A129575 -id:A129575 - cp's OEIS frontend

A321146 Exponential weird numbers: numbers that are exponential abundant (A129575) but not exponential pseudoperfect (A318100).

4900, 14700, 53900, 63700, 83300, 93100, 112700, 142100, 151900, 161700, 181300, 191100, 200900, 210700, 230300, 249900, 259700, 279300, 289100, 298900, 328300, 338100, 347900, 357700, 387100, 406700, 426300, 436100, 455700, 475300, 494900, 504700, 524300

Offset: 1

Amiram Eldar, Oct 28 2018

nonn

			4900 is in the sequence since its proper exponential divisors, {70, 140, 350, 490, 700, 980, 2450} sum to 5180 > 4900, yet no subset of its divisors sums to 4900.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert Israel)
Eric Weisstein's World of Mathematics, e-Divisor.
Eric Weisstein's World of Mathematics, e-Perfect Number.

The exponential version of A006037.

Cf. A129575, A318100.

filter:= proc(n)
  local L,m,P,i,j,T,S,t,v;
  L:= ifactors(n)[2];
  m:= nops(L);
  P:= map(t -> numtheory:-divisors(t[2]),L);
  if mul(add(L[i][1]^j, j=P[i]),i=1..m) <= 2*n then return false fi;
  T:= combinat:-cartprod(P);
  S:= {0}:
  while not T[finished] do
    t:= T:-nextvalue();
    v:= mul(L[i][1]^t[i],i=1..m);
    if v = n then next fi;
    if member(n-v,S) then return false fi;
    S:= S union select(`<=`,map(`+`,S,v),n);
  od;
  true
end proc:
select(filter, [$1..10^6]); # Robert Israel, Feb 19 2019

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]; eAbundantQ[n_] := esigma[n] > 2 n; a = {}; n = 0; While[Length[a] < 30, n++; If[!eAbundantQ[n], Continue[]]; d = Most[eDivs[n]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c < 1, AppendTo[a, n]]]; a

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]

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

A308053 Coreful abundant 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

72, 108, 144, 200, 216, 288, 324, 360, 400, 432, 504, 540, 576, 600, 648, 720, 756, 784, 792, 800, 864, 900, 936, 972, 1000, 1008, 1080, 1152, 1188, 1200, 1224, 1296, 1368, 1400, 1404, 1440, 1512, 1568, 1584, 1600, 1620, 1656, 1728, 1764, 1800, 1836, 1872, 1936

Offset: 1

Amiram Eldar, May 10 2019

nonn

Analogous to A005101 as A307958 is analogous to A000396.

The asymptotic density of this sequence is Sum_{n>=1} f(A356871(n)) = 0.0262215..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - Amiram Eldar, Sep 02 2022

			72 is in the sequence since its coreful divisors are 6, 12, 18, 24, 36, 72, whose sum is 168 > 2 * 72.

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

Cf. A005101, A057723, A307958.
A339940 and A356871 are subsequences.
Subsequence of A129575.

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, 2000}]; s

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
s(n) = rad(n)*sigma(n/rad(n)); \\ A057723
isok(k) = s(k) > 2*k; \\ Michel Marcus, May 11 2019

isok(k) = {my(f=factor(k)); prod(i = 1, #f~, (f[i,1]^(f[i,2]+1)-1) / (f[i,1]-1)-1) > 2*k}; \\ Amiram Eldar, Sep 02 2022

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

A348274 Noninfinitary abundant numbers: numbers k such that A348271(k) > k.

Original entry on oeis.org

36, 48, 80, 144, 180, 240, 252, 288, 300, 324, 336, 396, 400, 432, 468, 528, 560, 576, 588, 612, 624, 684, 720, 768, 784, 816, 828, 880, 900, 912, 960, 1008, 1040, 1044, 1104, 1116, 1200, 1232, 1260, 1280, 1296, 1332, 1360, 1392, 1440, 1456, 1476, 1488, 1520, 1548

Offset: 1

Amiram Eldar, Oct 09 2021

nonn

The first odd term is a(3577) = 99225.

The number of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 3, 31, 360, 3605, 36160, 360840, 3618980, 36144059, ... Apparently this sequence has an asymptotic density 0.0361...

			36 is a term since A348271(36) = 41 > 36.

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

Cf. A348271, A327633.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; s[n_] := DivisorSigma[1,n] - isigma[n]; Select[Range[1500], s[#] > # &]

A348604 Nonexponential abundant numbers: numbers k such that A160135(k) > k.

Original entry on oeis.org

24, 30, 42, 48, 54, 60, 66, 70, 72, 78, 84, 90, 96, 102, 114, 120, 126, 132, 138, 150, 156, 160, 162, 168, 174, 180, 186, 192, 198, 210, 216, 222, 224, 240, 246, 258, 264, 270, 280, 282, 288, 294, 300, 312, 318, 320, 330, 336, 352, 354, 360, 366, 378, 384, 390

Offset: 1

Amiram Eldar, Oct 25 2021

nonn

The smallest odd term is a(1357) = 8505.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 13, 148, 1595, 15688, 158068, 1578957, 15762209, 157745113, 1577808429, ... Apparently this sequence has an asymptotic density 0.157...

			24 is a term since A160135(24) = 30 > 24.

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

Cf. A160135, A348601.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; q[n_] := DivisorSigma[1, n] - esigma[n] > n; Select[Range[400], q]

A380929 Numbers k such that A380845(k) > 2*k.

Original entry on oeis.org

36, 72, 84, 140, 144, 168, 180, 264, 270, 280, 288, 300, 336, 360, 372, 392, 450, 520, 528, 532, 540, 558, 560, 576, 594, 600, 612, 620, 672, 720, 744, 756, 780, 784, 840, 900, 930, 1036, 1040, 1050, 1056, 1064, 1068, 1080, 1092, 1116, 1120, 1134, 1152, 1170, 1180, 1188, 1200

Offset: 1

Amiram Eldar, Feb 08 2025

Analogous to abundant numbers (A005101) with A380845 instead of A000203.

			36 is a term since A380845(36) = 84 > 2 * 36 = 72.

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

Cf. A000203, A380845, A380846.
Subsequence of A005101.
Subsequences: A380847, A380848, A380930, A380931.
Similar sequences: A034683, A064597, A087248, A129575, A129656, A292982, A348274, A348604, A379029.

q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k]; Select[Range[1200], q]

isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k;}

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

A323343 Numbers k whose exponential divisors have an even sum which is larger than 2k, but they cannot be partitioned into two disjoint parts whose sums are equal.

Original entry on oeis.org

1910412, 9552060, 21014532, 24835356, 32477004, 43939476, 55401948, 59222772, 70685244, 78326892, 82147716, 89789364, 101251836, 105072660, 112714308, 116535132, 124176780, 127997604, 135639252, 139460076, 150922548, 158564196, 162385020, 170026668, 185309964

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The exponential version of A171641.

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

Cf. A051377, A129575, A171641, A323341, A323342, A323344.

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]]]} ]]; ediv[n_] := Module[ {d=Rest[Divisors[n]]}, Select[ d, expDivQ[n, #]&]]; esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[ Last[#]]}] &) /@ FactorInteger[n]; seq={}; Do[s=esigma[n]; If[OddQ[s] || s<=2n, Continue[]]; div = ediv[n]; If[Coefficient[Times @@ (1 + x^div) // Expand, x, s/2] == 0, AppendTo[seq, n]], {n, 1, 10000}]; seq

cp's OEIS Frontend

A321146 Exponential weird numbers: numbers that are exponential abundant (A129575) but not exponential pseudoperfect (A318100).

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

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

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A308053 Coreful abundant 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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A321147 Odd exponential abundant numbers: odd numbers k whose sum of exponential divisors A051377(k) > 2*k.

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A348274 Noninfinitary abundant numbers: numbers k such that A348271(k) > k.

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A348604 Nonexponential abundant numbers: numbers k such that A160135(k) > k.

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A380929 Numbers k such that A380845(k) > 2*k.