A005934 Highly powerful numbers: numbers with record value of the product of the exponents in prime factorization (A005361).
1, 4, 8, 16, 32, 64, 128, 144, 216, 288, 432, 864, 1296, 1728, 2592, 3456, 5184, 7776, 10368, 15552, 20736, 31104, 41472, 62208, 86400, 108000, 129600, 194400, 216000, 259200, 324000, 432000, 518400, 648000, 972000, 1296000, 1944000, 2592000
Offset: 1
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Amiram Eldar, Table of n, a(n) for n = 1..609 (terms 1..300 from T. D. Noe)
- R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991.
- G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)
- C. B. Lacampagne and J. L. Selfridge, Large highly powerful numbers are cubeful, Proc. Amer. Math. Soc., Vol. 91, No. 2 (1984), pp. 173-181.
- Wikipedia, Highly powerful number.
- Index entries for sequences related to powerful numbers
Crossrefs
Programs
-
Mathematica
a = {1}; b = {1}; f[n_] := Times @@ Last /@ FactorInteger[n]; Do[If[f@ n > Max[b], And[AppendTo[b, f@ n], AppendTo[a, n]]], {n, 1000000}]; a (* Michael De Vlieger, Aug 28 2015 *) With[{s = Array[Times @@ FactorInteger[#][[All, -1]] &, 3*10^6]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Oct 15 2017 *) DeleteDuplicates[Table[{n,Times@@FactorInteger[n][[All,2]]},{n,26*10^5}],GreaterEqual[#1[[2]],#2[[2]]]&][[All,1]] (* Harvey P. Dale, May 13 2022 *) -
PARI
{prdex(n)=local(s,fac); s=1; fac=factor(n); for(k=1,matsize(fac)[1],s=s*fac[k,2]); return(s)} {hp(m)=local(rec); rec=0; for(n=1,m,if(prdex(n)>rec,rec=prdex(n); print1(n",")))}
Formula
For n = Product p_i^e_i, let b(n) = Product e_i; then n is highly powerful if b(n) sets a new record.
Extensions
Hardy and Subbarao give an extensive table.
Corrected and extended by Jason Earls, Jul 10 2003