The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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). (Formerly M3325 N1335) 358
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 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. 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. C. K. Caldwell, Powerful Numbers. Tsz Ho Chan, Arithmetic Progressions Among Powerful Numbers, J. Int. Seq., Vol. 26 (2023), Article 23.1.1. J.-M. de Koninck, N. Doyon, and F. Luca, Powerful Values of Quadratic Polynomials, J. Int. Seq. 14 (2011), Article 11.3.3. P. Erdős and G. 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] S. W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (1970), 848-852. K. Schneider, PlanetMath.org, Squarefull Number. V. 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. 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 EXAMPLE 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 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 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 *) PROG (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 (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 (PARI) is=ispowerful \\ Charles R Greathouse IV, Nov 13 2012 (Haskell) a001694 n = a001694_list !! (n-1) a001694_list = filter ((== 1) . a112526) [1..] -- Reinhard Zumkeller, Nov 30 2012 (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 (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 (Sage) sloane.A001694.list(54) # Peter Luschny, Feb 08 2015 CROSSREFS 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). Sequence in context: A339497 A348121 A080366 * A317102 A157985 A001597 Adjacent sequences: A001691 A001692 A001693 * A001695 A001696 A001697 KEYWORD nonn,nice,easy,changed AUTHOR N. J. A. Sloane EXTENSIONS More terms from Henry Bottomley, Mar 16 2000 Definition expanded by Jonathan Sondow, Jan 03 2016 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 15 15:30 EDT 2024. Contains 375938 sequences. (Running on oeis4.)