A051377 -id:A051377 - cp's OEIS frontend

A336680 Exponential admirable numbers: numbers k such that there is a proper exponential divisor d of k such that esigma(k) - 2d = 2k, where esigma is the sum of exponential divisors function (A051377).

900, 1764, 4356, 4500, 4900, 6084, 6300, 7056, 8820, 9900, 10404, 11700, 12348, 12996, 14700, 15300, 17100, 19044, 19404, 20700, 21780, 22932, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 35280, 36900, 38700, 40572, 42300, 42588, 47700

Offset: 1

Amiram Eldar, Jul 30 2020

nonn

Equivalently, numbers that are equal to the sum of their proper exponential divisors, with one of them taken with a minus sign.

			900 is a term since 900 = 30 + 60 + 90 + 150 - 180 + 300 + 450 is the sum of its proper exponential divisors with one of them, 180, taken with a minus sign.

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

The exponential version of A111592.
Subsequence of A129575.
Similar sequences: A328328, A334972, A334974.
Cf. A051377, A322791.

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]]]}]]; esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; expAdmQ[n_] := (ab = esigma[n] - 2*n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && expDivQ[n, ab/2]; Select[Range[50000], expAdmQ]

A357015 Nonsquarefree numbers whose sum of exponential divisors (A051377) is odd.

Original entry on oeis.org

81, 405, 567, 625, 891, 1053, 1377, 1539, 1863, 1875, 2349, 2401, 2511, 2835, 2997, 3321, 3483, 3807, 4293, 4375, 4455, 4779, 4941, 5265, 5427, 5751, 5913, 6237, 6399, 6723, 6875, 6885, 7203, 7209, 7371, 7695, 7857, 8125, 8181, 8343, 8667, 8829, 9153, 9315, 9639

Offset: 1

Amiram Eldar, Sep 09 2022

nonn

The squarefree numbers are excluded from this sequence since the sum of the exponential divisors of any squarefree number k is A005117(k) = k, so the sum of the exponential divisors of any odd squarefree number (A056911) is odd.
Equivalently, odd nonsquarefree numbers whose exponents in their prime factorization are squares.
The asymptotic density of this sequence is A357017 - 4/Pi^2 = 0.0045127121... .

			81 = 3^4 is a term since it is not squarefree and A051377(81) = 93 is odd.

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

Intersection of A013929 and A357014.

Cf. A005117, A051377, A056911, A185199.

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^4], ! SquareFreeQ[#] && OddQ[esigma[#]] &]

A236474 Numbers such that the sum of unitary divisors (A034448) is equal to the sum of exponential divisors (A051377).

Original entry on oeis.org

1, 20, 45, 320, 6615, 382200, 680890228200, 8169778639360, 27445575588992, 56626123593600, 1235050901504640

Offset: 1

Michel Marcus, Jan 29 2014

Following numbers also belongs to this sequence, however their actual positions are unknown: 3640527948039840, 181552482521182080, 19736989888296320640, 108455561012908979640, 796015410768776072160, 4220107447484548287360, 39697147230528075361920, 202868762331595335655680, 668431747385354202124160, 124402428235930297906738935, 2456687209744634987008753664.

			The e-divisors of 20 are 10 and 20, sum 30, and its unitary divisors are 1, 4, 5, and 20, also sum 30.
For n=320=2^6*5 we have A051377(n)=(2^6+2^3+2^2+2)*5 = 390 and A034448(n)=(2^6+1)*(5+1) = 390 again.

Tim Trudgian, The sum of the unitary divisor function, arXiv:1312.4615 [math.NT], 2013-2014 (see page 6).
Tim Trudgian, The sum of the unitary divisor function, Publications de l'Institut Mathématique (Beograd), Vol. 97(111), 2015.

Cf. A034448, A051377.

More terms from Andrew Lelechenko, Feb 06 2014

A328132 Exponential (2,3)-perfect numbers: numbers m such that esigma(esigma(m)) = 3m, where esigma(m) is the sum of exponential divisors of m (A051377).

Original entry on oeis.org

300, 2100, 3300, 3900, 5100, 5700, 6900, 8700, 9300, 11100, 12100, 12300, 12900, 14100, 15900, 17700, 18300, 20100, 21300, 21900, 23100, 23700, 23760, 24900, 26700, 27300, 29100, 30300, 30900, 32100, 32700, 33900, 35700, 38100, 39300, 39900, 41100, 41700, 42900

Offset: 1

Amiram Eldar, Oct 04 2019

nonn

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 A019281.

Cf. A051377, A064012.

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; espQ[n_] := esigma[esigma[n]] == 3n; Select[Range[50000], 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)) == 3*k; \\ Amiram Eldar, Jan 09 2025

300 is in the sequence since esigma(300) = 540, and esigma(540) = 900 = 3*300.

A328133 Exponential (2,4)-perfect numbers: numbers m such that esigma(esigma(m)) = 4m, where esigma(m) is the sum of exponential divisors of m (A051377).

Original entry on oeis.org

540, 3780, 5940, 7020, 9180, 10260, 12420, 15660, 16740, 19980, 22140, 23220, 25380, 28620, 31860, 32940, 36180, 38340, 39420, 41580, 42660, 44820, 48060, 49140, 52380, 54540, 55620, 57780, 58860, 61020, 64260, 68580, 70740, 71820, 73980, 75060, 77220, 80460

Offset: 1

Amiram Eldar, Oct 04 2019

nonn

Conjecturally, a subsequence of A083207 (tested for the first 659 terms of this sequence). - Ivan N. Ianakiev, Oct 05 2019

			540 is in the sequence since esigma(540) = 900, and esigma(900) = 2160 = 4*540.

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 A019282.

Cf. A051377, A083207.

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; espQ[n_] := esigma[esigma[n]] == 4n; Select[Range[80000], 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)) == 4*k; \\ Amiram Eldar, Jan 09 2025

A335396 Numbers m such that sigma(m)/esigma(m) > sigma(k)/esigma(k) for all k < m, where sigma(m) is the sum of divisors of m (A000203) and esigma(m) is the sum of exponential divisors of m (A051377).

Original entry on oeis.org

1, 2, 6, 30, 96, 210, 480, 1920, 3360, 13440, 36960, 147840, 480480, 1921920, 8168160, 11975040, 32672640, 155675520, 620780160, 1401079680, 2490808320, 2646483840

Offset: 1

Amiram Eldar, Jun 05 2020

			The ratio sigma(m)/esigma(m) for m = 1, 2, ..., 6 is 1, 3/2, 4/3, 7/6, 6/5, 2. The record values occur at m = 1, 2 and 6.

Cf. A000203, A051377, A285906, A335400.

f[n_] := DivisorSigma[1,n]/( Times @@ (Sum[ First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]); seq = {}; fm = 0; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[seq, n]], {n, 1, 10^4}]; seq

A322446 The number of solutions to usigma(k) > esigma(k) below 10^n, where usigma(k) is the sum of unitary divisors of k (A034448) and esigma(k) is the sum of exponential divisors of k (A051377).

Original entry on oeis.org

5, 74, 776, 7770, 77794, 778337, 7784712, 77833385, 778307928, 7783494530

Offset: 1

Amiram Eldar, Aug 28 2019

The value of the asymptotic density of these solutions was asked in the paper by Trudgian.

			Below 10^1 there are 5 numbers k with usigma(k) > esigma(k): 2, 3, 5, 6, and 7. Thus a(1) = 5.

Tim Trudgian, The sum of the unitary divisor function, Publications de l'Institut Mathématique (Beograd), Vol. 97, No. 111 (2015), pp. 175-180.

Cf. A034448, A051377, A236474.

aQ[1] = False; fun[p_, e_] := DivisorSum[e, p^# &]; aQ[n_] := Times @@ (1 + Power @@@ (f = FactorInteger[n])) > Times @@ (fun @@@ f); c = 0; k = 1; s = {}; Do[While[k < 10^n, If[aQ[k], c++]; k++]; AppendTo[s, c], {n, 1, 6}]; s

Lim_{n->oo} a(n)/10^n = 0.778...

A049419 a(1) = 1; for n > 1, a(n) = number of exponential divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 4, 1, 1

Offset: 1

Yasutoshi Kohmoto

The exponential divisors of a number 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.
Wu gives a complicated Dirichlet g.f.
a(1) = 1 by convention. This is also required for a function to be multiplicative. - N. J. A. Sloane, Mar 03 2009
The inverse Moebius transform seems to be in A124315. The Dirichlet inverse appears to be related to A166234. - R. J. Mathar, Jul 14 2014

			a(8)=2 because 2 and 2^3 are e-divisors of 8.
The sets of e-divisors start as:
  1:{1}
  2:{2}
  3:{3}
  4:{2, 4}
  5:{5}
  6:{6}
  7:{7}
  8:{2, 8}
  9:{3, 9}
  10:{10}
  11:{11}
  12:{6, 12}
  13:{13}
  14:{14}
  15:{15}
  16:{2, 4, 16}
  17:{17}
  18:{6, 18}
  19:{19}
  20:{10, 20}
  21:{21}
  22:{22}
  23:{23}
  24:{6, 24}

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Andrew V. Lelechenko, Exponential and infinitary divisors, arXiv:1405.7597 [math.NT], 2014, sequence tau^(e).
David Moews, A database of aliquot cycles.
J. O. M. Pedersen, Tables of Aliquot Cycles.
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]
László Tóth and Nicuşor Minculete, Exponential unitary divisors, arXiv:0910.2798 [math.NT], 2009.
Tim Trudgian, The sum of the unitary divisor function, arXiv:1312.4615 [math.NT], 2013-2014, Section 3.
Eric Weisstein's World of Mathematics, e-Divisor.
Jie Wu, Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, J. Theor. Nombr. Bordeaux 7 (1) (1995) 133-141.

Row lengths of A322791.
Cf. A049599, A061389, A051377 (sum of e-divisors).
Partial sums are in A099593.
Cf. A124010, A000005, A049599, A072911, A124315, A166234, A327837, A361012.

A049419:=n->Product(List(Collected(Factors(n)), p -> Tau(p[2]))); List([1..10^4], n -> A049419(n)); # Muniru A Asiru, Oct 29 2017

a049419 = product . map (a000005 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012

A049419 := proc(n)
    local a;
    a := 1 ;
    for pf in ifactors(n)[2] do
        a := a*numtheory[tau](op(2,pf)) ;
    end do:
    a ;
end proc:
seq(A049419(n),n=1..20) ; # R. J. Mathar, Jul 14 2014

a[1] = 1; a[p_?PrimeQ] = 1; a[p_?PrimeQ, e_] := DivisorSigma[0, e]; a[n_] := Times @@ (a[#[[1]], #[[2]]] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Jan 30 2012, after Vladeta Jovovic *)

a(n) = vecprod(apply(numdiv, factor(n)[,2])); \\ Amiram Eldar, Mar 27 2023

Multiplicative with a(p^e) = tau(e). - Vladeta Jovovic, Jul 23 2001

Sum_{k=1..n} a(k) ~ A327837 * n. - Vaclav Kotesovec, Feb 27 2023

More terms from Jud McCranie, May 29 2000

A061742 a(n) is the square of the product of first n primes.

Original entry on oeis.org

1, 4, 36, 900, 44100, 5336100, 901800900, 260620460100, 94083986096100, 49770428644836900, 41856930490307832900, 40224510201185827416900, 55067354465423397733736100, 92568222856376731590410384100, 171158644061440576710668800200900

Offset: 0

Jason Earls, Jun 21 2001

nonn

Squares of primorials (first definition, A002110).
Exponential superabundant numbers: numbers k with a record value of the exponential abundancy index, A051377(k)/k > A051377(m)/m for all m < k. - Amiram Eldar, Apr 13 2019
Numbers k with a record value of A056170(k), or least number k with A056170(k) = n. - Amiram Eldar, Apr 15 2019
Empirically, these are possibly the denominators for 1 - Sum_{k=1..n} (-1)^(k+1)/prime(k)^2. The numerators are listed in A136370. - Petros Hadjicostas, May 14 2020
a(n) = least k such that rad(k/rad(k)) = A002110(n). - David James Sycamore, Jun 10 2024

			a(4) = 2^2 * 3^2 * 5^2 * 7^2 = 44100.

Harry J. Smith, Table of n, a(n) for n = 0..100

Equals A002110^2.
Cf. A001248, A051377, A056170, A129575, A136370, A322887.
Cf. A007947.

[n eq 0 select 1 else (&*[NthPrime(j)^2: j in [1..n]]): n in [0..20]]; // G. C. Greubel, Apr 19 2019

a:= proc(n) option remember; `if`(n=0, 1, ithprime(n)^2*a(n-1)) end:
seq(a(n), n=0..15);  # Alois P. Heinz, May 14 2020

a[n_]:=Product[Prime[i]^2, {i, 1, n}]; (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008 *)

for(n=0,20,print1(prod(k=1,n, prime(k)^2), ", "))

{ n=-1; m=1; forprime (p=2, prime(101), write("b061742.txt", n++, " ", m^2); m*=p ) } \\ Harry J. Smith, Jul 27 2009

[product(nth_prime(j)^2 for j in (1..n)) for n in (0..20)] # G. C. Greubel, Apr 19 2019

a(n) = Product_{j=1..n} A001248(j). - Alois P. Heinz, May 14 2020

a(n) = A228593(n) * A000040(n), for n>0. - Marco Zárate, Jun 11 2024

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

Original entry on oeis.org

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

cp's OEIS Frontend

A336680 Exponential admirable numbers: numbers k such that there is a proper exponential divisor d of k such that esigma(k) - 2*d = 2*k, where esigma is the sum of exponential divisors function (A051377).

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A357015 Nonsquarefree numbers whose sum of exponential divisors (A051377) is odd.

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A236474 Numbers such that the sum of unitary divisors (A034448) is equal to the sum of exponential divisors (A051377).

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A328132 Exponential (2,3)-perfect numbers: numbers m such that esigma(esigma(m)) = 3m, where esigma(m) is the sum of exponential divisors of m (A051377).

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A328133 Exponential (2,4)-perfect numbers: numbers m such that esigma(esigma(m)) = 4m, where esigma(m) is the sum of exponential divisors of m (A051377).

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A335396 Numbers m such that sigma(m)/esigma(m) > sigma(k)/esigma(k) for all k < m, where sigma(m) is the sum of divisors of m (A000203) and esigma(m) is the sum of exponential divisors of m (A051377).

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A322446 The number of solutions to usigma(k) > esigma(k) below 10^n, where usigma(k) is the sum of unitary divisors of k (A034448) and esigma(k) is the sum of exponential divisors of k (A051377).

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A049419 a(1) = 1; for n > 1, a(n) = number of exponential divisors of n.

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A336680 Exponential admirable numbers: numbers k such that there is a proper exponential divisor d of k such that esigma(k) - 2d = 2k, where esigma is the sum of exponential divisors function (A051377).