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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, 3888000, 5184000

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OFFSET

1,2

COMMENTS

From Amiram Eldar, Jan 19 2026: (Start)

Numbers whose number of powerful divisors (A379545) increases to a record.

Numbers whose number of coreful divisors (A284318) increases to a record.

Analogous to highly composite numbers (A002182) with these two types of divisors counted by A005361. (End)

REFERENCES

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, p. 334.

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..1001 (terms below 2^200; terms 1..300 from T. D. Noe)

Richard 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)

Carole B. Lacampagne and John 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.

A005361(a(n)) = A036965(n). - Amiram Eldar, Jan 19 2026

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

Subsequence of A181800.

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

Cf. A002182, A085629, A284318, A379545.

Sequence in context: A111073 A298807 A353500 * A085629 A349111 A387732

Adjacent sequences: A005931 A005932 A005933 * A005935 A005936 A005937

KEYWORD

nonn,nice

AUTHOR