-e2; if(valuation(abi,p)!=d, return(0), abi /= (p^d)); n = n\p; p = nextprime(1+p)); (abi==1); }; \\ (This implementation does not require the construction of largish intermediate numbers, A276086, but might still be slower and return a few false positives on the long run, so please check the results with the above program). - Antti Karttunen, Feb 19 2022

A065997 Numbers n such that sigma(n) / n is prime.

Original entry on oeis.org

6, 28, 120, 496, 672, 8128, 523776, 33550336, 459818240, 1476304896, 8589869056, 14182439040, 31998395520, 51001180160, 137438691328, 518666803200, 13661860101120, 30823866178560, 740344994887680, 796928461056000, 212517062615531520, 2305843008139952128

Offset: 1

Joseph L. Pe, Dec 10 2001

This is a subsequence of the sequence of multiply perfect numbers A007691.
The prime values of sigma(n) / n are A219545.
Numbers whose abundancy index is a prime. There are two visible bends (sudden changes in the growth rate) in the scatter plot. Compare also to the scatter plot of A336702. - Antti Karttunen, Feb 25 2022

Antti Karttunen, Table of n, a(n) for n = 1..597 (computed from the b-file of A007691 prepared by T. D. Noe, using Flammenkamp's data)

Subsequence of A007691 and of A342924.
Cf. A000396, A005820, A046060 (subsequences).
Cf. A219545, A336702.

isA065997(n) = { my(p=sigma(n)/n); (1==denominator(p) && isprime(p)); }; \\ Antti Karttunen, Feb 25 2022

Terms a(10) to a(14) from Jonathan Sondow, Nov 22 2012

Extended by T. D. Noe, Nov 26 2012

A351551 Numbers k such that the largest unitary divisor of sigma(k) that is coprime with A003961(k) is also a unitary divisor of k.

Original entry on oeis.org

1, 2, 10, 34, 106, 120, 216, 260, 340, 408, 440, 580, 672, 696, 820, 1060, 1272, 1666, 1780, 1940, 2136, 2340, 2464, 3320, 3576, 3960, 4280, 4536, 5280, 5380, 5860, 6456, 6960, 7520, 8746, 8840, 9120, 9632, 10040, 10776, 12528, 12640, 13464, 14560, 16180, 16660, 17400, 17620, 19040, 19416, 19992, 21320, 22176, 22968

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

Numbers k for which A351546(k) is a unitary divisor of k.
The condition guarantees that A351555(k) = 0, therefore this is a subsequence of A351554.
The condition is also a necessary condition for A349745, therefore it is a subsequence of this sequence.
All six known 3-perfect numbers (A005820) are included in this sequence.
All 65 known 5-multiperfects (A046060) are included in this sequence.
Not all multiperfects (A007691) are present (only 587 of the first 1600 are), but all 23 known terms of A323653 are terms, while none of the (even) terms of A046061 or A336702 are.

			For n = 672 = 2^5 * 3^1 * 7^1, and the largest unitary divisor of the sigma(672) [= 2^5 * 3^2 * 7^1] coprime with A003961(672) [= 13365 = 3^5 * 5^1 * 11^1] is 2^5 * 7^1 = 224, therefore A351546(672) is a unitary divisor of 672, and 672 is included in this sequence.

Cf. A000203, A000396, A003961, A007691, A046061, A065997, A336702, A351546, A351555, A353633 (characteristic function).
Subsequence of A351552 and of A351554.
Cf. A349745, A351550 (subsequences), A005820, A046060, A323653 (very likely subsequences).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351546(n) = { my(f=factor(sigma(n)),u=A003961(n)); prod(k=1,#f~,f[k,1]^((0!=(u%f[k,1]))*f[k,2])); };
isA351551(n) =  { my(u=A351546(n)); (!(n%u) && 1==gcd(u,n/u)); };

A351554 Numbers k such that there are no odd prime factors p of sigma(k) such that p does not divide A003961(k) and the valuation(k, p) is different from valuation(sigma(k), p), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 3, 6, 7, 10, 14, 15, 20, 21, 22, 24, 27, 28, 30, 31, 33, 34, 40, 42, 46, 54, 57, 60, 62, 66, 69, 70, 84, 87, 91, 93, 94, 102, 105, 106, 110, 114, 120, 127, 130, 138, 140, 141, 142, 154, 160, 168, 170, 174, 177, 182, 186, 189, 190, 195, 198, 210, 214, 216, 217, 220, 224, 230, 231, 237, 238, 254, 260, 264, 270, 273

Offset: 1

Antti Karttunen, Feb 16 2022

Numbers k for which A351555(k) = 0. This is a necessary condition for the terms of A349169 and of A349745, therefore they are subsequences of this sequence.
All six known 3-perfect numbers (A005820) are included in this sequence.
All 65 known 5-multiperfects (A046060) are included in this sequence.
Moreover, all multiperfect numbers (A007691) seem to be in this sequence.
From Antti Karttunen, Aug 27 2025: (Start)
Multiperfect number m is included in this sequence only if its abundancy sigma(m)/m has only such odd prime factors p that prevprime(p) [A151799] divides m for each p.  E.g., all 65 known 5-multiperfects are multiples of 3, and all known terms of A005820 and A046061 are even.
This sequence contains natural numbers k such that the odd primes in the prime factorization of sigma(k) have the same valuation there as in k, except that the primes in A003961(k) [or equally in A003961(A007947(k))] stand for "don't care primes", that are "masked off" from the comparison.
(End)

Cf. A000203, A003961, A151799, A351546.
Positions of zeros in A351555.
Subsequences: A000396, A351553 (even terms), A386430 (odd terms), A351551, A349169, A349745, A387160 (terms of the form prime * m^2), also these, at least all the currently (Feb 2022) known terms: A005820, A007691, A046060.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351555(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); sum(k=1,#f~,if((f[k,1]%2) && 0!=(u%f[k,1]), (valuation(n,f[k,1])!=f[k,2]), 0)); };
isA351554(n) = (0==A351555(n));

isA351554(n) = { my(sh=A351546(n),f=factor(sh)); for(i=1,#f~, if((f[i,1]%2)&&valuation(n,f[i,1])!=f[i,2],return(0))); (1); }; \\ Uses also program given in A351546.

Definition corrected by Antti Karttunen, Aug 22 2025

A348942 a(n) = A326042(n) / gcd(n, A326042(n)).

Original entry on oeis.org

1, 1, 2, 11, 1, 1, 2, 3, 29, 1, 5, 11, 4, 1, 2, 49, 3, 29, 2, 11, 4, 5, 6, 1, 34, 2, 22, 11, 1, 1, 17, 55, 10, 3, 2, 319, 10, 1, 8, 3, 7, 2, 2, 5, 29, 3, 8, 49, 85, 17, 2, 11, 6, 11, 1, 3, 4, 1, 29, 11, 13, 17, 58, 1091, 4, 5, 4, 33, 4, 1, 31, 29, 3, 5, 68, 11, 10, 4, 10, 49, 469, 7, 12, 11, 3, 1, 2, 15, 25, 29, 8

Offset: 1

Antti Karttunen, Nov 04 2021

Numerator of ratio A326042(n)/n. Ratio A326042(n)/n is multiplicative because both A326042 and A000027 are.

Cf. A000203, A003961, A005820, A007691, A046060, A064989, A326042, A348739, A348940, A348941 (denominators), A348943.

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_] := (sn = s2[n])/GCD[n, sn]; 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)));
A348942(n) = { my(u=A326042(n)); (u / gcd(n, u)); };

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

For all n >= 1, A348943(A064989(n)) = 1.

A216780 Numbers n such that numerator(sigma(n)/n) and denominator(sigma(n)/n) are both odd.

Original entry on oeis.org

1, 9, 10, 12, 25, 26, 34, 44, 49, 56, 58, 74, 76, 81, 82, 90, 106, 120, 121, 122, 146, 169, 172, 178, 184, 194, 202, 216, 218, 225, 226, 234, 236, 260, 268, 274, 289, 298, 300, 306, 312, 314, 332, 340, 346, 361, 362, 386, 394, 396, 408, 428, 440, 441, 458

Offset: 1

Michel Marcus, Sep 16 2012

a(n) contains odd squares (A016754), 3-perfect numbers (A005820) and 5-perfect numbers (A046060).

This is also the sequence of numbers x such that A243473(x) is even. - Michel Marcus, Jun 06 2014

			sigma(10)/10 = 9/5; both 9 and 5 are odd, so 10 is in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A216781, A216782, A016754, A005820, A046060.

Select[Range[500], OddQ[Numerator[DivisorSigma[1, #]/#]] && OddQ[Denominator[DivisorSigma[1, #]/#]] &] (* Alonso del Arte, Sep 16 2012 *)

ooab(k) = {for (i=1, k, ab = sigma(i)/i; if ((numerator(ab) % 2 == 1) && (denominator(ab) % 2 == 1), print1(i, ", ")););}

A249670 a(n) = A017665(n)*A017666(n).

Original entry on oeis.org

1, 6, 12, 28, 30, 2, 56, 120, 117, 45, 132, 21, 182, 84, 40, 496, 306, 78, 380, 210, 672, 198, 552, 10, 775, 273, 1080, 2, 870, 60, 992, 2016, 176, 459, 1680, 3276, 1406, 570, 2184, 36, 1722, 112, 1892, 231, 390, 828, 2256, 372, 2793, 4650, 408, 1274, 2862

Offset: 1

Michel Marcus, Nov 03 2014

nonn

If n is a k-multiperfect, then a(n) = k.

Allan C. Wechsler, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma(n)).
Cf. A017665/A017666 (abundancy of n).
Cf. A009194 (gcd(n, sigma(n))), A064987 (n*sigma(n)).
Cf. A000396, A005820, A027687, A046060, A046061.

a249670 n = div (n * s) (gcd n s ^ 2)
 where s = sum (filter (\k -> mod n k == 0) [1..n])
-- Allan C. Wechsler, Mar 31 2023

a249670[n_Integer] := Numerator[DivisorSigma[-1, n]]*Denominator[DivisorSigma[-1, n]]; a249670 /@ Range[80] (* Michael De Vlieger, Nov 10 2014 *)

a(n) = my(ab = sigma(n)/n); numerator(ab)*denominator(ab);

a(n) = A064987(n)/A009194(n)^2.
a(A000396(n)) = 2 (perfect).
a(A005820(n)) = 3 (tri-perfect).
For p prime, a(p) = p*(p+1).

A386425 Odd composites k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

Original entry on oeis.org

153, 801, 1773, 3725, 4689, 4753, 5013, 6957, 8577, 8725, 9549, 9873, 11493, 13437, 14409, 15381, 18621, 19269, 21213, 21537, 23481, 25101, 26073, 26225, 28989, 29161, 29313, 29961, 32229, 33849, 34173, 36117, 38061, 39033, 40653, 42597, 43893, 47457, 47781, 48725, 48753, 51669, 52317, 54261, 56953, 57177, 57501

Offset: 1

Antti Karttunen, Aug 17 2025

nonn

By definition, the sequence contains all odd perfect numbers, and also includes any hypothetical odd triperfect number that is not a multiple of 3 (see A005820 and A347391), and similarly, any odd term of A046060 that is not a multiple of 5, etc. If there are no squares in this sequence (see conjecture in A386424), then the latter categories of numbers certainly do not exist, and this is then a subsequence of A228058.

The first nondeficient term is a(32315) = 81022725. See A386426.

Intersection of A071904 and A386424.
Nonsquare terms form a subsequence of A228058.
Cf. A000203, A003557, A057521, A386426 (nondeficient terms).
Cf. also A324647, A349749.

rad[n_] := Times @@ First /@ FactorInteger[n];a057521[n_] := n/Denominator[n/rad[n]^2];Select[Range[9,57501,2],!PrimeQ[#]&&a057521[DivisorSigma[1,#]]==a057521[#]&] (* James C. McMahon, Aug 18 2025 *)

A057521(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1))
isA386425(n) = ((n>1) && (n%2) && !isprime(n) && (A057521(sigma(n))==A057521(n)));

{k | k is odd composite and A003557(A000203(k)) = A003557(k)}.

A066289 Numbers k such that k divides DivisorSigma(2*j-1, k) for all j; i.e., all odd-power-sums of divisors of k are divisible by k.

Original entry on oeis.org

1, 6, 120, 672, 30240, 32760, 31998395520, 796928461056000, 212517062615531520, 680489641226538823680000, 13297004660164711617331200000, 1534736870451951230417633280000, 6070066569710805693016339910206758877366156437562171488352958895095808000000000

Offset: 1

Labos Elemer, Dec 12 2001

nonn

Tested for each k and j < 200. Otherwise the proof for all j seems laborious, since the number of divisors of terms of sequence rapidly increases: {1, 4, 16, 24, 96, 96, 2304, ...}.
Tested for each k and j <= 1000. - Thomas Baruchel, Oct 10 2003
The given terms have been tested for all j. - Don Reble, Nov 03 2003
This is a proper subset of the multiply perfect numbers A007691. E.g., 8128 from A007691 is not here because its remainder at Sigma[odd,8128]/8128 division is 0 or 896 depending on odd exponent.

Cf. A066135, A066284, A007691, A066290.

DivisorSigma(2*j-1, k)/k is an integer for all j = 1, 2, 3, ..., 200, ...

The following numbers belong to the sequence, but there may be missing terms in between: 796928461056000 (also belongs to A046060); 212517062615531520 (also belongs to A046060); 680489641226538823680000 (also belongs to A046061); 13297004660164711617331200000 (also belongs to A046061). - Thomas Baruchel, Oct 10 2003

Extended to 13 confirmed terms by Don Reble, Nov 04 2003. There is a question whether there are other members below a(13). However, there are none in Achim's list of multiperfect numbers (see A007691); Richard C. Schroeppel has suggested that that list is complete to 10^70 - if so, a(1..12) are correct; as for a(13), Rich says there's only "an epsilon chance that some undiscovered MPFN lies in the gap." So it is very likely to be correct. - Don Reble

A342924 Composite numbers k such that A003415(sigma(k)) = k + p*A003415(k), for some prime p, where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

Original entry on oeis.org

6, 28, 120, 496, 672, 963, 1036, 5871, 8128, 10479, 164284, 264768, 523776, 2308203, 6511664, 33550336, 41240261, 75384301, 400902412, 459818240, 581013140, 1253768516, 1476304896, 2114464203, 8589869056

Offset: 1

Antti Karttunen, Apr 08 2021

Composite numbers k for which A342926(k) = p*A003415(k), for some prime p.

Corresponding prime p for the first 25 terms is: 2, 2, 3, 2, 3, 3, 3, 11, 2, 11, 2, 3, 3, 5, 2, 2, 101, 397, 2, 3, 5, 7, 3, 5, 2. - Antti Karttunen, Feb 25 2022

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Odd terms in this sequence form a subsequence of A347884.
Cf. A342925, A342926.
Cf. A000396, A005820, A046060, A065997 (subsequences).
Cf. also A342922, A342923, A007691.

Block[{f}, f[n_] := If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Select[Range[4, 10^6], And[CompositeQ[#], PrimeQ[(f[DivisorSigma[1, #]] - #)/f[#] ]] &]] (* Michael De Vlieger, Apr 08 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342925(n) = A003415(sigma(n));
isA342924(n) = if((n<2)||isprime(n),0,my(q=(A342925(n)-n)/A003415(n)); ((1==denominator(q))&&isprime(q)));

Terms a(21) - a(25) from Antti Karttunen, Feb 25 2022

cp's OEIS Frontend

A351458 Numbers k for which k * gcd(sigma(k), A276086(k)) is equal to sigma(k) * gcd(k, A276086(k)), where A276086 is the primorial base exp-function, and sigma gives the sum of divisors of its argument.

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A065997 Numbers n such that sigma(n) / n is prime.

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A351551 Numbers k such that the largest unitary divisor of sigma(k) that is coprime with A003961(k) is also a unitary divisor of k.

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A351554 Numbers k such that there are no odd prime factors p of sigma(k) such that p does not divide A003961(k) and the valuation(k, p) is different from valuation(sigma(k), p), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

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A348942 a(n) = A326042(n) / gcd(n, A326042(n)).

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A216780 Numbers n such that numerator(sigma(n)/n) and denominator(sigma(n)/n) are both odd.

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A249670 a(n) = A017665(n)*A017666(n).

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A386425 Odd composites k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

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