A051377 -id:A051377 - 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

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

A328134 Exponential highly abundant numbers: numbers m such that esigma(m) > esigma(k) for all k < m, where esigma(m) is the sum of exponential divisors of m (A051377).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 12, 16, 18, 20, 28, 36, 52, 60, 68, 72, 84, 92, 100, 124, 132, 140, 144, 180, 244, 252, 300, 324, 360, 396, 468, 588, 612, 684, 828, 900, 1116, 1260, 1332, 1476, 1548, 1692, 1764, 2124, 2196, 2340, 2412, 2556, 2628, 2700, 2772, 2844, 2988

Offset: 1

Amiram Eldar, Oct 04 2019

nonn

The exponential version of A002093.

			The first 10 values of esigma(k) for k = 1 to 10 are 1, 2, 3, 6, 5, 6, 7, 10, 12, 10. The record values are reached for 1, 2, 3, 4, 7, 8, 9.

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

Cf. A002093, A051377, A285614 (unitary), A292983 (bi-unitary), A327634 (infinitary).

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; s = {}; em = 0; Do[e = esigma[n]; If[e > em, em = e; AppendTo[s, n]], {n, 1, 3000}]; s

A307042 Partial sums of the exponential divisors sum function: Sum_{k=1..n} esigma(k), where esigma is A051377.

Original entry on oeis.org

1, 3, 6, 12, 17, 23, 30, 40, 52, 62, 73, 91, 104, 118, 133, 155, 172, 196, 215, 245, 266, 288, 311, 341, 371, 397, 427, 469, 498, 528, 559, 593, 626, 660, 695, 767, 804, 842, 881, 931, 972, 1014, 1057, 1123, 1183, 1229, 1276, 1342, 1398, 1458, 1509, 1587, 1640

Offset: 1

Amiram Eldar, Mar 21 2019

nonn

Amiram Eldar, 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.

Cf. A024916, A051377, A064609, A275480.

esigma[n_] := Times @@ (Sum[ First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; Accumulate[Array[esigma, 60]] (* after Jean-François Alcover at A051377 *)

a(n) ~ B * n^2, where B = 0.5682854937... (A275480).

A328135 Exponential 3-abundant numbers: numbers m such that esigma(m) >= 3m, where esigma(m) is the sum of exponential divisors of m (A051377).

Original entry on oeis.org

901800900, 1542132900, 1926332100, 2153888100, 2690496900, 2822796900, 3942584100, 4487660100, 4600908900, 5127992100, 6267888900, 6742052100, 7162236900, 7305120900, 8421732900, 8969984100, 9866448900, 10203020100, 10718460900, 11723411700, 11787444900, 12528324900

Offset: 1

Amiram Eldar, Oct 04 2019

nonn

Aiello et al. found bounds on e-multiperfect numbers, i.e., numbers m such that esigma(m) = k * m for k > 2: 2 * 10^7 for k = 3, and 10^85, 10^320, and 10^1210 for k = 4, 5, and 6. The data of this sequence raise the bound for exponential 3-perfect numbers to 3 * 10^10.
The least odd term is (59#/2)^2 = 924251841031287598942273821762233522616225. The least term which is coprime to 6 is (239#/6)^2 = 3.135... * 10^190.
The least exponential 4-abundant number (esigma(m) >= 4m) is (31#)^2 = 40224510201185827416900. In general, the least exponential k-abundant number (esigma(m) >= k*m), for k > 2, is (A002110(A072986(k)))^2.
The asymptotic density of this sequence is Sum_{n>=1} f(A383699(n)) = 1.325...*10^(-9), where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - Amiram Eldar, May 06 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
W. Aiello, G. E. Hardy, and M. V. Subbarao, On the existence of e-multiperfect numbers, Fibonacci Quarterly, Vol. 25, No. 1 (1987), pp. 65-71.

Subsequence of A129575.
A383699 is a subsequence.
Cf. A002110, A051377, A072986, A321147.
Cf. A023197, A307112, A285615 (unitary), A293187 (bi-unitary), A300664 (infinitary).

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^10], esigma[#] >= 3 # &]

A307043 Numbers n such that A307042(n) = Sum_{k=1..n} esigma(k) is divisible by n, where esigma(k) is sum of exponential divisors of k (A051377).

Original entry on oeis.org

1, 3, 4, 8, 13, 78, 94, 481, 511, 4819, 13557, 23083, 84245, 204744, 562243, 591105, 614339, 617675, 656263, 1545716, 6370802, 34882737, 534034248, 601990019, 1153304776, 2064184733, 3570196547, 10572032882

Offset: 1

Amiram Eldar, Mar 21 2019

The exponential version of A056550.

The corresponding quotients are 1, 2, 3, 5, 8, 45, ... (see the link for more values).

			3 is in the sequence since esigma(1) + esigma(2) + esigma(3) = 1 + 2 + 3 = 6 is divisible by 3.

Amiram Eldar, Table of n, a(n), A307042(a(n))/a(n) for n=1..28.

Cf. A051377, A056550, A064611, A307042.

esigma[n_] := Times @@ (Sum[ First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; seq={};s = 0; Do[s = s + esigma [n]; If[Divisible[s,n], AppendTo[seq,n]], {n, 1, 10^6}]; seq (* after Jean-François Alcover at A051377 *)

A322887 Decimal expansion of the asymptotic mean value of the exponential abundancy index A051377(k)/k.

Original entry on oeis.org

1, 1, 3, 6, 5, 7, 0, 9, 8, 7, 4, 9, 3, 6, 1, 3, 9, 0, 8, 6, 5, 2, 0, 7, 3, 1, 5, 2, 3, 8, 3, 8, 3, 2, 5, 9, 3, 4, 4, 8, 8, 0, 9, 0, 1, 8, 6, 3, 9, 5, 7, 2, 7, 6, 7, 8, 9, 0, 5, 2, 6, 5, 4, 4, 3, 1, 6, 2, 3, 9, 7, 2, 0, 3, 1, 5, 1, 5, 2, 8, 8, 3, 6, 8, 7, 6, 1, 3, 9, 2, 7, 2, 7, 4, 8, 9, 8, 5, 5, 2, 6, 2, 1, 9, 2

Offset: 1

Amiram Eldar, Dec 29 2018

			1.13657098749361390865207315238383259344880901863957...

Peter Hagis, Jr., Some results concerning exponential divisors, International Journal of Mathematics and Mathematical Sciences, Vol. 11, No. 2, (1988), pp. 343-349.

Cf. A013661 (all divisors), A051377.

default(realprecision, 120); default(parisize, 2000000000);
my(kmax = 135); prodeulerrat(1 + (1 - 1/p) * sum(k = 1, kmax, 1/(p^(3*k)-p^k))) \\ Amiram Eldar, Mar 09 2024 (The calculation takes a few minutes.)

Equals lim_{n->oo} (1/n) * Sum_{k=1..n} esigma(k)/k, where esigma(k) is the sum of exponential divisors of k (A051377).

Equals Product_{p prime} (1 + (1 - 1/p) * Sum_{k>=1} 1/(p^(3*k)-p^k)).

a(7)-a(22) from Jon E. Schoenfield, Dec 30 2018

More terms from Amiram Eldar, Mar 09 2024

A357014 Numbers whose sum of exponential divisors (A051377) is odd.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 29, 31, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 127, 129, 131, 133, 137, 139, 141, 143, 145, 149

Offset: 1

Amiram Eldar, Sep 09 2022

nonn

Includes all the odd squarefree numbers (A056911). First differs from this sequence at n = 34.
Equivalently, the odd terms of A197680, i.e., odd numbers with an odd number of exponential divisors (A049419).
The asymptotic density of this sequence is 0.409797... (A357017).

			1 is a term since A051377(1) = 1 is odd.
3 is a term since A051377(3) = 3 is odd.

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

Cf. A049419, A051377, A357017.
Subsequence of A197680.
Subsequences: A056911, A357015.
Similar sequences: A000079 (numbers with an odd sum of unitary divisors), A028982 (numbers with an odd sum of divisors).

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[150], OddQ[esigma[#]] &]

A348961 Exponential harmonic (or e-harmonic) numbers of type 1: numbers k such that esigma(k) | k * d_e(k), where d_e(k) is the number of exponential divisors of k (A049419) and esigma(k) is their sum (A051377).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105

Offset: 1

Amiram Eldar, Nov 05 2021

nonn

First differs from A005117 at n = 24, from A333634 and A348499 at n = 47, and from A336223 at n = 63.
Sándor (2006) proved that all the squarefree numbers are e-harmonic of type 1, and that an e-perfect number (A054979) is a term of this sequence if and only if at least one of the exponents in its prime factorization is not a perfect square.
Since all the e-perfect numbers are products of a primitive e-perfect number (A054980) and a coprime squarefree number, and all the known primitive e-perfect numbers have a nonsquare exponent in their prime factorizations, there is no known e-perfect number that is not in this sequence.

			3 is a term since esigma(3) = 3, 3 * d_e(3) = 3 * 1, so esigma(3) | 3 * d_e(3).

Amiram Eldar, Table of n, a(n) for n = 1..10000
József Sándor, On exponentially harmonic numbers, Scientia Magna, Vol. 2, No. 3 (2006), pp. 44-47.
József Sándor, Selected Chapters of Geomety, Analysis and Number Theory, 2005, pp. 141-145.

A005117 and A348962 are subsequences.

Cf. A049419, A051377, A322791, A333634, A336223, A348499, A348964.

f[p_, e_] := p^e * DivisorSigma[0, e] / DivisorSum[e, p^# &]; ehQ[1] = True; ehQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], ehQ]

A328120 Exponential superperfect numbers (or e-superperfect numbers): numbers m such that esigma(esigma(m)) = 2m, where esigma(m) is the sum of exponential divisors of m (A051377).

Original entry on oeis.org

9, 12, 45, 60, 63, 84, 99, 117, 132, 153, 156, 171, 204, 207, 228, 261, 270, 276, 279, 315, 333, 348, 369, 372, 387, 420, 423, 444, 477, 492, 495, 516, 531, 549, 564, 585, 603, 636, 639, 657, 660, 693, 708, 711, 732, 747, 765, 780, 801, 804, 819, 852, 855, 873

Offset: 1

Amiram Eldar, Oct 04 2019

nonn

Hanumanthachari et al. proved that:
1) The only e-superperfect number of the form p^q with p and q primes is 9 = 3^2.
2) If p prime, m squarefree coprime to m with gcd(p+1, m) > 1 then p^2 * m is e-superperfect only if p = 2.
3) If k is squarefree coprime to esigma(m) then m*k is e-superperfect if and only if m is e-superperfect.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 53.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. Hanumanthachari, V. V. Subrahmanya Sastri, and V. Srinivasan, On e-perfect numbers, Math. Student, Vol. 46, No. 1 (1978), pp. 71-80; entire issue.

The exponential version of A019279.

Cf. A038843, A051377.

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; espQ[n_] := esigma[esigma[n]] == 2n; Select[Range[1000], espQ]

esigma(n) = {my(f = factor(n)); prod(k = 1, #f~, sumdiv(f[k, 2], d, f[k, 1]^d));}
isok(k) = esigma(esigma(k)) == 2*k; \\ Amiram Eldar, Jan 09 2025

9 is in the sequence since esigma(9) = 12 and esigma(12) = 18 = 2*9.

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

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A328134 Exponential highly abundant numbers: numbers m such that esigma(m) > esigma(k) for all k < m, where esigma(m) is the sum of exponential divisors of m (A051377).

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A307042 Partial sums of the exponential divisors sum function: Sum_{k=1..n} esigma(k), where esigma is A051377.

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A328135 Exponential 3-abundant numbers: numbers m such that esigma(m) >= 3m, where esigma(m) is the sum of exponential divisors of m (A051377).

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A307043 Numbers n such that A307042(n) = Sum_{k=1..n} esigma(k) is divisible by n, where esigma(k) is sum of exponential divisors of k (A051377).

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A322887 Decimal expansion of the asymptotic mean value of the exponential abundancy index A051377(k)/k.

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A357014 Numbers whose sum of exponential divisors (A051377) is odd.

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