A005934 Highly powerful numbers: numbers with record value of the product of the exponents in prime factorization (A005361). (Formerly M3333)

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

1,2

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

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.

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 *)

Prog

(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", ")))}

Crossrefs

Cf. A036965, A001694, A007532, A005361, A005188, A003321, A014576, A023052, A046074.

Cf. A085629.

Sequence in context: A111073 A298807 A353500 * A085629 A349111 A307870

Adjacent sequences: A005931 A005932 A005933 * A005935 A005936 A005937

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

Hardy and Subbarao give an extensive table.

Corrected and extended by Jason Earls, Jul 10 2003

Status

approved