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A005934 Highly powerful numbers: numbers with record value of the product of the exponents in prime factorization (A005361).
(Formerly M3333)
22

%I M3333

%S 1,4,8,16,32,64,128,144,216,288,432,864,1296,1728,2592,3456,5184,7776,

%T 10368,15552,20736,31104,41472,62208,86400,108000,129600,194400,

%U 216000,259200,324000,432000,518400,648000,972000,1296000,1944000,2592000

%N Highly powerful numbers: numbers with record value of the product of the exponents in prime factorization (A005361).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Amiram Eldar, <a href="/A005934/b005934.txt">Table of n, a(n) for n = 1..609</a> (terms 1..300 from T. D. Noe)

%H R. K. Guy, <a href="/A001599/a001599_1.pdf">Letter to N. J. A. Sloane with attachment, Jun. 1991</a>

%H G. E. Hardy and M. V. Subbarao, <a href="/A005934/a005934.pdf">Highly powerful numbers</a>, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)

%H C. B. Lacampagne and J. L. Selfridge, <a href="https://doi.org/10.1090/S0002-9939-1984-0740165-6">Large highly powerful numbers are cubeful</a>, Proc. Amer. Math. Soc., Vol. 91, No. 2 (1984), pp. 173-181.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Highly_powerful_number">Highly powerful number</a>

%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>

%F 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.

%t 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 *)

%t With[{s = Array[Times @@ FactorInteger[#][[All, -1]] &, 3*10^6]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* _Michael De Vlieger_, Oct 15 2017 *)

%o (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",")))}

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

%Y Cf. A085629.

%K nonn,nice,easy

%O 1,2

%A _N. J. A. Sloane_

%E Hardy and Subbarao give an extensive table.

%E Corrected and extended by _Jason Earls_, Jul 10 2003

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Last modified April 5 06:13 EDT 2020. Contains 333238 sequences. (Running on oeis4.)