A034897 -id:A034897 - cp's OEIS frontend

A034898 n-th term of A034897 is an a(n)-hyperperfect number.

1, 2, 1, 6, 3, 1, 12, 18, 18, 12, 2, 30, 1, 11, 6, 2, 60, 48, 19, 132, 132, 10, 192, 2, 31, 168, 108, 66, 35, 252, 78, 132, 342, 366, 390, 168, 348, 282, 498, 540, 546, 59, 12, 378, 438, 4, 222, 336, 18, 660, 138, 798, 810, 528, 450, 75, 252, 150, 948, 162

Offset: 1

Jud McCranie

nonn

R. K. Guy, Unsolved Problems in Number Theory, Sect. B2.
Roberts, Lure of the Integers, see Integer 28, p. 177.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Judson S. McCranie, A Study of Hyperperfect Numbers, Journal of Integer Sequences, Vol. 3 (2000), Article 00.1.3.
Eric Weisstein's World of Mathematics, Hyperperfect Number.

Cf. A034897.

forcomposite(n=2, 10^8, if(1==Mod(n, sigma(n)-n-1), print1((n-1)/(sigma(n)-n-1)", "))) \\ Hans Loeblich, May 10 2019

More terms from Donovan Johnson, Nov 20 2012

A174226 Partial sums of A034897.

Original entry on oeis.org

6, 27, 55, 356, 681, 1177, 1874, 3207, 5116, 7157, 9290, 13191, 21319, 32012, 48525, 68046, 92647, 119624, 170925, 267286, 397439, 557280, 720481, 897142, 1111415, 1361736, 1637569, 1933910, 2240091, 2629684, 3116561, 3612090, 4154503

Offset: 1

Jonathan Vos Post, Mar 12 2010

nonn

Partial sums of hyperperfect numbers. The subsequence of prime values in this partial sum begins: 21319, 92647, 720481.

			a(x) = 6 + 21 + 28 + 301 + 325 + 496 + 697 + 1333 + 1909 + 2041 + 2133 + 3901 + 8128 + 10693 + 16513 + 19521 + 24601 = 92647 is prime.

Cf. A034897, A034898, A007592, A019279.

a(n) = SUM[i=1..n] A034897(i) = SUM[i=1..n] {m such that k*sigma(m) = (k+1)*m + k - 1 for some positive integer k, and sigma being the divisor function A000005; noting that k=1 gives perfect numbers}.

A019279 Superperfect numbers: numbers k such that sigma(sigma(k)) = 2*k where sigma is the sum-of-divisors function (A000203).

Original entry on oeis.org

2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, 81129638414606681695789005144064, 85070591730234615865843651857942052864

Offset: 1

N. J. A. Sloane

Let sigma_m(n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives (2,2)-perfect numbers.
Even values of these are 2^(p-1) where 2^p-1 is a Mersenne prime (A000043 and A000668). No odd superperfect numbers are known. Hunsucker and Pomerance checked that there are no odd ones below 7 * 10^24. - Jud McCranie, Jun 01 2000
The number of divisors of a(n) is equal to A000043(n), if there are no odd superperfect numbers. - Omar E. Pol, Feb 29 2008
The sum of divisors of a(n) is the n-th Mersenne prime A000668(n), provided that there are no odd superperfect numbers. - Omar E. Pol, Mar 11 2008
Largest proper divisor of A072868(n) if there are no odd superperfect numbers. - Omar E. Pol, Apr 25 2008
This sequence is a divisibility sequence if there are no odd superperfect numbers. - Charles R Greathouse IV, Mar 14 2012
For n>1, sigma(sigma(a(n))) + phi(phi(a(n))) = (9/4)*a(n). - Farideh Firoozbakht, Mar 02 2015
The term "super perfect number" was coined by Suryanarayana (1969). He and Kanold (1969) gave the general form of even superperfect numbers. - Amiram Eldar, Mar 08 2021

			sigma(sigma(4))=2*4, so 4 is in the sequence.

Dieter Bode, Über eine Verallgemeinerung der vollkommenen Zahlen, Dissertation, Braunschweig, 1971.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B9, pp. 99-100.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, pp. 110-111.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, pp. 38-42.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

P. Bundschuh, Aufgabe 601, Elem. Math., Vol. 24 (1969), p. 69; alternative link.
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
G. G. Dandapat, J. L. Hunsucker, and Carl Pomerance, Some new results on odd perfect numbers, Pacific J. Math. Volume 57, Number 2 (1975), 359-364.
A. Hoque and H. Kalita, Generalized perfect numbers connected with arithmetic functions, Math. Sci. Lett. 3, No. 3, 249-253 (2014).
J. L. Hunsucker and Carl Pomerance, There are no odd superperfect number less than 7*10^24, Indian J. Math., Vol. 17 (1975), pp. 107-120.
H.-J. Kanold, Über "Super perfect numbers", Elem. Math., Vol. 24 (1969), pp. 61-62; alternative link.
Graham Lord, Even Perfect and Superperfect Numbers, Elem. Math., Vol. 30 (1975), pp. 87-88.
H. G. Niederreiter, Solution of Aufgabe 601, Elem. Math., Vol. 25 (1970), pp. 66-67; alternative link.
Paul Shubhankar, Ten Problems of Number Theory, International Journal of Engineering and Technical Research (IJETR), ISSN: 2321-0869, Volume-1, Issue-9, November 2013.
D. Suryanarayana, Super perfect numbers, Elem. Math., Vol. 24 (1969), pp. 16-17; alternative link.
D. Suryanarayana, There is no superperfect number of the form p^(2*alpha), Elem. Math., Vol. 28 (1973), pp. 148-150; alternative link.
László Tóth, The alternating sum-of-divisors function, 9th Joint Conf. on Math. and Comp. Sci., February 9-12, 2012, Siófok, Hungary.
László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.
Eric Weisstein's World of Mathematics, Superperfect Number.
Wikipedia, Superperfect number.
Tomohiro Yamada, On finiteness of odd superperfect numbers, Journal de Théorie des Nombres de Bordeaux, Vol. 32, No. 1 (2020), pp. 259-274.

Cf. A019280, A000203, A000396, A000668, A000043, A034897, A061652, A032742, A072868.

sigma = DivisorSigma[1, #]&;
For[n = 2, True, n++, If[sigma[sigma[n]] == 2 n, Print[n]]] (* Jean-François Alcover, Sep 11 2018 *)

is(n)=sigma(sigma(n))==2*n \\ Charles R Greathouse IV, Nov 20 2012

from itertools import count, islice
def A019279_gen(): # generator of terms
    return (n for n in count(1) if divisor_sigma(divisor_sigma(n)) == 2*n)
A019279_list = list(islice(A019279_gen(),6)) # Chai Wah Wu, Feb 18 2022

a(n) = (1 + A000668(n))/2, if there are no odd superperfect numbers. - Omar E. Pol, Mar 11 2008
Also, if there are no odd superperfect numbers then a(n) = 2^A000043(n)/2 = A072868(n)/2 = A032742(A072868(n)). - Omar E. Pol, Apr 25 2008
a(n) = 2^A090748(n), if there are no odd superperfect numbers. - Ivan N. Ianakiev, Sep 04 2013

a(8)-a(9) from Jud McCranie, Jun 01 2000

Corrected by Michel Marcus, Oct 28 2017

A007592 Hyperperfect numbers: k = m*(sigma(k) - k - 1) + 1 for some m > 1.

Original entry on oeis.org

21, 301, 325, 697, 1333, 1909, 2041, 2133, 3901, 10693, 16513, 19521, 24601, 26977, 51301, 96361, 130153, 159841, 163201, 176661, 214273, 250321, 275833, 296341, 306181, 389593, 486877, 495529, 542413, 808861, 1005421, 1005649, 1055833, 1063141, 1232053

Offset: 1

N. J. A. Sloane

D. Minoli, Sufficient Forms For Generalized Perfect Numbers, Ann. Fac. Sciences, Univ. Nation. Zaire, Section Mathem; Vol. 4, No. 2, Dec 1978, pp. 277-302.
D. Minoli, New Results For Hyperperfect Numbers, Abstracts American Math. Soc., October 1980, Issue 6, Vol. 1, pp. 561.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 177.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Donovan Johnson, Table of n, a(n) for n = 1..10000
Mariano Garcia, Hyperperfect Numbers with Five and Six Different Prime Factors, Fib. Quart. 42, no. 4, Nov. 2004, pp. 292-294.
J. S. McCranie, A study of hyperperfect numbers, J. Int. Seqs. Vol. 3 (2000) #P00.1.3.
D. Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
Daniel Minoli, W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
Daniel Minoli, Issues In Non-Linear Hyperperfect Numbers, Mathematics of Computation, Vol. 34, No. 150, April 1980, pp. 639-645.
D. Minoli, Structural Issues For Hyperperfect Numbers, Fibonacci Quarterly, Feb. 1981, Vol. 19, No. 1, pp. 6-14.
Herman J. J. te Riele, Hyperperfect numbers with three different prime factors, Math. Comp. 36 (1981), 297-298.
Eric Weisstein's World of Mathematics, Hyperperfect Number.

See A034897 (for m >= 1).

hpnQ[n_]:=Module[{den=DivisorSigma[1,n]-n-1,c},If[den!=0,c=(n-1)/den, c=Pi];IntegerQ[c]&&c>1]; Select[Range[1250000],hpnQ] (* Harvey P. Dale, Aug 11 2012 *)

More terms from Jud McCranie, Oct 15 1997

A059047 Numbers x such that sigma(x)-x divides x-1, other than prime powers.

Original entry on oeis.org

77, 611, 1073, 2033, 5293, 6031, 9983, 13969, 15947, 23489, 25241, 40301, 49901, 50249, 51101, 56759, 65017, 71677, 85079, 97217, 97783, 98099, 99101, 131237, 142091, 160133, 165101, 180767, 189281, 210367, 213053, 228719, 259741, 303239

Offset: 1

Jud McCranie, Dec 18 2000

nonn

Primes and prime powers (A000961) also satisfy this equation. A059046 is the union of A000961 and A059047. These numbers are related to hyperperfect numbers (A034897) in the cited paper by te Riele.

			For x=77, sigma(77)=96, 96-77=19, which divides 77-1.

Donovan Johnson, Table of n, a(n) for n = 1..5263 (terms < 10^11)
H. J. J. te Riele, Rules for constructing hyperperfect numbers, Fibonacci Quarterly, 22(1)1984, 50-60. See equation (3), the set M*.

Cf. A059046, A000203, A000961, A034897.

is(n)=n>1 && !isprimepower(n) && (n-1)%(sigma(n)-n)==0 \\ Charles R Greathouse IV, Oct 21 2015

Offset corrected by Donovan Johnson, Nov 03 2011

A059046 Numbers n such that sigma(n)-n divides n-1.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 77, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211

Offset: 1

Jud McCranie, Dec 18 2000

nonn

Primes and prime powers (A000961) satisfy this equation, but other numbers do also (A059047). This sequence is the union of A000961 and A059047. These are related to hyperperfect numbers (A034897) in the cited paper by te Riele. [Mentions this sequence]

			For x=77, sigma(77)=96, 96-77=19, which divides 77-1.

G. C. Greubel, Table of n, a(n) for n = 1..1000
JRM Antalan, JAB Dris, Some New Results On Even Almost Perfect Numbers Which Are Not Powers Of Two, arXiv preprint arXiv:1602.04248, 2016
H. J. J. te Riele, Rules for constructing hyperperfect numbers, Fibonacci Quarterly, 22(1)1984, 50-60. See equation (3), the set M*.

Cf. A059047, A000203, A000961, A034897.

[n : n in [2..1000] | (n-1) mod (SumOfDivisors(n)-n) eq 0 ]; /* N. J. A. Sloane, Dec 23 2006 */

Select[Range[2,250],Divisible[#-1,DivisorSigma[1,#]-#]&] (* Harvey P. Dale, Jan 18 2011 *)

is(n)=n>1 && (n-1)%(sigma(n)-n)==0 \\ Charles R Greathouse IV, Oct 21 2015

A225150 Unitary hyperperfect numbers.

Original entry on oeis.org

6, 21, 40, 52, 60, 90, 288, 301, 657, 697, 1333, 1909, 2041, 2176, 3856, 3901, 5536, 6517, 15025, 24601, 26977, 30105, 87360, 96361, 105301, 130153, 163201, 250321, 275833, 296341, 389593, 486877, 495529, 524961, 542413, 808861, 1005421, 1005649, 1055833

Offset: 1

Michel Lagneau, Apr 30 2013

nonn

A k-unitary hyperperfect number is an integer n for which the equality n = 1 + k(usigma(n) - n - 1) holds, where usigma(n) is the sum of all positive unitary divisors of n for some integer k. (See the definition of the k-hyperperfect number in the links, and the sequence A034897.)
A squarefree number is hyperperfect if, and only if this number is a unitary hyperperfect number.
In this sequence, the corresponding k are 1, 2, 3, 3, 1, 1, 7, 6, 8, 12, 18, 18, 12, 15, 15, 30, 27, 18, 24, 60, 48, 4, ...
Peter Hagis, Jr. calculated all the unitary hyperperfect numbers below 10^6. - Amiram Eldar, Aug 24 2018

			21 is in the sequence because 1 + k(usigma(21) - 21 - 1) = 1 + 2(32 - 21 - 1) = 21 where k = 2 and usigma(21) = A034448 (21) = 32.

J. M. De Koninck, Ces nombres qui nous fascinent, Ellipses 2008, Entry 288 p. 74.

Donovan Johnson, Table of n, a(n) for n = 1..1000
Peter Hagis, Jr., Unitary Hyperperfect Numbers, Mathematics of Computation, Vol. 36, No. 153 (1981), pp. 299-301.
Eric Weisstein's World of Mathematics, Hyperperfect Number
Wikipedia, Hyperperfect number

Cf. A034897, A034448, A034444.

with(numtheory) :for n from 1 to 100000 do :it:=1:x:=divisors(n):n1:=nops(x):s:=1:for i from 2 to n1 do:d:=x[i]:if gcd(d,n/d)=1 then s:=s+d:else fi:od: ii:=0:for k from 1 to 2000 while (ii=0) do:z:=1+k*(s-n-1):if z=n then ii:=1:printf(`%d, `,n):else fi:od: od:

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; hpnQ[n_]:=Module[{c= usigma[n]-n-1}, c>0&&IntegerQ[(n-1)/c]]; Select[Range[2, 1100000], hpnQ]

A133447 Nonsemiprime hyperperfect numbers.

Original entry on oeis.org

325, 2133, 10693, 16513, 19521, 51301, 159841, 176661, 214273, 306181, 1433701, 1570153, 1950625, 2469601, 2924101, 5199013, 9398593, 10445221, 15407173, 23548753, 28600321, 39147301, 60110701, 62722153, 88347781, 112803841

Offset: 1

Jonathan Vos Post, Dec 22 2007

nonn

The other 25 of the first 35 values of A007592 are all semiprimes A001358.

This sequence excludes perfect numbers (A000396), which are 1-hyperperfect numbers. - Jud McCranie, Mar 23 2025.

			a(1) = 325 = 5^2 * 13.
a(2) = 2133 = 3^3 * 79.
a(3) = 10693 = 17^2 * 37.
a(4) = 16513 = 7^2 * 337.
a(5) = 19521 = 3^4 * 341.
a(6) = 51301 = 29^2 * 61.
a(7) = 159841 = 11^2 * 1321.
a(8) = 176661 = 3^5 * 727.
a(9) = 214273 = 47^2 * 97.
a(10) = 306181 = 53^2 * 109.
a(12) = 1570153 = 13 * 269 * 449. - _Jud McCranie_, Mar 23 2025

Jud McCranie, Table of n, a(n) for n = 1..1167 (terms < 10^15). Earlier terms from Jonathan Vos Post and R. J. Mathar, and a b-file with 192 terms (to 10^12) by Donovan Johnson.

Cf. A001358, A007592, A100959.

a034897 := [] : fd := fopen("b034897.txt",READ) : bf := fscanf(fd,"%d %d") : while nops(bf) <> 0 do a034897 := [op(a034897), op(2,bf) ] ; bf := fscanf(fd,"%d %d") ; od: a007592 := [] : for n in a034897 do m := (n-1)/( numtheory[sigma](n)-n-1) ; if m > 1 then a007592 := [op(a007592),n] ; fi ; od: isA100959 := proc(n) if numtheory[bigomega](n) <> 2 then true ; else false ; end: end: for n in a007592 do if isA100959(n) then printf("%d, ",n) ; fi ; od: # R. J. Mathar, Jan 08 2008

A100959 INTERSECTION A007592.

More terms from R. J. Mathar, Jan 08 2008

A309568 Bi-unitary k-hyperperfect numbers: numbers m such that m = 1 + k * (bsigma(m) - m - 1) where bsigma(m) is the sum of bi-unitary divisors of m (A188999) and k >= 1 is an integer.

Original entry on oeis.org

6, 21, 52, 60, 90, 301, 657, 697, 1333, 1909, 2041, 2133, 3901, 15025, 24601, 26977, 96361, 130153, 163201, 176661, 250321, 275833, 296341, 389593, 486877, 495529, 542413, 808861, 1005421, 1005649, 1055833, 1063141, 1232053, 1246417, 1284121, 1357741, 1403221

Offset: 1

Amiram Eldar, Aug 08 2019

nonn

The bi-unitary version of A034897.
The only bi-unitary 1-hyperperfect numbers are 6, 60, and 90 (the bi-unitary perfect numbers).
The corresponding k values are 1, 2, 3, 1, 1, 6, 8, 12, 18, 18, 12, 2, 30, 24, 60, 48, 132, 132, 192, 2, 168, 108, 66, 252, 78, 132, 342, 366, 390, 168, 348, 282, 498, 552, 540, 30, 546, ...

			21 is in the sequence since bsigma(21) = 32 and 21 = 1 + 2 * (32 - 21 - 1).

Amiram Eldar, Table of n, a(n) for n = 1..1000
József Sándor and Mihály Bencze, On modified hyperperfect numbers, Research report collection, Vol. 8, No. 2 (2005).

Cf. A034897, A188999, A225150.

fun[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); hpnQ[n_] := (c = bsigma[n]-n-1) > 0 && Divisible[n-1, c]; Select[Range[10^5],  hpnQ]

cp's OEIS Frontend

A034898 n-th term of A034897 is an a(n)-hyperperfect number.

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A174226 Partial sums of A034897.

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A019279 Superperfect numbers: numbers k such that sigma(sigma(k)) = 2*k where sigma is the sum-of-divisors function (A000203).

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A007592 Hyperperfect numbers: k = m*(sigma(k) - k - 1) + 1 for some m > 1.

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A059047 Numbers x such that sigma(x)-x divides x-1, other than prime powers.

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A059046 Numbers n such that sigma(n)-n divides n-1.

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Magma

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A225150 Unitary hyperperfect numbers.

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A133447 Nonsemiprime hyperperfect numbers.