A001694 -id:A001694 - cp's OEIS frontend

A286708 Powerful numbers (A001694) that are not prime powers (A000961).

36, 72, 100, 108, 144, 196, 200, 216, 225, 288, 324, 392, 400, 432, 441, 484, 500, 576, 648, 675, 676, 784, 800, 864, 900, 968, 972, 1000, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1352, 1372, 1444, 1521, 1568, 1600, 1728, 1764, 1800, 1936, 1944, 2000, 2025, 2116, 2304, 2312, 2500, 2592, 2601, 2700, 2704, 2744

Offset: 1

Ilya Gutkovskiy, May 13 2017

nonn

If a prime p divides a(n) then p^2 must also divide a(n) and number of distinct primes dividing a(n) > 1.

Intersection of A001694 and A024619.

			-------------------------------
| n | a(n) | prime            |
|   |      | factorization    |
|------------------------------
| 1 | 36   | {{2, 2}, {3, 2}} |
| 2 | 72   | {{2, 3}, {3, 2}} |
| 3 | 100  | {{2, 2}, {5, 2}} |
| 4 | 108  | {{2, 2}, {3, 3}} |
| 5 | 144  | {{2, 4}, {3, 2}} |
| 6 | 196  | {{2, 2}, {7, 2}} |
| 7 | 200  | {{2, 3}, {5, 2}} |
| 8 | 216  | {{2, 3}, {3, 3}} |
| 9 | 225  | {{3, 2}, {5, 2}} |
-------------------------------
a(n) = p_1^e_1*p_2^e_2*... : {{p_1, e_1}, {p_2, e_2}, ...}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 5997 terms from Robert Israel)
Eric Weisstein's World of Mathematics, Prime Power.
Eric Weisstein's World of Mathematics, Powerful Number.
Index entries for sequences related to powerful numbers

Cf. A000961, A001221, A001694, A024619, A082695, A136141, A131605.

N:= 10000:
S:= {1}: P:= {1}:
p:= 1:
do
  p:= nextprime(p);
  if p^2 > N then break fi;
  S:= map(s -> (s, seq(s*p^k, k = 2 .. floor(log[p](N/s)))), S);
  P:= P union {seq(p^k, k=2..floor(log[p](N)))}:
od:
sort(convert(S minus P, list)); # Robert Israel, May 14 2017

Select[Range@2750, Min@FactorInteger[#][[All, 2]] > 1 && ! PrimePowerQ[#] &]
(* Second program *)
nn = 2^25; Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], ! PrimePowerQ[#] &] (* Michael De Vlieger, Jun 22 2022 *)

from sympy import primefactors, factorint
print([n for n in range(4,2745) if len(primefactors(n)) > 1 and min(list(factorint(n).values())) > 1]) # Karl-Heinz Hofmann, Feb 07 2023

from math import isqrt
from sympy import integer_nthroot, primepi, mobius
def A286708(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l
        return c+1+sum(primepi(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))
    return bisection(f,n,n) # Chai Wah Wu, Sep 10 2024

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{p prime} 1/(p*(p-1)) - 1 = A082695 - A136141 - 1 = 0.17043976777096407719... - Amiram Eldar, Feb 12 2021

A076445 The smaller of a pair of powerful numbers (A001694) that differ by 2.

Original entry on oeis.org

25, 70225, 130576327, 189750625, 512706121225, 13837575261123, 99612037019889, 1385331749802025, 3743165875258953025, 10114032809617941274225, 8905398244301708746029223, 27328112908421802064005625, 73840550964522899559001927225

Offset: 1

Jud McCranie, Oct 15 2002

nonn

Erdos conjectured that there aren't three consecutive powerful numbers and no examples are known. There are an infinite number of powerful numbers differing by 1 (cf. A060355). A requirement for three consecutive powerful numbers is a pair that differ by 2 (necessarily odd). These pairs are much more rare.

Sentance gives a method for constructing families of these numbers from the solutions of Pell equations x^2-my^2=1 for certain m whose square root has a particularly simple form as a continued fraction. Sentance's result can be generalized to any m such that A002350(m) is even. These m, which generate all consecutive odd powerful numbers, are in A118894. - T. D. Noe, May 04 2006

			25=5^2 and 27=3^3 are powerful numbers differing by 2, so 25 is in the sequence.

R. K. Guy, Unsolved Problems in Number Theory, B16

Max Alekseyev, Conjectured table of n, a(n) for n = 1..33 [These terms certainly belong to the sequence, but they are not known to be consecutive.]
R. A. Mollin and P. G. Walsh, On powerful numbers, IJMMS 9:4 (1986), 801-806.
W. A. Sentance, Occurrences of consecutive odd powerful numbers, Amer. Math. Monthly, 88 (1981), 272-274.
Eric Weisstein's World of Mathematics, Powerful numbers

Cf. A001694, A060355.

a(8)-a(10) from Geoffrey Reynolds (geoff(AT)hisplace.co.nz), Feb 15 2005

More terms from T. D. Noe, May 04 2006

A076871 Sum of two powerful numbers (definition (1), A001694).

Original entry on oeis.org

2, 5, 8, 9, 10, 12, 13, 16, 17, 18, 20, 24, 25, 26, 28, 29, 31, 32, 33, 34, 35, 36, 37, 40, 41, 43, 44, 45, 48, 50, 52, 53, 54, 57, 58, 59, 61, 63, 64, 65, 68, 72, 73, 74, 76, 80, 81, 82, 85, 88, 89, 90, 91, 96, 97, 98, 99, 100, 101, 104, 106, 108, 109, 112, 113, 116, 117

Offset: 1

N. J. A. Sloane, Nov 25 2002

Complement of A085253. - Reinhard Zumkeller, Jun 23 2003

Aleksandar Ivić, The Riemann Zeta-Function, Wiley, NY, 1985, see p. 439.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Valentin Blomer, Binary quadratic forms with large discriminants and sums of two squareful numbers II, Journal of the London Mathematical Society 71:1 (2005), pp. 69-84.
Alexander Kalmynin and Segei Konyagin, Large gaps between sums of two squareful numbers, arXiv:2303.14833 [math.NT], 2023.
Eric Weisstein's World of Mathematics, Powerful Number.

Cf. A001694, A076872, A085252, A085253, A261883.

Different from A070049.

With[{m = 120}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; Select[Union[Plus @@@ Tuples[pow, {2}]], # <= m &]] (* Amiram Eldar, Jan 30 2023 *)

A085252(a(n)) > 0. - Reinhard Zumkeller, Jun 23 2003

Blomer shows that there are x/log^k x sums of two powerful numbers up to x, where k = 0.20629947... is A261883. - Charles R Greathouse IV, Sep 04 2015

More terms from Vladeta Jovovic, Nov 25 2002

A320966 Powerful numbers A001694 divisible by a cube > 1.

Original entry on oeis.org

8, 16, 27, 32, 64, 72, 81, 108, 125, 128, 144, 200, 216, 243, 256, 288, 324, 343, 392, 400, 432, 500, 512, 576, 625, 648, 675, 729, 784, 800, 864, 968, 972, 1000, 1024, 1125, 1152, 1296, 1323, 1331, 1352, 1372, 1568, 1600, 1728, 1800, 1936, 1944, 2000, 2025, 2048, 2187, 2197, 2304, 2312, 2401, 2500

Offset: 1

Hugo Pfoertner, Oct 25 2018

nonn

Powerful numbers that are not squares of squarefree numbers. - Amiram Eldar, Jun 25 2022

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000578, A320965.
Intersection of A001694 and A046099.
A001694 \ A062503.

Select[Range[2500], (m = MinMax[FactorInteger[#][[;; , 2]]])[[1]] > 1 && m[[2]] > 2 &] (* Amiram Eldar, Jun 25 2022 *)

isA001694(n)=n=factor(n)[, 2]; for(i=1, #n, if(n[i]==1, return(0))); 1 \\ from Charles R Greathouse IV
isA046099(n)=n=factor(n)[, 2]; for(i=1, #n, if(n[i]>2, return(1)));0
for (k=1,2500,if(isA001694(k)&&isA046099(k),print1(k,", ")))

from math import isqrt
from sympy import mobius, integer_nthroot
def A320966(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x+squarefreepi(isqrt(x))-squarefreepi(integer_nthroot(x,3)[0]), 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    return bisection(f,n,n) # Chai Wah Wu, Sep 15 2024

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - 15/Pi^2 = 0.4237786821... . - Amiram Eldar, Jun 25 2022

A085253 Numbers having no representation as sum of two powerful numbers (A001694).

Original entry on oeis.org

1, 3, 4, 6, 7, 11, 14, 15, 19, 21, 22, 23, 27, 30, 38, 39, 42, 46, 47, 49, 51, 55, 56, 60, 62, 66, 67, 69, 70, 71, 75, 77, 78, 79, 83, 84, 86, 87, 92, 93, 94, 95, 102, 103, 105, 107, 110, 111, 114, 115, 118, 119, 120, 123, 131, 138, 139, 142, 143, 147, 151, 154

Offset: 1

Reinhard Zumkeller, Jun 23 2003

nonn

Complement of A076871.

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

Cf. A001694, A076871, A085252.

Different from A075434.

With[{m = 160}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; Complement[Range[m], Select[Union[Plus @@@ Tuples[pow, {2}]], # <= m &]]] (* Amiram Eldar, Jan 30 2023 *)

A085252(a(n)) = 0.

A119241 Number of powerful numbers (A001694) between consecutive squares n^2 and (n+1)^2.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 2, 0, 0, 2, 1, 1, 0, 1, 1, 1, 0, 2, 0, 0, 2, 0, 0, 1, 1, 0, 3, 0, 2, 0, 0, 3, 1, 0, 1, 0, 1, 1, 0, 2, 1, 2, 0, 1, 0, 1, 1, 1, 1, 0, 2, 1, 1, 2, 1, 0, 0, 2, 1, 0, 1, 0, 3, 0, 0, 2, 0, 2, 2, 1, 0, 1, 1, 1, 1, 0, 0, 2, 1, 1, 0, 0, 1, 2, 1, 1, 0, 1, 3, 1, 0, 2, 0, 2, 0, 1, 1, 1, 2, 2, 0

Offset: 1

T. D. Noe, May 09 2006

nonn

Is there an upper bound on the number of powerful numbers between consecutive squares? Pettigrew conjectures that there is no bound. See A119242.

This sequence is unbounded. For each k >= 0 the sequence of solutions to a(x) = k has a positive asymptotic density (Shiu, 1980). - Amiram Eldar, Jul 10 2020

			a(5) = 2 because the two powerful numbers 27 and 32 are between 25 and 36.

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter VI, p. 226.

Amiram Eldar, Table of n, a(n) for n = 1..10000
P. Shiu, On the number of square-full integers between successive squares, Mathematika, Vol. 27, No. 2 (1980), pp. 171-178.

Cf. A001694, A090699, A119242.

Powerful[n_] := (n==1) || Min[Transpose[FactorInteger[n]][[2]]]>1; Table[Length[Select[Range[k^2+1, k^2+2k], Powerful[ # ]&]], {k,130}]

from math import isqrt
from sympy import integer_nthroot, factorint
def A119241(n):
    def f(x): return int(sum(isqrt(x//k**3) for k in range(1, integer_nthroot(x, 3)[0]+1) if all(d<=1 for d in factorint(k).values())))
    return f((n+1)**2-1)-f(n**2) # Chai Wah Wu, Sep 10 2024

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = zeta(3/2)/zeta(3) - 1 = A090699 - 1 = 1.173254... - Amiram Eldar, Oct 24 2020

A180114 a(n) = sigma(A001694(n)), sum of divisors of the powerful number A001694(n).

Original entry on oeis.org

1, 7, 15, 13, 31, 31, 40, 63, 91, 57, 127, 195, 121, 217, 280, 133, 156, 255, 403, 183, 399, 465, 600, 403, 364, 511, 819, 307, 847, 400, 381, 855, 961, 1240, 741, 931, 1092, 1023, 553, 1651, 781, 1815, 1240, 1281, 1093, 1767, 1953, 871, 2520, 2821, 993, 1995

Offset: 1

Walter Kehowski, Aug 10 2010

			Sigma(2^2) = 7, sigma(2^3) = 15, sigma(3^2) = 13.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (7).

Cf. A001694, A000203, A090699, A180090, A180097, A180098, A362984.

emin := proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); min(L) else 0 fi end: L:=[]: for w to 1 do for n from 1 to 144 do sn:=sigma(n); if emin(n)>1 then L:=[op(L),sn]; print(n,ifactor(n),sn,ifactor(sn)) fi; od; od;

pwfQ[n_] := n == 1 || Min[Last /@ FactorInteger[n]] > 1; DivisorSigma[1, Select[ Range@ 1000, pwfQ]] (* Giovanni Resta, Feb 06 2018 *)

lista(kmax) = for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 1, print1(sigma(k), ", "))); \\ Amiram Eldar, May 12 2023

from itertools import count, islice
from math import prod
from sympy import factorint
def A180114_gen(): # generator of terms
    for n in count(1):
        f = factorint(n)
        if all(e>1 for e in f.values()):
            yield prod((p**(e+1)-1)//(p-1) for p,e in f.items())
A180114_list = list(islice(A180114_gen(),20)) # Chai Wah Wu, May 21 2023

from math import isqrt
from sympy import mobius, integer_nthroot, divisor_sigma
def A180114(n):
    def squarefreepi(n):
        return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2, 3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x, 3)[0])-l
        return c
    return divisor_sigma(bisection(f, n, n)) # Chai Wah Wu, Sep 10 2024

From Amiram Eldar, May 12 2023: (Start)
Sum_{A001694(k) < x} a(k) = c * x^(3/2) + O(x^(23/18 + eps)), where c = A362984 * A090699 / 3 = 1.5572721108... (Jakimczuk and Lalín, 2022). [corrected Sep 21 2024]
Sum_{k=1..n} a(k) ~ c * n^3, where c = A362984 / (3 * A090699^2) = 0.151716514097... . (End)

a(1)=1 prepended by and more terms from Giovanni Resta, Feb 06 2018

A258599 a(n) is the index m such that A001694(m) = prime(n)^2.

Original entry on oeis.org

2, 4, 6, 10, 16, 20, 28, 31, 39, 48, 51, 65, 71, 75, 84, 94, 107, 110, 120, 129, 133, 145, 152, 163, 180, 187, 191, 199, 202, 212, 238, 246, 258, 261, 282, 286, 297, 309, 319, 330, 342, 344, 366, 372, 377, 382, 407, 431, 440, 443, 450, 463, 468, 487, 498

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

			.   n |  p |  a(n) | A001694(a(n)) = A001248(n) = p^2
. ----+----+-------+---------------------------------
.   1 |  2 |     2 |             4
.   2 |  3 |     4 |             9
.   3 |  5 |     6 |            25
.   4 |  7 |    10 |            49
.   5 | 11 |    16 |           121
.   6 | 13 |    20 |           169
.   7 | 17 |    28 |           289
.   8 | 19 |    31 |           361
.   9 | 23 |    39 |           529
.  10 | 29 |    48 |           841
.  11 | 31 |    51 |           961
.  12 | 37 |    65 |          1369
.  13 | 41 |    71 |          1681
.  14 | 43 |    75 |          1849
.  15 | 47 |    84 |          2209
.  16 | 53 |    94 |          2809
.  17 | 59 |   107 |          3481
.  18 | 61 |   110 |          3721
.  19 | 67 |   120 |          4489
.  20 | 71 |   129 |          5041
.  21 | 73 |   133 |          5329
.  22 | 79 |   145 |          6241
.  23 | 83 |   152 |          6889
.  24 | 89 |   163 |          7921
.  25 | 97 |   180 |          9409  .

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A258567, A000040, A001248, A001694, A258600, A258601, A258602, A258603.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a258599 = (+ 1) . fromJust . (`elemIndex` a258567_list) . a000040

With[{m = 60}, c = Select[Range[Prime[m]^2], Min[FactorInteger[#][[;; , 2]]] > 1 &]; 1 + Flatten[FirstPosition[c, #] & /@ (Prime[Range[m]]^2)]] (* Amiram Eldar, Feb 07 2023 *)

from math import isqrt
from sympy import prime, integer_nthroot, factorint
def A258599(n):
    m = prime(n)**2
    return int(sum(isqrt(m//k**3) for k in range(1, integer_nthroot(m, 3)[0]+1) if all(d<=1 for d in factorint(k).values()))) # Chai Wah Wu, Sep 10 2024

A001694(a(n)) = A001248(n) = prime(n)^2.
A001694(m) mod prime(n) > 0 for m < a(n).
Also smallest number m such that A258567(m) = prime(n):
A258567(a(n)) = A000040(n) and A258567(m) != A000040(n) for m < a(n).

A323333 The Euler phi function values of the powerful numbers, A000010(A001694(n)).

Original entry on oeis.org

1, 2, 4, 6, 8, 20, 18, 16, 12, 42, 32, 24, 54, 40, 36, 110, 100, 64, 48, 156, 84, 80, 72, 120, 162, 128, 96, 272, 108, 294, 342, 168, 160, 144, 252, 220, 200, 256, 506, 192, 500, 216, 360, 312, 486, 336, 320, 812, 288, 240, 930, 440, 324, 400, 512, 660, 600

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The sum of the reciprocals of all the terms of this sequence is Murata's constant Product_{p prime}(1 + 1/(p-1)^2) (A065485).

Sequence is injective: no value occurs more than once. - Amiram Eldar and Antti Karttunen, Sep 30 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, A property of Dedekind's psi-function, Proceedings of the American Mathematical Society, Vol. 12, No. 6 (1961), p. 996.

Cf. A000010, A001694, A002618 (a subsequence), A065485, A082695, A112526, A323332.

EulerPhi /@ Join[{1}, Select[Range@ 1200, Min@ FactorInteger[#][[All, 2]] > 1 &]] (* after Harvey P. Dale at A001694 *)

lista(nn) = apply(x->eulerphi(x), select(x->ispowerful(x), vector(nn, k, k))); \\ Michel Marcus, Jan 11 2019

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

cp's OEIS Frontend

A286708 Powerful numbers (A001694) that are not prime powers (A000961).

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A076445 The smaller of a pair of powerful numbers (A001694) that differ by 2.

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A076871 Sum of two powerful numbers (definition (1), A001694).

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A320966 Powerful numbers A001694 divisible by a cube > 1.

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A085253 Numbers having no representation as sum of two powerful numbers (A001694).

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Mathematica

Formula

A119241 Number of powerful numbers (A001694) between consecutive squares n^2 and (n+1)^2.

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Mathematica

Python

Formula

A180114 a(n) = sigma(A001694(n)), sum of divisors of the powerful number A001694(n).

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