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A005934
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Highly powerful numbers: numbers with record value of the product of the exponents in prime factorization (A005361).
(Formerly M3333)
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29
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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
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OFFSET
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1,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)
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FORMULA
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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.
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MATHEMATICA
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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 *)
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PROG
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(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", ")))}
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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Hardy and Subbarao give an extensive table.
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STATUS
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approved
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