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URL: https://oeis.org/A005934

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A005934
Highly powerful numbers: numbers with record value of the product of the exponents in prime factorization (A005361).
32
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
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)
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.
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", ")))}
KEYWORD
nonn,nice
EXTENSIONS
Hardy and Subbarao give an extensive table.
Corrected and extended by Jason Earls, Jul 10 2003
STATUS
approved