A027687 -id:A027687 - cp's OEIS frontend

A171264 Primes of the form 1 + 5-multiperfect numbers.

30823866178561, 796928461056001, 1802582780370364661761, 2827987212986831882236723201, 68688966922031309945174465761834751373920047004215278394826366933532673

Offset: 1

M. F. Hasler and Farideh Firoozbakht, Apr 15 2010

A subsequence of A093034. If p is a term of this sequence then for each positive integer k, x=p^k is a solution for the equation sigma(phi(x))=5(x-1). See comment lines of the sequence A093034.

			p=30823866178561 is prime and sigma(p-1)/(p-1)=5, so p is in the sequence.

Cf. A027687, A093034, A171263, A171265.

A332208 Numbers k such that the squarefree kernel of sigma(k) is equal to the squarefree kernel of 2*k.

Original entry on oeis.org

6, 28, 120, 135, 270, 496, 672, 891, 1080, 1638, 1782, 3780, 8128, 18600, 20580, 24948, 26208, 30240, 32640, 32760, 35640, 41850, 44226, 55860, 66960, 164640, 167400, 185220, 199584, 200655, 273000, 293760, 307125, 401310, 441936, 446880, 502740, 523776, 544635, 614250, 707616, 802620, 819000, 884520

Offset: 1

Antti Karttunen, Feb 07 2020

nonn

Numbers k such that sigma(k) has the same set of distinct prime factors as 2*k.
Numbers k such that A007947(sigma(k)) is equal to A007947(2*k), or equally, that A087207(sigma(k)) is equal to A087207(2*k).
Of the first 256 terms 44 are odd, and none occurs in A228058. Compare also to A331752.

Antti Karttunen, Table of n, a(n) for n = 1..256
Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A007947, A080398, A087207, A228058, A331751, A331752.

Subsequences: A000396, A027687.

Select[Range[10^6], SameQ @@ Map[Times @@ FactorInteger[#][[All, 1]] &, {DivisorSigma[1, #], 2 #}] &] (* Michael De Vlieger, Feb 08 2020 *)

A007947(n) = factorback(factorint(n)[, 1]);
isA332208(n) = (A007947(sigma(n)) == A007947(2*n));

{n: A080398(n) == A007947(2n)}.

A094701 Smallest linear combination of phi(n) and sigma(n) with nonnegative coefficients: a(n) = Min_{x>=0,y>=0} (x+y) for which xphi(n) + ysigma(n) is a multiple of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 1, 2, 2, 3, 3, 2, 3, 2, 4, 5, 2, 2, 3, 2, 4, 7, 4, 2, 2, 5, 4, 3, 1, 2, 5, 2, 2, 9, 4, 18, 3, 2, 4, 9, 4, 2, 7, 2, 6, 3, 4, 2, 3, 7, 5, 9, 5, 2, 3, 7, 3, 9, 4, 2, 5, 2, 4, 7, 2, 11, 7, 2, 9, 9, 10, 2, 3, 2, 4, 12, 4, 10, 7, 2, 5, 3, 4, 2, 3, 13, 4, 9, 4, 2, 5, 9, 9, 9, 4, 19, 3, 2, 7, 5, 5, 2, 7

Offset: 1

Walter Nissen, May 20 2004

nonn

a(n) is a generalization of the multiperfect numbers in A007691.

			a(6) = 1 as 1*sigma(6) is a multiple of 6.
a(4) = 2 as 2*phi(4) + 0*sigma(4) = 4. - Example added by _Antti Karttunen_, Feb 24 2020
a(14) = 4 as 3*phi(14) + 1*sigma(14) = 3*6 + 24 = 3*14, where 3+1 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..12857
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000010, A000203, A000396, A005820, A007691 (positions of ones), A027687, A046060, A046061.

A094701(n) = { my(x=eulerphi(n),y=sigma(n)); for(s=1,oo,for(t=0,s,if(!(((t*x)+((s-t)*y))%n),return(s)))); }; \\ Antti Karttunen, Feb 24 2020

a(multiperfect) = 1.
a(prime) = 2 as 1*phi(prime) + 1*sigma(prime) and 1+1 = 2.
For primes > 5, a(2*prime) = 4.

Name clarified by Antti Karttunen, Feb 24 2020

A134639 Conjectured number of numbers k such that sigma(k)/k = n.

Original entry on oeis.org

6, 36, 65, 245, 516

Offset: 3

T. D. Noe, Nov 05 2007

These numbers come from Guy and Flammenkamp. Sequences A000396, A005820, A027687, A046060 and A046061 give the k for which the abundancy sigma(k)/k is 2, 3, 4, 5 and 6, respectively. Sequence A054030 gives the abundancy of each multiperfect number A007691.

R. K. Guy, Unsolved Problems in Number Theory, 3rd Ed., New York, Springer-Verlag, 2004, Section B2.

Achim Flammenkamp, The Multiply Perfect Numbers Page

A317681 a(n) = smallest m such that sigma(m) = n*m/2.

Original entry on oeis.org

1, 2, 6, 24, 120, 4320, 30240, 8910720, 14182439040, 17116004505600, 154345556085770649600, 170974031122008628879954060917200710847692800, 141310897947438348259849402738485523264343544818565120000, 12749472205565550032020636281352368036406720997031277595140988449695952806020854579200000

Offset: 2

Jianing Song, Aug 04 2018

Interleaving of A007539 and A088912.
For even n, a(n) is a multiply perfect number; for odd n it is a hemiperfect number.
Note that 1 is the only number with abundancy 1, and 2 is the only number with abundancy 3/2 (in other words, 1 and 2 are solitary numbers; see A014567). For k >= 4 it is not known whether there are finitely many or infinitely many numbers with abundancy k/2. Also it is not known whether a(n) < a(n+1) always holds.
On the Riemann Hypothesis (RH), a(n) > exp(exp(n/(2*exp(gamma)))), where gamma = 0.5772156649... is the Euler-Mascheroni constant (A001620).

			a(7) = 4320 since sigma(4320) = 15120 = 7/2*4320 and 4320 is the smallest m such that sigma(m)/m = 7/2.

Max Alekseyev, Table of n, a(n) for n = 2..16
Achim Flammenkamp, The Multiply Perfect Numbers Page
Fred Helenius, A glossary of MPFN-related terms.
G. P. Michon, Multiplicative functions: Abundancy = sigma(n)/n
G. P. Michon, Multiperfect and hemiperfect numbers
Walter Nissen, Abundancy: Some Resources

Cf. A007539, A088912, A246454.

Numbers with abundancy k/2: A000396 (k=4), A141643 (k=5), A005820 (k=6), A055153 (k=7), A027687 (k=8), A141645 (k=9), A046060 (k=10), A159271 (k=11), A046061 (k=12), A160678 (k=13).

Nest[Append[#, Block[{m = #1[[-1]] + 1}, While[DivisorSigma[1, m] != #2 m/2, m++]; m]] & @@ {#, Length@ # + 2} &, {1}, 6] (* Michael De Vlieger, Aug 05 2018 *)

for(n=2, 10, for(m=1, 10^12, if(sigma(m)/m==n/2, print1(m, ", "); break())))

a(n) = my(k=1); while (sigma(k) != k*n/2, k++); k; \\ Michel Marcus, May 15 2025

a(2n) = A007539(n), a(2n+1) = A088912(n), n > 0.

a(15) = A088912(7) added by Max Alekseyev, Jun 05 2025

A335267 Composite numbers whose harmonic mean of their divisors that are larger than 1 is an integer.

Original entry on oeis.org

6, 15, 28, 30, 91, 117, 135, 252, 270, 496, 703, 864, 936, 1891, 1989, 2295, 2701, 4284, 4590, 5733, 8128, 8432, 12403, 18721, 19872, 21528, 38503, 41580, 49141, 51319, 56896, 79003, 88831, 104653, 121920, 146611, 188191, 218791, 226801, 235053, 269011, 286903

Offset: 1

Amiram Eldar, May 29 2020

nonn

The primes are excluded from this sequence since they are trivial terms.
The corresponding harmonic means are 3, 5, 5, 5, 13, 9, 9, 9, 9, 9, 37, ...
Equivalently, composite numbers m such that (sigma(m)-m) | m*(tau(m)-1), or A001065(m) | A168014(m).
The semiprimes terms of this sequence are of the form p*q where p and q = 2*p - 1 are primes (A129521).
If m is a k-perfect numbers, k = 2, 3, ... (i.e., sigma(m) = k*m), then sigma(m)-m = (k-1)*m. If (k-1)*m | m*(tau(m)-1) then (k-1) | (tau(m)-1). If k is odd then tau(m) is also odd, so m is a square, and sigma(m) is odd. Since m | sigma(m) this means that m is also odd. Since there is no known odd multiply-perfect number except for 1 (A007691), there are no known k-perfect numbers with odd k in this sequence.
The perfect numbers (k=2, A000396) are terms: if m is a perfect number then sigma(m)-m = m.
The 4-perfect number (k=4, A027687) m are terms if 3 | (tau(m)-1). Of the first 36 terms of A027687 there are 8 such terms, the first is A027687(26).
The 6-perfect number (k=6, A046061) m are terms if 5 | (tau(m)-1). Of the first 245 terms of A046061 there are 20 such terms, the first is A046061(19).
Hemiperfect numbers that are terms of this sequence include A055153(i) for i = 10, 18 and 20, A141645(21), and A159271(i) for i = 97 and 103.

			6 is a term since its divisors other than 1 are 2, 3 and 6, and their harmonic mean, 3/(1/2 + 1/3 + 1/6) = 3, is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..1000

A000396 and A129521 are subsequences.
Similar sequences: A001599, A247077, A247078.
Cf. A000005 (tau), A000203 (sigma).
Cf. A001065, A032741, A168014,
Cf. A005382, A005383.
Cf. A027687, A046061, A055153, A141645, A159271.

Select[Range[10^6], CompositeQ[#] && Divisible[# * (DivisorSigma[0, #] - 1), DivisorSigma[1, #] - #] &]
Select[Range[287000],CompositeQ[#]&&IntegerQ[HarmonicMean[ Rest[ Divisors[ #]]]]&] (* Harvey P. Dale, Jan 21 2021 *)

A336745 Numbers m that divide the product phi(m) * sigma(m) * tau(m), where phi is the Euler totient function (A000010), sigma is the sum of divisors function (A000203) and tau is the number of divisors function (A000005).

Original entry on oeis.org

1, 2, 6, 8, 9, 12, 18, 24, 28, 32, 36, 40, 54, 72, 80, 84, 96, 108, 117, 120, 128, 135, 144, 162, 196, 200, 216, 224, 234, 240, 243, 252, 270, 288, 324, 360, 384, 400, 405, 448, 468, 486, 496, 512, 540, 576, 588, 600, 625, 640, 648, 672, 675, 720, 756, 768, 775, 810, 819

Offset: 1

Bernard Schott, Aug 02 2020

nonn

If s and t are terms with gcd(s, t) = 1, then s*t is another term as phi, sigma and tau are multiplicative functions.

The only prime term is 2 because prime p must divide 2*(p-1)*(p+1) to be a term.

			For 24, phi(24) = 8, sigma(24) = 60 and tau(24) = 8, then 8*60*8 / 24 = 160, hence 24 is a term.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A000203, A062355.

Subsequences: A000396 (perfect numbers), A005820 (tri-perfect), A027687 (4-perfect), A046060 (5-multiperfect), A046061 (6-multiperfect), A007691 (multiply-perfect numbers), A336715 (m divides phi(m)*tau(m)), A004171, A005010.

with(numtheory):
filter:= m -> irem(tau(m)*phi(m)*sigma(m), m) =0:
select(filter,[$1..850]);

Select[Range[1000], Divisible[Times @@ DivisorSigma[{0, 1}, #] * EulerPhi[#], #] &] (* Amiram Eldar, Aug 02 2020 *)

isok(m) = !(eulerphi(m)*sigma(m)*numdiv(m) % m); \\ Michel Marcus, Aug 05 2020

A113286 Numbers k such that S(S(k))=k, with S(n)=sigma(n)/4: 1/4-sociable numbers of order 1 or 2.

Original entry on oeis.org

30240, 32760, 859320, 898560, 2096640, 2178540, 2234232, 23569920, 45532800, 54996480, 61281792, 142990848, 422688000, 436205952, 1379454720

Offset: 1

Yasutoshi Kohmoto, Jan 27 2006

a(1) and a(2) are 4-perfect numbers (A027687). a(4) = S(a(3)).

a(6), a(8), a(9), a(12), and a(15) are also 4-perfect numbers (A027687). - Michel Marcus, Feb 23 2014

Eric Weisstein's World of Mathematics, Sociable Numbers

A027687 is a subsequence.
Cf. A113546 (1/3-sociable numbers).
Cf. A355279 (1/4-sociable numbers of order 1 or 3).

isok(n) = (type(s= sigma(n)/4) == "t_INT") && (sigma(s)/4 == n); \\ Michel Marcus, Feb 20 2014

a(5)-a(15) added by Michel Marcus, Feb 24 2014

A211677 First number k whose value of sigma(k)/k appears n times.

Original entry on oeis.org

1, 28, 496, 1638, 24384, 2886100, 13035330, 29410290, 4426793280

Offset: 1

T. D. Noe, May 09 2012

The values of sigma(k)/k are 1, 2, 2, 8/3, 8/3, 96/35, 32/9, 32/9, 32/7. Note that these values are nondecreasing. Is that always the case? In the table below, all numbers in the same row are friendly to each other.
a(10) <= 27477725184. a(11) <= 88071903612. a(12) <= A027687(12). - Donovan Johnson, Aug 06 2012
For n>1, these are the smallest numbers to appear consecutively (n-1) times in A050973. - Michel Marcus, Jan 28 2014

			These are the values of k such that sigma(k)/k appears n times:
n   k values
1:  1
2:  6, 28
3:  6, 28, 496
4:  84, 270, 1488, 1638
5:  84, 270, 1488, 1638, 24384
6:  210, 17360, 43400, 284480, 2229500, 2886100
7:  3780, 66960, 167400, 406224, 1097280, 6656832, 13035330
8:  3780, 66960, 167400, 406224, 1097280, 6656832, 13035330, 29410290
9:  164989440, 270138960, 318729600, 326781000, 481572000, 623397600, 675347400, 995248800, 4426793280 - _Donovan Johnson_, Aug 06 2012
These numbers appear in A211679.

Claude W. Anderson and Dean Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84 (1977) pp. 65-66.
Achim Flammenkamp, Multiply Perfect Numbers (sigma(k)/k is an integer)
Tom De Medts, MathOverflow: Recovering n from sigma(n)/n
Carl Pomerance, Multiply perfect numbers, Mersenne primes and effective computability, Math. Ann. 226 (1977), 195-206.
Eric W. Weisstein, Friendly number
Eric W. Weisstein, Friendly pair
Wikipedia, Friendly number

Cf. A000203 (sigma), A050973, A211679.

a(7)-a(8) from Donovan Johnson, May 10 2012

a(9) from Michel Marcus and Donovan Johnson, Aug 06 2012

A364976 3-abundant numbers k such that k/(sigma(k)-3*k) is an integer.

Original entry on oeis.org

180, 240, 360, 420, 540, 600, 780, 1080, 1344, 1872, 1890, 2016, 2184, 2352, 2376, 2688, 3192, 3276, 3744, 4284, 4320, 4680, 5292, 5376, 5796, 6048, 6552, 7128, 7440, 8190, 10416, 13776, 14850, 18600, 19824, 19872, 20496, 21528, 22932, 25056, 26208, 26496, 26784

Offset: 1

Amiram Eldar, Aug 15 2023

nonn

Analogous to A153501 as 3-abundant numbers (A068403) are analogous to abundant numbers (A005101).

Numbers k such that the sum of the divisors of k except for one of them is equal to 3*k.

			180 is a term since sigma(180) - 3*180 = 6 > 0 and 180 is divisible by 6.

Amiram Eldar, Table of n, a(n) for n = 1..576 (terms below 2.5*10^10)

Subsequence of A068403.
A027687 is a subsequence.
Cf. A000203 (sigma), A005101, A005820, A055153, A153501, A160320, A218404, A218406, A329189, A364977.

Select[Range[27000], (d = DivisorSigma[1, #] - 3*#) > 0 && Divisible[#, d] &]

is(n) = {my(d = sigma(n) - 3*n); d > 0 && n%d == 0;}

cp's OEIS Frontend

A171264 Primes of the form 1 + 5-multiperfect numbers.

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A332208 Numbers k such that the squarefree kernel of sigma(k) is equal to the squarefree kernel of 2*k.

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A094701 Smallest linear combination of phi(n) and sigma(n) with nonnegative coefficients: a(n) = Min_{x>=0,y>=0} (x+y) for which x*phi(n) + y*sigma(n) is a multiple of n.

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A134639 Conjectured number of numbers k such that sigma(k)/k = n.

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A317681 a(n) = smallest m such that sigma(m) = n*m/2.

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A335267 Composite numbers whose harmonic mean of their divisors that are larger than 1 is an integer.

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A336745 Numbers m that divide the product phi(m) * sigma(m) * tau(m), where phi is the Euler totient function (A000010), sigma is the sum of divisors function (A000203) and tau is the number of divisors function (A000005).

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A113286 Numbers k such that S(S(k))=k, with S(n)=sigma(n)/4: 1/4-sociable numbers of order 1 or 2.

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A094701 Smallest linear combination of phi(n) and sigma(n) with nonnegative coefficients: a(n) = Min_{x>=0,y>=0} (x+y) for which xphi(n) + ysigma(n) is a multiple of n.