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
Examples
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
References
- 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).
Links
- 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.
Crossrefs
Programs
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Haskell
a001694 n = a001694_list !! (n-1) a001694_list = filter ((== 1) . a112526) [1..] -- Reinhard Zumkeller, Nov 30 2012
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Maple
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
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Mathematica
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 *) -
PARI
isA001694(n)=n=factor(n)[,2];for(i=1,#n,if(n[i]==1,return(0)));1 \\ Charles R Greathouse IV, Feb 11 2011
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PARI
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
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PARI
is=ispowerful \\ Charles R Greathouse IV, Nov 13 2012
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Python
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
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Python
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
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Sage
sloane.A001694.list(54) # Peter Luschny, Feb 08 2015
Formula
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
Extensions
More terms from Henry Bottomley, Mar 16 2000
Definition expanded by Jonathan Sondow, Jan 03 2016
Comments