A183097 -id:A183097 - cp's OEIS frontend

A349064 Numbers k such that A183097(k) > 2*k.

144, 216, 288, 432, 576, 648, 864, 1152, 1296, 1600, 1728, 1944, 2000, 2304, 2592, 3200, 3456, 3600, 3888, 4000, 4608, 5000, 5184, 5400, 5488, 5832, 6272, 6400, 6912, 7056, 7200, 7776, 8000, 9000, 9216, 10000, 10368, 10584, 10800, 10976, 11664, 12544, 12800, 13500

Offset: 1

Amiram Eldar, Nov 07 2021

nonn

The least odd term is a(934) = A349065(1) = 3472875.

Not all the terms are powerful. E.g., (prime(44)#)^3/4 and (prime(22)#)^6/32 are nonpowerful terms. What is the least nonpowerful term?

			144 is a term since A183097(144) = 290 > 2*144 = 288.

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

Subsequence of A005101.
A349065 is a subsequence.
Cf. A001694, A183097.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := s[n] > 2*n; Select[Range[15000], q]

isok(k) = sumdiv(k, d, ispowerful(d)*d) > 2*k; \\ Michel Marcus, Nov 07 2021

A349065 Odd numbers k such that A183097(k) > 2*k.

Original entry on oeis.org

3472875, 10418625, 17364375, 24310125, 31255875, 52093125, 72930375, 86821875, 93767625, 121550625, 156279375, 170170875, 202145625, 218791125, 260465625, 281302875, 364651875, 420217875, 434109375, 468838125, 510512625, 586915875, 606436875, 607753125, 656373375

Offset: 1

Amiram Eldar, Nov 07 2021

nonn

The odd terms of A349064.

			3472875 is a term since A183097(3472875) = 7002474 > 2*3472875 = 6945750.

Amiram Eldar, Table of n, a(n) for n = 1..100

Subsequence of A005101, A005231 and A349064.

Cf. A001694, A183097.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := s[n] > 2*n; Select[Range[1, 2.5*10^7, 2], q]

A349112 Powerful highly abundant numbers: numbers m such that psigma(m) > psigma(k) for all k < m, where psigma(k) is the sum of powerful divisors of k (A183097).

Original entry on oeis.org

1, 4, 8, 16, 27, 32, 64, 72, 108, 128, 144, 200, 216, 256, 288, 392, 400, 432, 576, 648, 800, 864, 1152, 1296, 1728, 1944, 2304, 2592, 3456, 3888, 5184, 6912, 7776, 10000, 10368, 11664, 13824, 15552, 20000, 20736, 23328, 27000, 27648, 31104, 34992, 40000, 41472

Offset: 1

Amiram Eldar, Nov 08 2021

nonn

The corresponding record values are 1, 5, 13, 29, 37, 61, 125, 130, 185, 253, ...

			The first 8 terms of A183097 are 1, 1, 1, 5, 1, 1, 1 and 13. The record values, 1, 5 and 13, occur at 1, 4 and 8, the first 3 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..287 (terms below 10^10)

A349111 is a subsequence.
Cf. A005934, A183097.
Similar sequences: A285614, A292983, A327634, A328134, A329883, A348272.

f[p_,e_] := (p^(e+1)-1)/(p-1) - p; s[1] = 1; s[n_] := Times @@ f @@@FactorInteger[n]; seq = {}; sm = 0; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 10^5}]; seq

A349063 Numbers k such that k and k+1 have the same sum of powerful divisors (A183097) and this sum is larger than 1.

Original entry on oeis.org

2988, 4067, 7595, 13572, 14651, 24156, 25235, 27684, 28763, 34740, 35819, 38268, 39347, 41327, 46403, 48852, 49931, 56987, 59436, 66492, 70020, 78155, 81683, 87660, 88739, 91188, 98244, 99323, 101772, 102851, 108828, 109907, 112356, 113435, 119412, 120491, 122940

Offset: 1

Amiram Eldar, Nov 07 2021

nonn

Numbers k such that A183097(k) = A183097(k+1) > 1.

			2988 is a term since = A183097(2988) = A183097(2989) = 50 > 1.

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

Cf. A183097.

Similar sequences: A002961, A064115, A064125, A293183, A306985.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := (s1 = s[n]) > 1 && s1 == s[n + 1]; Select[Range[10^5], q]

A349111 Powerful superabundant numbers: numbers m such that psigma(m)/m > psigma(k)/k for all k < m, where psigma(k) is the sum of powerful divisors of k (A183097).

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 128, 144, 216, 432, 864, 1296, 1728, 2592, 5184, 10368, 15552, 31104, 54000, 108000, 162000, 216000, 324000, 648000, 1296000, 1944000, 3240000, 3888000, 6480000, 9720000, 19440000, 38880000, 58320000, 74088000, 111132000, 222264000, 444528000, 666792000

Offset: 1

Amiram Eldar, Nov 08 2021

nonn

The corresponding record values are 1, 5/4, 13/8, 29/16, 61/32, 125/64, ...

The least term k with psigma(k)/k > m, for m = 2, 3, ..., is 144, 54000, 666792000, ...

Subsequence of A349112.
Cf. A005934, A183097.
Similar sequences: A002110 (unitary), A037992 (infinitary), A061742, A292984, A329882, A348273.

f[p_,e_] := (p^(e+1)-1)/(p-1) - p; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; rm = 0; Do[r1 = s[n]/n; If[r1 > rm, rm = r1; AppendTo[seq, n]], {n, 1, 10^6}]; seq

A360720 a(n) is the sum of unitary divisors of n that are powerful (A001694).

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 9, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 9, 26, 1, 28, 5, 1, 1, 1, 33, 1, 1, 1, 50, 1, 1, 1, 9, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 28, 1, 9, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 90, 1, 1, 26, 5, 1, 1, 1, 17, 82

Offset: 1

Amiram Eldar, Feb 18 2023

The number of these divisors is given by A323308.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^(3/2) for n = 1..10^8.
Index entries for sequences related to divisors of numbers.

Cf. A001694, A004709, A034444, A034448, A057521, A077610, A092261, A323308.

Similar sequences: A183097, A360722.

f[p_, e_] := If[e == 1, 1, p^e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, f[i, 1]^f[i, 2] + 1));}

for(n=1, 100, print1(direuler(p=2, n, (1 - p^3*X^4 - p^2*X^3 + p^3*X^3) / ((1 - X) * (1 - p^2*X^2)))[n], ", ")) \\ Vaclav Kotesovec, Feb 18 2023

Multiplicative with a(p) = 1 and a(p^e) = p^e + 1 for e > 1.
a(n) <= A034448(n), with equality if and only if n is powerful (A001694).
a(n) <= A183097(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(s)*zeta(s-1)*Product_{p prime} (1 - p^(1-s) + p^(2-2*s) - p^(2-3*s)).
From Vaclav Kotesovec, Feb 18 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{primes p} (1 - p^(3-4*s) - p^(2-3*s) + p^(3-3*s)).
Sum_{k=1..n} a(k) ~ c * zeta(3/2) * n^(3/2) / 3, where c = Product_{primes p} (1 + 1/p^(3/2) - 1/p^(5/2) - 1/p^3) = 1.48039182258752809541724060173644... (End)
a(n) = A034448(A057521(n)) (the sum of unitary divisors of the powerful part of n). - Amiram Eldar, Dec 12 2023
a(n) = A034448(n)/A092261(n). - Amiram Eldar, Jun 19 2025

A385006 The sum of the biquadratefree divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 15, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 15, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 60, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 15, 84, 144

Offset: 1

Amiram Eldar, Jun 15 2025

First differs from A365682 and A366992 at n = 32.

The number of these divisors is A252505(n), and the largest of them is A058035(n).

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

Cf. A046100, A048250, A058035, A073185, A252505, A365682, A366992.

The sum of divisors d of n such that d is: A000593 (odd), A033634 (exponentially odd), A035316 (square), A038712 (power of 2), A048250 (squarefree), A072079 (3-smooth), A073185 (cubefree), A113061 (cube), A162296 (nonsquarefree), A183097 (powerful), A186099 (5-rough), A353900 (exponentially 2^n), A385005 (cubefull), this sequence (biquadratefree).

f[p_, e_] := (p^Min[e+1, 4] - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; (p^min(e+1, 4) - 1)/(p - 1));}

Multiplicative with a(p^e) = (p^min(e+1, 4) - 1)/(p - 1).
In general, the sum of the k-free (numbers that are not divisible by a k-th power larger than 1) divisors of n is multiplicative with a(p^e) = (p^min(e+1, k) - 1)/(p - 1).
Dirichlet g.f.: zeta(s) * zeta(s-1) /zeta(4*s-4).
In general, the sum of the k-free divisors of n has Dirichlet g.f.: zeta(s)*zeta(s-1)/zeta(k*s-k).
Sum_{k=1..n} a(k) ~ (15/(2*Pi^2)) * n^2.
In general, the sum of the k-free divisors of n has an average order (Pi^2/(12*zeta(k))) * n^2.

A091051 Sum of divisors of n that are perfect powers.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 13, 10, 1, 1, 5, 1, 1, 1, 29, 1, 10, 1, 5, 1, 1, 1, 13, 26, 1, 37, 5, 1, 1, 1, 61, 1, 1, 1, 50, 1, 1, 1, 13, 1, 1, 1, 5, 10, 1, 1, 29, 50, 26, 1, 5, 1, 37, 1, 13, 1, 1, 1, 5, 1, 1, 10, 125, 1, 1, 1, 5, 1, 1, 1, 58, 1, 1, 26, 5, 1, 1, 1, 29, 118, 1, 1, 5, 1, 1, 1, 13, 1, 10

Offset: 1

Reinhard Zumkeller, Dec 15 2003

nonn

a(n) = 1 iff n is squarefree: a(A005117(n))=1, a(A013929(n))>1;
a(p^k) = 1+(p^2)*(p^(k-1)-1)/(p-1) for p prime, k>0.
a(A000961(n)) = A086455(n)-A025473(n).

			Divisors of n=108: {1,2,3,4,6,9,12,18,27,36,54,108}, a(108) = 1^2 + 2^2 + 3^2 + 3^3 + 6^2 = 1+4+9+27+36 = 77.

Antti Karttunen, Table of n, a(n) for n = 1..16385
Eric Weisstein's World of Mathematics, Perfect Power
Eric Weisstein's World of Mathematics, Divisor Function
Index entries for sequences related to sums of divisors

Cf. A091050, A001597, A000203, A183104.

Differs from A183097 for the first time at n=72.

a[n_] := DivisorSum[n, #*Boole[# == 1 || GCD @@ FactorInteger[#][[All, 2]] > 1]&]; Array[a, 90] (* Jean-François Alcover, May 09 2017 *)

a(n) = sumdiv(n, d, d*((d==1) || ispower(d))); \\ Michel Marcus, Oct 02 2014

G.f.: Sum_{k=i^j, i>=1, j>=2, excluding duplicates} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 20 2017

A183098 a(1) = 0, a(n) = sum of divisors d of n such that if d = Product_{i} (p_i^e_i) then not all e_i are > 1.

Original entry on oeis.org

0, 2, 3, 2, 5, 11, 7, 2, 3, 17, 11, 23, 13, 23, 23, 2, 17, 29, 19, 37, 31, 35, 23, 47, 5, 41, 3, 51, 29, 71, 31, 2, 47, 53, 47, 41, 37, 59, 55, 77, 41, 95, 43, 79, 68, 71, 47, 95, 7, 67, 71, 93, 53, 83, 71, 107, 79, 89, 59, 163, 61, 95, 94, 2, 83, 143, 67, 121, 95, 143, 71, 65, 73, 113, 98, 135, 95, 167, 79, 157, 3, 125, 83, 219, 107, 131, 119, 167, 89, 224, 111, 163, 127, 143, 119, 191, 97, 121, 146, 87

Offset: 1

Jaroslav Krizek, Dec 25 2010

a(n) = sum of non-powerful divisors d of n where powerful numbers are numbers from A001694(m) for m >= 2.

Sequence is not the same as A183101(n): a(72) = 65, A183101(72) = 137.

			For n = 12, the set of such divisors is {2, 3, 6, 12}; a(12) = 2+3+6+12 = 23.

Antti Karttunen, Table of n, a(n) for n = 1..16385
Index entries for sequences related to sums of divisors.

Cf. A000203, A001694, A183097, A183100, A183101, A183103.

f1[p_, e_] := (p^(e+1)-1)/(p-1); f2[p_, e_] := f1[p, e] - p; a[1] = 0; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Aug 29 2023 *)

A183098(n) = sumdiv(n, d, d*(!ispowerful(d))); \\ Antti Karttunen, Oct 07 2017

a(n) = A000203(n) - A183097(n) = A183100(n) - 1.

a(1) = 0, a(p) = p, a(p*q) = p+q+p*q, a(p*q*...*z) = (p+1)*(q+1)*...*(z+1) - 1, a(p^k) = p, for p, q = primes, k = natural numbers, p*q*...*z = product of k (k > 2) distinct primes p, q, ..., z.

Name corrected by Jon E. Schoenfield, Aug 29 2023

A183102 a(n) = product of powerful divisors d of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 32, 9, 1, 1, 4, 1, 1, 1, 512, 1, 9, 1, 4, 1, 1, 1, 32, 25, 1, 243, 4, 1, 1, 1, 16384, 1, 1, 1, 1296, 1, 1, 1, 32, 1, 1, 1, 4, 9, 1, 1, 512, 49, 25, 1, 4, 1, 243, 1, 32, 1, 1, 1, 4, 1, 1, 9, 1048576, 1, 1, 1, 4, 1, 1, 1, 746496, 1

Offset: 1

Jaroslav Krizek, Dec 25 2010

nonn

a(n) = product of divisors d of n from set A001694 - powerful numbers.
Sequence is not the same as A183104(n): a(72) = 746496, A183104(72) = 10368.
Not multiplicative: a(4)*a(9) = 4*9=36 <> a(36) = 1296. - R. J. Mathar, Jun 07 2011

			For n = 12, set of such divisors is {1, 4}; a(12) = 1*4 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..16385

Cf. A001694, A007955, A183097, A183103, A183104.

isA001694 := proc(n) for p in ifactors(n)[2] do if op(2,p) = 1 then return false; end if; end do; return true; end proc:
A183102 := proc(n) local a,d; a := 1 ; for d in numtheory[divisors](n) do if isA001694(d) then a := a*d; end if; end do; a ; end proc:
seq(A183102(n),n=1..70) ; # R. J. Mathar, Jun 07 2011

powerfulQ[n_] := Min[FactorInteger[n][[All, 2]]] > 1;
a[n_] := Times @@ Select[Divisors[n], powerfulQ];
Table[a[n], {n, 1, 73}] (* Jean-François Alcover, Jun 01 2024 *)

A183102(n) = { my(m=1); fordiv(n, d, if(ispowerful(d), m *= d)); m; }; \\ Antti Karttunen, Oct 07 2017

a(n) = A007955(n) / A183103(n).

a(1) = 1, a(p) = 1, a(pq) = 1, a(pq...z) = 1, a(p^k) = p^((1/2*k*(k+1))-1), for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.

cp's OEIS Frontend

A349064 Numbers k such that A183097(k) > 2*k.

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A349065 Odd numbers k such that A183097(k) > 2*k.

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A349112 Powerful highly abundant numbers: numbers m such that psigma(m) > psigma(k) for all k < m, where psigma(k) is the sum of powerful divisors of k (A183097).

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A349063 Numbers k such that k and k+1 have the same sum of powerful divisors (A183097) and this sum is larger than 1.

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A349111 Powerful superabundant numbers: numbers m such that psigma(m)/m > psigma(k)/k for all k < m, where psigma(k) is the sum of powerful divisors of k (A183097).

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A360720 a(n) is the sum of unitary divisors of n that are powerful (A001694).

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A385006 The sum of the biquadratefree divisors of n.

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A091051 Sum of divisors of n that are perfect powers.

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