A076871 -id:A076871 - cp's OEIS frontend

A001694 Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000

Offset: 1

N. J. A. Sloane

Numbers of the form a^2*b^3, a >= 1, b >= 1.
In other words, if the prime factorization of n is Product_k p_k^e_k then all e_k are greater than 1.
Numbers n such that Sum_{d|n} phi(d)*phi(n/d)*mu(d) > 0; places of nonzero A300717. - Benoit Cloitre, Nov 30 2002
This sequence is closed under multiplication. The primitive elements are A168363. - Franklin T. Adams-Watters, May 30 2011
Complement of A052485. - Reinhard Zumkeller, Sep 16 2011
The number of terms less than or equal to 10^k beginning with k = 0: 1, 4, 14, 54, 185, 619, 2027, 6553, 21044, ...: A118896. - Robert G. Wilson v, Aug 11 2014
a(10^n): 1, 49, 3136, 253472, 23002083, 2200079025, 215523459072, 21348015504200, 2125390162618116, ... . - Robert G. Wilson v, Aug 15 2014
a(m) mod prime(n) > 0 for m < A258599(n); a(A258599(n)) = A001248(n) = prime(n)^2. - Reinhard Zumkeller, Jun 06 2015
From Des MacHale, Mar 07 2021: (Start)
A number m is powerful if and only if |R/Z(R)| = m, for some finite non-commutative ring R.
A number m is powerful if and only if |G/Z(G)| = m, for some finite nilpotent class two group G (Reference Aine Nishe). (End)
Numbers n such that Sum_{k=1..n} phi(gcd(n,k))*mu(gcd(n,k)) > 0. - Richard L. Ollerton, May 09 2021

			1 is a term because for every prime p that divides 1, p^2 also divides 1.
2 is not a term since 2 divides 2 but 2^2 does not.
4 is a term because 2 is the only prime that divides 4 and 2^2 does divide 4. - _N. J. A. Sloane_, Jan 16 2022

G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307.
Aleksandar Ivić, The Riemann Zeta-Function, Wiley, NY, 1985, see p. 407.
Richard A. Mollin, Quadratics, CRC Press, 1996, Section 1.6.
Aine NiShe, Commutativity and Generalisations in Finite Groups, Ph.D. Thesis, University College Cork, 2000.
Paulo Ribenboim, Meine Zahlen, meine Freunde, 2009, Springer, 9.1 Potente Zahlen, pp. 241-247.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 54, exercise 10 (in the third edition 2015, p. 63, exercise 70).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe, terms 1001..5000 from G. C. Greubel)
Paul T. Bateman and Emil Grosswald, On a theorem of Erdős and Szekeres, Illinois J. Math. 2:1 (1958), pp. 88-98.
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
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.
Chris K. Caldwell, Powerful Numbers.
Tsz Ho Chan, Arithmetic Progressions Among Powerful Numbers, J. Int. Seq., Vol. 26 (2023), Article 23.1.1.
Tsz Ho Chan, A note on three consecutive powerful numbers, arXiv:2503.21485 [math.NT], 2025.
Jean-Marie De Koninck, Nicolas Doyon, and Florian Luca, Powerful Values of Quadratic Polynomials, J. Int. Seq. 14 (2011), Article 11.3.3.
Paul Erdős and George Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), 7 (1935), 95-102. [Zahlen i-ter Art, p. 101]
Solomon W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (1970), 848-852.
K. Schneider, PlanetMath.org, Squarefull Number.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.
D. Suryanarayana and R. Sita Rama Chandra Rao, The distribution of square-full integers, Ark. Mat., Volume 11, Number 1-2 (1973), 195-201.
Eric Weisstein's World of Mathematics, Powerful Number.
Eric Weisstein's World of Mathematics, Squareful.
Wikipedia, Powerful number.
Index entries for sequences related to powerful numbers.

Disjoint union of A062503 and A320966.
Cf. A007532 (Powerful numbers, definition (2)), A005934, A005188, A003321, A014576, A023052 (Powerful numbers, definition (3)), A046074, A013929, A076871, A258599, A001248, A112526, A168363, A224866, A261883, A300717.
Cf. A052485 (complement), A076446 (first differences), A376361, A376362.

a001694 n = a001694_list !! (n-1)
a001694_list = filter ((== 1) . a112526) [1..]
-- Reinhard Zumkeller, Nov 30 2012

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:
A001694 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if isA001694(a) then return a; end if; end do; end if; end proc:
seq(A001694(n),n=1..20) ; # R. J. Mathar, Jun 07 2011

Join[{1}, Select[ Range@ 1250, Min@ FactorInteger[#][[All, 2]] > 1 &]]
(* Harvey P. Dale, Sep 18 2011; modified by Robert G. Wilson v, Aug 11 2014 *)
max = 10^3; Union@ Flatten@ Table[a^2*b^3, {b, max^(1/3)}, {a, Sqrt[max/b^3]}] (* Robert G. Wilson v, Aug 11 2014 *)
nextPowerfulNumber[n_] := Block[{r = Range[ Floor[1 + n^(1/3)]]^3}, Min@ Select[ Sort[ r*Floor[1 + Sqrt[n/r]]^2], # > n &]]; NestList[ nextPowerfulNumber, 1, 55] (* Robert G. Wilson v, Aug 16 2014 *)

isA001694(n)=n=factor(n)[,2];for(i=1,#n,if(n[i]==1,return(0)));1 \\ Charles R Greathouse IV, Feb 11 2011

list(lim,mn=2)=my(v=List(),t); for(m=1,sqrtnint(lim\1,3), t=m^3; for(n=1,sqrtint(lim\t), listput(v,t*n^2))); Set(v) \\ Charles R Greathouse IV, Jul 31 2011; edited Sep 22 2015

is=ispowerful \\ Charles R Greathouse IV, Nov 13 2012

from sympy import factorint
A001694 = [1]+[n for n in range(2,10**6) if min(factorint(n).values()) > 1]
# Chai Wah Wu, Aug 14 2014

from math import isqrt
from sympy import mobius, integer_nthroot
def A001694(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 bisection(f,n,n) # Chai Wah Wu, Sep 09 2024

sloane.A001694.list(54) # Peter Luschny, Feb 08 2015

A112526(a(n)) = 1. - Reinhard Zumkeller, Sep 16 2011
Bateman & Grosswald prove that there are zeta(3/2)/zeta(3) x^{1/2} + zeta(2/3)/zeta(2) x^{1/3} + O(x^{1/6}) terms up to x; see section 5 for a more precise error term. - Charles R Greathouse IV, Nov 19 2012
a(n) = A224866(n) - 1. - Reinhard Zumkeller, Jul 23 2013
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6). - Ivan Neretin, Aug 30 2015
Sum_{n>=1} 1/a(n)^s = zeta(2*s)*zeta(3*s)/zeta(6*s), s > 1/2 (Golomb, 1970). - Amiram Eldar, Oct 02 2022

More terms from Henry Bottomley, Mar 16 2000

Definition expanded by Jonathan Sondow, Jan 03 2016

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.

A085252 Number of ways to write n as sum of two powerful numbers (A001694).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jun 23 2003

nonn

			a(81) = 2: 81 = 9 + 72 = A001694(4) + A001694(12) = 32 + 49 = A001694(8) + A001694(10).

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

Cf. A001694, A076871, A085253, A085254, A085255.

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

a(A085253(n)) = 0.
a(A076871(n)) > 0.
a(A085254(n)) = 1.
a(A085255(n)) > 1.

A085254 Numbers having a unique representation as sum of two powerful numbers (A001694).

Original entry on oeis.org

2, 5, 8, 9, 10, 12, 13, 16, 18, 20, 24, 25, 26, 28, 29, 31, 32, 34, 35, 37, 43, 44, 45, 48, 53, 54, 58, 59, 61, 63, 64, 74, 82, 88, 90, 91, 96, 98, 99, 100, 101, 106, 112, 121, 122, 124, 126, 127, 128, 134, 135, 140, 141, 146, 149, 150, 155, 161, 162, 169, 171

Offset: 1

Reinhard Zumkeller, Jun 23 2003

nonn

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

Cf. A001694, A076871, A085252, A085253, A085255.

With[{m = 180}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; Position[BinCounts[Select[Plus @@@ Union[Sort /@ Tuples[pow, {2}]], # <= m &], {1, m, 1}], 1] // Flatten] (* Amiram Eldar, Jan 30 2023 *)

A085252(a(n)) = 1.

A085255 Numbers having at least two representations as a sum of two powerful numbers (A001694).

Original entry on oeis.org

17, 33, 36, 40, 41, 50, 52, 57, 65, 68, 72, 73, 76, 80, 81, 85, 89, 97, 104, 108, 109, 113, 116, 117, 125, 129, 130, 132, 133, 136, 137, 144, 145, 148, 152, 153, 157, 160, 164, 170, 172, 177, 180, 185, 189, 193, 197, 200, 201, 204, 205, 208, 209, 216, 221

Offset: 1

Reinhard Zumkeller, Jun 23 2003

nonn

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

Cf. A001694, A076871, A085252, A085253, A085254.

With[{m = 222}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; Position[BinCounts[Select[Plus @@@ Union[Sort /@ Tuples[pow, {2}]], # <= m &], {1, m, 1}], ?(# > 1 &)] // Flatten] (* _Amiram Eldar, Jan 30 2023 *)

A085252(a(n)) > 1.

A056828 Numbers that are not the sum of at most three powerful (1) numbers.

Original entry on oeis.org

7, 15, 23, 87, 111, 119

Offset: 1

Henry Bottomley, Aug 30 2000

Mollin and Walsh conjectured that there are no further terms.
Heath-Brown proved that the sequence is finite.
No other terms less than 40000000. - Paul.Jobling(AT)WhiteCross.com, May 14 2001

			Smallest powerful numbers are 1, 4, 8, 9, 16, 25,... so 7, 15 and 23 are not the sum of one, two or three of them.

D. R. Heath-Brown, "Ternary Quadratic Forms and Sums of Three Square-Full Numbers." In Séminaire de Théorie des Nombres, Paris 1986-87 (Ed. C. Goldstein). Boston, MA: Birkhauser, pp. 137-163, 1988.

R. A. Mollin and P. G. Walsh, On Powerful Numbers, Intern. J. Math. and Math. Sci, 9:801-806, 1986.
Eric Weisstein's World of Mathematics, Powerful Number.

Cf. A001694, A076871, A113505.

A076872 a(n) = number of numbers <= n that are the sum of two squarefull numbers.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 3, 4, 5, 5, 6, 7, 7, 7, 8, 9, 10, 10, 11, 11, 11, 11, 12, 13, 14, 14, 15, 16, 16, 17, 18, 19, 20, 21, 22, 23, 23, 23, 24, 25, 25, 26, 27, 28, 28, 28, 29, 29, 30, 30, 31, 32, 33, 33, 33, 34, 35, 36, 36, 37, 37, 38, 39, 40, 40, 40, 41, 41, 41, 41, 42, 43, 44, 44

Offset: 1

N. J. A. Sloane, Nov 25 2002

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

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

Cf. A001694, A076871.

With[{m = 120}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; Accumulate @ BinCounts[Select[Union[Plus @@@ Tuples[pow, {2}]], # <= m &], {1, m, 1}]] (* Amiram Eldar, Feb 12 2023 *)

More terms from Vladeta Jovovic, Nov 26 2002

A261883 Decimal expansion of 1 - 2^(-1/3).

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, Sep 04 2015

Blomer shows that there are x/log^k x powerful numbers up to x, where k = 0.20629947... is this constant.

			0.20629947401590026262414718036384586980425333605007349509585711908739174...

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.
Index entries for algebraic numbers, degree 3.
Index entries for sequences related to powerful numbers

Cf. A076871, A001694.

First@RealDigits[N[1 - 2^(-1/3), 120]] (* Michael De Vlieger, Sep 04 2015 *)
RealDigits[1-1/Surd[2,3],10,120][[1]] (* Harvey P. Dale, Dec 05 2023 *)

PARI
```
1 - 2^(-1/3)
```

A331802 Integers having no representation as sum of two nonsquarefree numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 14, 15, 19, 23

Offset: 1

Bernard Schott, Feb 23 2020

This sequence is finite with 14 terms and 23 is the largest term (see Prime Curios link); a proof can be found in comments of A331801.

			With the two smallest nonsquarefree numbers 4 and 8, it is not possible to get 1, 2, 3, 4, 5, 6, 7, 9, 10 and 11 as sum of two nonsquarefree numbers.

G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 23

Cf. A005117 (squarefree), A013929 (nonsquarefree), A331801 (complement).
Cf. A000404 (sum of 2 nonzero squares), A018825 (not the sum of 2 nonzero squares).
Cf. A001694 (squareful), A052485 (not squareful), A076871 (sum of 2 squareful), A085253 (not the sum of 2 squareful).

max = 25; Complement[Range[max], Union @ Select[Total /@ Tuples[Select[Range[max], !SquareFreeQ[#] &], 2], # <= max &]] (* Amiram Eldar, Feb 24 2020 *)

A143813 Sum of two powerful numbers greater than 1.

Original entry on oeis.org

8, 12, 13, 16, 17, 18, 20, 24, 25, 29, 31, 32, 33, 34, 35, 36, 40, 41, 43, 44, 45, 48, 50, 52, 53, 54, 57, 58, 59, 61, 63, 64, 65, 68, 72, 73, 74, 76, 80, 81, 85, 88, 89, 90, 91, 96, 97, 98, 99, 100, 104, 106, 108, 109, 112, 113, 116, 117, 121, 124, 125, 127

Offset: 1

Charles R Greathouse IV, Jul 06 2011

nonn

A076871 is the primary sequence.

			8 = 4 + 4, 12 = 4 + 8, 13 = 4 + 9.

isP(n)={
  n>3 && vecmin(factor(n)[,2])>1
};
sumset(a,b)={
  my(c=vector(#a*#b));
  for(i=1,#a,
    for(j=1,#b,
      c[(i-1)*#b+j]=a[i]+b[j]
    )
  );
  vecsort(c,,8)
};
upto(lim)={
  my(v=select(isP, vector(floor(lim),i,i)));
  select(n->n<=lim, sumset(v,v))
};

cp's OEIS Frontend

A001694 Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).

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

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A085252 Number of ways to write n as sum of two powerful numbers (A001694).

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A085254 Numbers having a unique representation as sum of two powerful numbers (A001694).

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A085255 Numbers having at least two representations as a sum of two powerful numbers (A001694).

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A056828 Numbers that are not the sum of at most three powerful (1) numbers.

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A076872 a(n) = number of numbers <= n that are the sum of two squarefull numbers.

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A261883 Decimal expansion of 1 - 2^(-1/3).