A054980 -id:A054980 - cp's OEIS frontend

A054979 e-perfect numbers: numbers k such that the sum of the e-divisors (exponential divisors) of k equals 2*k.

36, 180, 252, 396, 468, 612, 684, 828, 1044, 1116, 1260, 1332, 1476, 1548, 1692, 1800, 1908, 1980, 2124, 2196, 2340, 2412, 2556, 2628, 2700, 2772, 2844, 2988, 3060, 3204, 3276, 3420, 3492, 3636, 3708, 3852, 3924, 4068, 4140, 4284, 4572, 4716

Offset: 1

Jud McCranie, May 29 2000

nonn

The e-divisors (or exponential divisors) of x=Product p(i)^r(i) are all numbers of the form Product p(i)^s(i) where s(i) divides r(i) for all i.
The number of e-divisors for n is A049419(n). - Jon Perry, Nov 13 2012
Conjecture: Every e-perfect number is divisible by 36, see A219016. - Jon Perry, Nov 13 2012

			The e-divisors of 36 are 2*3, 4*3, 2*9 and 4*9 and the sum of these = 2*36, so 36 is e-perfect.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B17, pp. 110-111.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 116-117.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
E. G. Straus and M. V. Subbarao, On exponential divisors, Duke Math. J., Vol. 41 (1974), pp. 465-471.
Eric Weisstein's World of Mathematics, e-Perfect Number.

Cf. A051377, A054980, A219016.

for n from 1 do
    if A051377(n) = 2*n then
        printf("%d,\n",n) ;
    end if;
end do: # R. J. Mathar, Oct 05 2017

ee[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; Select[Range[5000], ee[#] == 2 # &] (* T. D. Noe, Nov 14 2012 *)

is(n)=my(f=factor(n));prod(i=1,#f[,1],sumdiv(f[i,2],d, f[i,1]^d))==2*n \\ Charles R Greathouse IV, Nov 22 2011

{n: A051377(n) = 2*n}. - R. J. Mathar, Oct 05 2017

A307958 Coreful perfect numbers: numbers k such that csigma(k) = 2*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

Original entry on oeis.org

36, 180, 252, 392, 396, 468, 612, 684, 828, 1044, 1116, 1176, 1260, 1332, 1476, 1548, 1692, 1908, 1960, 1980, 2124, 2196, 2340, 2412, 2556, 2628, 2772, 2844, 2988, 3060, 3204, 3276, 3420, 3492, 3636, 3708, 3852, 3924, 4068, 4140, 4284, 4312, 4572, 4716, 4788

Offset: 1

Amiram Eldar, May 08 2019

nonn

Hardy and Subbarao defined a coreful divisor d of a number k as a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947). The number of these divisors is A005361(k) and their sum is csigma(k) = A057723(k). Since csigma(k) is multiplicative and csigma(p) = p for prime p, then if k is coreful perfect number, then also m*k is, for any squarefree number m coprime to k, gcd(m, k) = 1. Thus there are infinitely many coreful perfect numbers, and all of them can be generated from the sequence of primitive coreful perfect numbers (A307959), which is the subsequence of powerful terms of this sequence. This sequence and A307959 are analogous to e-perfect numbers (A054979) and primitive e-perfect numbers (A054980).

			36 is in the sequence since its coreful divisors are 6, 12, 18, 36, whose sum is 72 = 2 * 36.

Amiram Eldar, Table of n, a(n) for n = 1..10000
G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)

Cf. A005361, A007947, A054979, A054980, A057723, A307959, A307960, A307961.

f[p_,e_] := (p^(e+1)-1)/(p-1)-1; a[1]=1; a[n_] := Times @@ (f @@@ FactorInteger[n]); s={}; Do[If[a[n] == 2n, AppendTo[s,n]], {n, 1, 10^6}]; s

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
s(n) = rad(n)*sigma(n/rad(n)); \\ A057723
isok(n) = s(n) == 2*n; \\ Michel Marcus, May 14 2019

A129575 Exponential abundant numbers: integers k for which A126164(k) > k, or equivalently for which A051377(k) > 2k.

Original entry on oeis.org

900, 1764, 3600, 4356, 4500, 4900, 6084, 6300, 7056, 8100, 8820, 9900, 10404, 11700, 12348, 12996, 14700, 15300, 17100, 19044, 19404, 20700, 21780, 22500, 22932, 25200, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 35280, 36900, 38700, 39600

Offset: 1

Ant King, Apr 28 2007

There are only 52189 exponential abundant numbers less than 50 million, which suggests that these account for approximately 0.1% of all integers.
Includes 36*m for all m coprime to 6 that are not squarefree. - Robert Israel, Feb 19 2019
The asymptotic density of this sequence is Sum_{n>=1} f(A328136(n)) = 0.001043673..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - Amiram Eldar, Sep 02 2022

			The third integer that is exceeded by its proper exponential divisor sum is 3600. Hence a(3) = 3600.

Robert Israel, Table of n, a(n) for n = 1..10000
Peter Hagis, Jr., Some Results Concerning Exponential Divisors, Internat. J. Math. & Math. Sci., Vol. 11, No. 2, (1988), pp. 343-350.
Eric Weisstein's World of Mathematics, e-Divisor.

Cf. A126164, A051377, A049419, A054979, A054980, A328136.

filter:= proc(n)  local L,m,i,j;
  L:= ifactors(n)[2];
  m:= nops(L);
  mul(add(L[i][1]^j, j=numtheory:-divisors(L[i][2])),i=1..m)>2*n
end proc:
select(filter, [$1..10^5]); # Robert Israel, Feb 19 2019

ExponentialDivisors[1]={1};ExponentialDivisors[n_]:=Module[{}, {pr,pows}=Transpose@FactorInteger[n];divpowers=Distribute[Divisors[pows],List];Sort[Times@@(pr^Transpose[divpowers])]];properexponentialdivisorsum[k_]:=Plus@@ExponentialDivisors[k]-k;Select[Range[5 10^4],properexponentialdivisorsum[ # ]># &]
f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[40000], esigma[#] > 2*# &] (* Amiram Eldar, May 06 2025 *)

isok(k) = {my(f = factor(k)); prod(i = 1, #f~, sumdiv(f[i, 2], d, f[i, 1]^d)) > 2*k;} \\ Amiram Eldar, May 06 2025

A126164 Sum of the proper exponential divisors of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, 6, 0, 0, 0, 6, 0, 6, 0, 10, 0, 0, 0, 6, 5, 0, 3, 14, 0, 0, 0, 2, 0, 0, 0, 36, 0, 0, 0, 10, 0, 0, 0, 22, 15, 0, 0, 18, 7, 10, 0, 26, 0, 6, 0, 14, 0, 0, 0, 30, 0, 0, 21, 14, 0, 0, 0, 34, 0, 0, 0, 48, 0, 0, 15

Offset: 1

Ant King, Dec 21 2006

The e-divisors (or exponential divisors) of x=Product p(i)^r(i) are all numbers of the form Product p(i)^s(i) where s(i) divides r(i) for all i.

			The exponential divisors of 240 are 30, 60 and 240, so a(240) = 30+60 = 90.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Ant King, Mathematica programs for A126164 - A126166.
Eric Weisstein's World of Mathematics, e-Divisor.

Cf. A051377, A049419, A054979, A054980, A126168.

f[p_, e_] := DivisorSum[e, p^# &]; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* Amiram Eldar, Aug 13 2023 *)

A051377(n) = { my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2], d, f[i, 1]^d)); }; \\ This function from Charles R Greathouse IV, Nov 22 2011
A126164(n) = (A051377(n) - n); \\ Antti Karttunen, Oct 04 2017, after the given formula

a(n) = esigma(n) - n = A051377(n) - n.

A126166 Larger member of each exponential amicable pair.

Original entry on oeis.org

100548, 502740, 968436, 1106028, 1307124, 1709316, 2312604, 2915892, 3116988, 3720276, 4122468, 4323564, 4725756, 5027400, 4842180, 5329044, 5530140, 5932332, 6133428, 6535620, 6736716, 7138908, 7340004, 7943292, 8345484, 8546580, 8948772, 9753156, 10155348

Offset: 1

Ant King, Dec 21 2006

nonn

This sequence includes the largest member of all exponential amicable pairs and does not discriminate between primitive and nonprimitive pairs.

			a(3)= 968436 because (937692,968436) is the third exponential amicable pair

Hagis, Peter Jr.; Some Results Concerning Exponential Divisors, Internat. J. Math. & Math. Sci., Vol. 11, No. 2, (1988), pp. 343-350.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, Jr., Some results concerning exponential divisors, International Journal of Mathematics and Mathematical Sciences, Vol. 11, No. 2, (1988), pp. 343-349.
Ant King, Mathematica programs for A126164 - A126166
David Moews, Perfect, amicable and sociable numbers.
J. M. Pedersen, Known exponential amicable pairs.

Cf. A126165, A051377, A054979, A049419, A054980, A126164.

fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; s = {}; Do[m = esigma[n] - n; If[m > n && esigma[m] - m == n, AppendTo[s, m]], {n, 1, 10^7}]; s (* Amiram Eldar, May 09 2019 *)

The values of n for which esigma(m)=esigma(n)=m+n and mA051377

Link corrected and reference added by Andrew Lelechenko, Dec 04 2011

More terms from Amiram Eldar, May 09 2019

A126165 Smaller member of each exponential amicable pair.

Original entry on oeis.org

90972, 454860, 937692, 1000692, 1182636, 1546524, 2092356, 2638188, 2820132, 3365964, 3729852, 3911796, 4275684, 4548600, 4688460, 4821516, 5003460, 5367348, 5549292, 5913180, 6095124, 6459012, 6640956, 7186788, 7550676, 7732620, 8096508, 8824284, 9188172, 9370116

Offset: 1

Ant King, Dec 21 2006

nonn

This sequence includes the smallest member of all exponential amicable pairs and does not discriminate between primitive and nonprimitive pairs.

			a(3)=937692 because (937692,968436) is the third exponential amicable pair

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, Jr., Some results concerning exponential divisors, International Journal of Mathematics and Mathematical Sciences, Vol. 11, No. 2, (1988), pp. 343-349.
Ant King, Mathematica programs for A126164 - A126166
David Moews, Perfect, amicable and sociable numbers.
J. M. Pedersen, Known exponential amicable pairs.

Cf. A051377, A054979, A049419, A054980, A126164.

fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; s = {}; Do[m = esigma[n] - n; If[m > n && esigma[m] - m == n, AppendTo[s, n]], {n, 1, 10^7}]; s (* Amiram Eldar, May 09 2019 *)

The values of m for which esigma(m)=esigma(n)=m+n and mA051377.

More terms from Amiram Eldar, May 09 2019

A241405 Sum of modified exponential divisors: if n = Product p_i^r_i then me-sigma(x) = Product (sum p_i^s_i such that s_i+1 divides r_i+1).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 11, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 44, 26, 42, 31, 40, 30, 72, 32, 39, 48, 54, 48, 50, 38, 60, 56, 66, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 93, 72, 88, 80, 90, 60, 120, 62, 96, 80, 65, 84, 144, 68, 90, 96, 144, 72, 110, 74, 114, 104, 100, 96, 168, 80

Offset: 1

Andrew Lelechenko, May 06 2014

The modified exponential divisors of a number n = product p_i^r_i are all numbers of the form product p_i^s_i such that s_i+1 divides r_i+1 for each i.

The concept of modified exponential divisors simplifies combinatorial problems on the sum of exponential divisors A051377 such as a search of e-perfect numbers. Each primitive e-perfect number A054980 corresponds to a unique me-perfect number of smaller magnitude.

Antti Karttunen, Table of n, a(n) for n = 1..16384
David Moews, Perfect, amicable and sociable numbers.
Index entries for sequences related to sums of divisors.

Cf. A007947, A049419, A051377, A054980, A051378, A379027, A379028.

f[p_, e_] := DivisorSum[e+1, p^(#-1)&]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 03 2023 *)

A241405(n) = {my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1)))}

a(n / A007947(n)) = A051377(n).

Multiplicative with a(p^a) = sum p^b such that b+1 divides a+1.

More terms from Antti Karttunen, Nov 23 2017

Incorrect comment removed by Amiram Eldar, Dec 14 2024

A328136 Primitive exponential abundant numbers: the powerful terms of A129575.

Original entry on oeis.org

900, 1764, 3600, 4356, 4500, 4900, 6084, 7056, 8100, 10404, 12348, 12996, 19044, 22500, 30276, 34596, 44100, 47916, 49284, 60516, 66564, 79092, 79524, 86436, 88200, 101124, 108900, 112500, 125316, 132300, 133956, 152100, 161604, 176400, 176868, 181476, 191844

Offset: 1

Amiram Eldar, Oct 04 2019

nonn

For squarefree numbers k, esigma(k) = k, where esigma is the sum of exponential divisors function (A051377). Thus, if m is a term (esigma(m) > 2m) and k is a squarefree number coprime to m, then esigma(k*m) = esigma(k) * esigma(m) = k * esigma(m) > 2*k*m, so k*m is an exponential abundant number. Therefore the sequence of exponential abundant numbers (A129575) can be generated from this sequence by multiplying with coprime squarefree numbers.

			900 is a term since esigma(900) = 2160 > 2 * 900, and 900 = 2^2 * 3^2 * 5^2 is powerful.
6300 is exponential abundant, since esigma(6300) = 15120 > 2 * 6300, but it is not powerful, 6300 = 2^2 * 3^2 * 5^2 * 7, thus it is not in this sequence. It can be generated as a term of A129575 from 900 by 7 * 900 = 6300, since gcd(7, 900) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, Some results concerning exponential divisors, International Journal of Mathematics and Mathematical Sciences, Vol. 11, No. 2 (1988), pp. 343-349.
E. G. Straus and M. V. Subbarao, On exponential divisors, Duke Mathematical Journal, Vol. 41, No. 2 (1974), pp. 465-471.

Intersection of A001694 and A129575.

Cf. A051377, A054979, A054980, A126164.

fun[p_, e_] := DivisorSum[e, p^# &];aQ[n_] := Min[(f = FactorInteger[n])[[;;,2]]] > 1 && Times @@ fun @@@ f > 2n; Select[Range[200000], aQ]

A322858 List of e-perfect numbers that are not e-unitary perfect.

Original entry on oeis.org

17424, 87120, 121968, 226512, 296208, 331056, 400752, 505296, 540144, 609840, 644688, 714384, 749232, 818928, 923472, 1028016, 1062864, 1132560, 1167408, 1237104, 1271952, 1306800, 1376496, 1446192, 1481040, 1550736, 1585584, 1655280, 1690128, 1759824

Offset: 1

Amiram Eldar, Dec 29 2018

nonn

The e-unitary perfect numbers are numbers k such that the sum of their exponential unitary divisors (A322857) equals 2k. Apparently most of the e-perfect numbers (A054979) are also e-unitary perfect numbers: the first 150 e-perfect numbers are also the first 150 e-unitary perfect numbers. But A054979(151) = 17424 is not e-unitary perfect.
Minculete and Tóth ask if there is any e-unitary perfect number which is not e-perfect.
The asymptotic density of this sequence is Sum_{n>=1} f(b(n)) = 0.000016169..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)) and b = {17424, 1306800, 54531590400, ...} is the sequence of primitive e-perfect numbers (A054980) that are not e-unitary perfect. - Amiram Eldar, May 06 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
Nicuşor Minculete and László Tóth, Exponential unitary divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 35 (2011), pp. 205-216.

Cf. A051377, A054979, A054980, A322857.

f[p_, e_] := DivisorSum[e, p^# &]; esigma[n_] := Times @@ f @@@ FactorInteger[n]; ePerfectQ[n_] := esigma[n] == 2n; fu[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#]==1 &]; eusigma[n_] := Times @@ fu @@@ FactorInteger[n]; euPerfectQ[n_] := eusigma[n] == 2n; aQ[n_] := ePerfectQ[n] && !euPerfectQ[n]; Select[Range[125000], aQ]

A323757 Modified exponential perfect numbers: numbers k such that A241405(k) = 2*k.

Original entry on oeis.org

6, 60, 90, 264, 3960, 8736, 87360, 131040, 1868160

Offset: 1

Amiram Eldar, Jan 26 2019

Each term of this sequence corresponds to a primitive e-perfect number (A054980, see formula and Andrew Lelechenko's comment in A241405).

Also in the sequence are 1028004440830371164160, 20546724596095746048000, and 146361946186458562560000 (corresponding to the 3 additional terms of A054980 given by Andrew Lelechenko). - Amiram Eldar, Jul 18 2019

A. V. Lelechenko, The Quest for the Generalized Perfect Numbers, Theoretical and Applied Aspects of Cybernetics, Proceedings, The 4th International Scientific Conference of Students and Young Scientists, Kyiv, 2014.
David Moews, Perfect, amicable and sociable numbers.

Cf. A003557, A054980, A241405.

f[p_, e_] := DivisorSum[e+1, p^(#-1)&]; mesigma[1]=1; mesigma[n_] := Times @@ f @@@FactorInteger@n; mePerfectQ[n_] := mesigma[n]==2n; Select[Range[10000], mePerfectQ]

f(n) = {my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1)));} \\ A241405
isok(n) = f(n) == 2*n; \\ Michel Marcus, Jan 30 2019

a(n) = A003557(A054980(n)).

cp's OEIS Frontend

A054979 e-perfect numbers: numbers k such that the sum of the e-divisors (exponential divisors) of k equals 2*k.

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A307958 Coreful perfect numbers: numbers k such that csigma(k) = 2*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

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A129575 Exponential abundant numbers: integers k for which A126164(k) > k, or equivalently for which A051377(k) > 2k.

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A126164 Sum of the proper exponential divisors of n.

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A126166 Larger member of each exponential amicable pair.

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A126165 Smaller member of each exponential amicable pair.

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A241405 Sum of modified exponential divisors: if n = Product p_i^r_i then me-sigma(x) = Product (sum p_i^s_i such that s_i+1 divides r_i+1).

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