A126164 -id:A126164 - cp's OEIS frontend

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

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

A051377 a(1)=1; for n > 1, a(n) = sum of exponential divisors (or e-divisors) of n.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 18, 13, 14, 15, 22, 17, 24, 19, 30, 21, 22, 23, 30, 30, 26, 30, 42, 29, 30, 31, 34, 33, 34, 35, 72, 37, 38, 39, 50, 41, 42, 43, 66, 60, 46, 47, 66, 56, 60, 51, 78, 53, 60, 55, 70, 57, 58, 59, 90, 61, 62, 84, 78, 65, 66, 67, 102, 69, 70, 71

Offset: 1

Yasutoshi Kohmoto

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.

a(n) = n if and only if n is squarefree. - Jon Perry, Nov 13 2012

			a(8)=10 because 2 and 2^3 are e-divisors of 8 and 2+2^3=10.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J. Fabrykowski and M. V. Subbarao, The maximal order and the average order of multiplicative function sigma^(e)(n), in Théorie des nombres/Number theory (Quebec, PQ, 1987), 201-206, de Gruyter, Berlin, 1989.
Nicussor Minculete, Concerning some arithmetic functions which use exponential divisors, Acta Universitatis Apulensis, No. 28/2011, pp. 125-133.
Y.-F. S. Pétermann and J. Wu, On the sum of exponential divisors of an integer, Acta Math. Hungar. 77 (1997), 159-175.
Eric Weisstein's World of Mathematics, e-Divisor

Cf. A051378, A049419 (number of e-divisors).
Cf. A027748, A124010, A027750.
Row sums of A322791.
See A307042 and A275480 where the formula and constant appear.

A051377:=n->Product(List(Collected(Factors(n)), p -> Sum(DivisorsInt(p[2]),d->p[1]^d))); List([1..10^4], n -> A051377(n)); # Muniru A Asiru, Oct 29 2017

a051377 n = product $ zipWith sum_e (a027748_row n) (a124010_row n) where
   sum_e p e = sum [p ^ d | d <- a027750_row e]
-- Reinhard Zumkeller, Mar 13 2012

A051377 := proc(n)
    local a,pe,p,e;
    a := 1;
    for pe in ifactors(n)[2] do
        p := pe[1] ;
        e := pe[2] ;
        add(p^d,d=numtheory[divisors](e)) ;
        a := a*% ;
    end do:
    a ;
end proc:
seq(A051377(n),n=1..100) ; # R. J. Mathar, Oct 05 2017

a[n_] := Times @@ (Sum[ First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, Apr 06 2012 *)

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

ediv(n,f=factor(n))=my(v=List(),D=apply(divisors,f[,2]~),t=#f~); forvec(u=vector(t,i,[1,#D[i]]), listput(v,prod(j=1,t,f[j,1]^D[j][u[j]]))); Set(v)
a(n)=vecsum(ediv(n)) \\ Charles R Greathouse IV, Oct 29 2018

Multiplicative with a(p^e) = Sum_{d|e} p^d. - Vladeta Jovovic, Apr 23 2002
a(n) = A126164(n)+n. - R. J. Mathar, Oct 05 2017
The average order of a(n) is Dn + O(n^e) for any e > 0, due to Fabrykowski & Subbarao, where D is about 0.568. (D >= 0.5 since a(n) >= n.) - Charles R Greathouse IV, Sep 22 2023

More terms from Jud McCranie, May 29 2000

Definition corrected by Jaroslav Krizek, Feb 27 2009

A126168 Sum of the proper infinitary divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 6, 1, 7, 1, 8, 1, 8, 1, 10, 9, 1, 1, 12, 1, 10, 11, 14, 1, 36, 1, 16, 13, 12, 1, 42, 1, 19, 15, 20, 13, 14, 1, 22, 17, 50, 1, 54, 1, 16, 15, 26, 1, 20, 1, 28, 21, 18, 1, 66, 17, 64, 23, 32, 1, 60, 1, 34, 17, 21, 19, 78, 1, 22, 27, 74, 1, 78, 1, 40, 29

Offset: 1

Ant King, Dec 21 2006

A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.

			As the infinitary divisors of 240 are 1, 3, 5, 15, 16, 48, 80, 240, we have a(240) = 1 + 3 + 5 + 15 + 16 + 48 + 80 = 168.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Infinitary Divisor.

Cf. A049417, A037445, A126164.

A049417 := proc(n)
    local a,pe,k,edgs,p ;
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        edgs := convert(op(2,pe),base,2) ;
        for k from 0 to nops(edgs)-1 do
            dk := op(k+1,edgs) ;
            a := a*(p^(2^k*(1+dk))-1)/(p^(2^k)-1) ;
        end do:
    end do:
    a ;
end proc:
A126168 := proc(n)
    A049417(n)-n ;
end proc:
seq(A126168(n),n=1..100) ; # R. J. Mathar, Jul 23 2021

ExponentList[n_Integer, factors_List] := {#, IntegerExponent[n, # ]} & /@ factors; InfinitaryDivisors[1] := {1}; InfinitaryDivisors[n_Integer?Positive] := Module[ { factors = First /@ FactorInteger[n], d = Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f, g}, BitOr[f, g] == g][ #, Last[ # ]]] & /@ Transpose[Last /@ ExponentList[ #, factors] & /@ d]], ?( And @@ # &), {1}]] ]] ] Null; properinfinitarydivisorsum[k] := Plus @@ InfinitaryDivisors[k] - k; properinfinitarydivisorsum /@ Range[75]
f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); a[n_] := isigma[n] - n; Array[a, 100] (* Amiram Eldar, Mar 20 2025 *)

A049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[, 2], b = binary(f[k, 2]); prod(j=1, #b, if(b[j], 1+f[k, 1]^(2^(#b-j)), 1)))} \\ This function from Andrew Lelechenko, Apr 22 2014
A126168(n) = (A049417(n) - n); \\ Antti Karttunen, Oct 04 2017, after the given formula.

a(n) = isigma(n) - n = A049417(n) - n.

A321147 Odd exponential abundant numbers: odd numbers k whose sum of exponential divisors A051377(k) > 2*k.

Original entry on oeis.org

225450225, 385533225, 481583025, 538472025, 672624225, 705699225, 985646025, 1121915025, 1150227225, 1281998025, 1566972225, 1685513025, 1790559225, 1826280225, 2105433225, 2242496025, 2466612225, 2550755025, 2679615225, 2930852925, 2946861225, 3132081225

Offset: 1

Amiram Eldar, Oct 28 2018

nonn

From Amiram Eldar, Jun 08 2020: (Start)
Exponential abundant numbers that are odd are relatively rare: there are 235290 even exponential abundant number smaller than the first odd term, i.e., a(1) = A129575(235291).
Odd exponential abundant numbers k such that k-1 or k+1 is also exponential abundant number exist (e.g. (73#/5#)^2-1 and (73#/5#)^2 are both exponential abundant numbers, where prime(k)# = A002110(k)). Which pair is the least?
The least exponential abundant number that is coprime to 6 is (31#/3#)^2 = 1117347505588495206025. In general, the least exponential abundant number that is coprime to A002110(k) is (A007708(k+1)#/A002110(k))^2. (End)
The asymptotic density of this sequence is Sum_{n>=1} f(A328136(n)) = 5.29...*10^(-9), where f(n) = (4/(Pi^2*n))*Product_{prime p|n}(p/(p+1)) if n is odd and 0 otherwise. - Amiram Eldar, Sep 02 2022

			225450225 is in the sequence since it is odd and A051377(225450225) = 484323840 > 2 * 225450225.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, e-Divisor.
Eric Weisstein's World of Mathematics, e-Perfect Number.

The exponential version of A005231.
The odd subsequence of A129575.
Cf. A002110, A007708, A051377, A126164, A129485, A293186, A328136.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s={};Do[If[esigma[n]>2n,AppendTo[s,n]],{n,1,10^10,2}]; s

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

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]

A318100 Exponential pseudoperfect numbers: numbers n equal to the sum of a subset of their proper exponential divisors.

Original entry on oeis.org

36, 180, 252, 396, 468, 612, 684, 828, 900, 1044, 1116, 1260, 1332, 1476, 1548, 1692, 1764, 1800, 1908, 1980, 2124, 2196, 2340, 2412, 2556, 2628, 2700, 2772, 2844, 2988, 3060, 3204, 3276, 3420, 3492, 3600, 3636, 3708, 3852, 3924, 4068, 4140, 4284, 4356, 4500

Offset: 1

Amiram Eldar, Oct 28 2018

nonn

			900 is in the sequence since its proper exponential divisors are 30, 60, 90, 150, 180, 300, 450 and 900 = 150 + 300 + 450.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, e-Divisor
Eric Weisstein's World of Mathematics, e-Perfect Number

The exponential version of A005835. A054979 is a subsequence.

Cf. A051377, A126164, A129575, A292985, A293188.

dQ[n_,m_] := (n>0&&m>0 &&Divisible[n,m]); expDivQ[n_,d_] := Module[ {ft=FactorInteger[n]}, And@@MapThread[dQ, {ft[[;;,2]], IntegerExponent[ d,ft[[;;,1]]]} ]]; eDivs[n_] := Module[ {d=Rest[Divisors[n]]}, Select[ d,expDivQ[n,#]&] ]; esigma[1]=1; esigma[n_] := Total@eDivs[n]; eDeficientQ[n_] := esigma[n] < 2n; a = {}; n = 0; While[Length[a] < 30, n++; If[eDeficientQ[n], Continue[]]; d = Most[eDivs[n]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c > 0, AppendTo[a, n]]]; a

ediv(n,f=factor(n))=my(v=List(),D=apply(divisors,f[,2]~),t=#f~); forvec(u=vector(t,i,[1,#D[i]]), listput(v,prod(j=1,t,f[j,1]^D[j][u[j]]))); Set(v)
is(n)=my(e=ediv(n)); e=e[1..#e-1]; forsubset(#e, v, if(vecsum(vecextract(e,v))==n, return(1))); 0 \\ Charles R Greathouse IV, Oct 29 2018

A127659 Exponential amicable numbers.

Original entry on oeis.org

90972, 100548, 454860, 502740, 937692, 968436, 1000692, 1106028, 1182636, 1307124, 1546524, 1709316, 2092356, 2312604, 2638188, 2820132, 2915892, 3116988, 3365964, 3720276, 3729852, 3911796, 4122468, 4275684, 4323564, 4548600, 4688460, 4725756, 4821516, 4842180

Offset: 1

Ant King, Jan 25 2007

nonn

Union of A126165 and A126166. The first 10 terms of this sequence are the same as the first 10 terms of A127660.

			a(5)=937692 because the fifth non-e-perfect integer that satisfies A126164(A126164(n))=n is 937692.

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
J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]

Cf. A126164, A126165, A126166, A127656, A127657, A127658, A127660, A051377.

ExponentialDivisors[1]={1};ExponentialDivisors[n_]:=Module[{}, {pr,pows}=Transpose@FactorInteger[n];divpowers=Distribute[Divisors[pows],List];Sort[Times@@(pr^Transpose[divpowers])]];se[n_]:=Plus@@ExponentialDivisors[n]-n;g[n_] := If[n > 0, se[n], 0];eTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];ExponentialAmicableNumberQ[k_]:=If[Nest[se,k,2]==k && !se[k]==k,True,False];Select[Range[5 10^6],ExponentialAmicableNumberQ[ # ] &]
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, 10^6}]; s (* Amiram Eldar, May 09 2019 *)

Non-e-perfect numbers for which A126164(A126164(n))=n.

Link corrected by Andrew Lelechenko, Dec 04 2011

More terms from Amiram Eldar, May 09 2019

A331970 The sum of the proper bi-unitary divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 6, 1, 7, 1, 8, 1, 8, 1, 10, 9, 11, 1, 12, 1, 10, 11, 14, 1, 36, 1, 16, 13, 12, 1, 42, 1, 31, 15, 20, 13, 14, 1, 22, 17, 50, 1, 54, 1, 16, 15, 26, 1, 60, 1, 28, 21, 18, 1, 66, 17, 64, 23, 32, 1, 60, 1, 34, 17, 55, 19, 78, 1, 22, 27, 74, 1, 78, 1

Offset: 1

Amiram Eldar, Feb 03 2020

nonn

First differs from A126168 at n = 16.

			a(6) = 6 since A188999(6) - 6 = 12 - 6 = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001065, A034460, A126164, A126168, A188999, A222266.

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_] := bsigma[n] = Times @@ (fun @@@ FactorInteger[n]); bs[n_] := bsigma[n] - n; Array[bs, 100]

a(n) = A188999(n) - n.

cp's OEIS Frontend

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

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A051377 a(1)=1; for n > 1, a(n) = sum of exponential divisors (or e-divisors) of n.

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A126168 Sum of the proper infinitary divisors of n.

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A321147 Odd exponential abundant numbers: odd numbers k whose sum of exponential divisors A051377(k) > 2*k.

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

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