A085232 -id:A085232 - cp's OEIS frontend

A085231 Numbers k in whose canonical factorization the power of the smallest prime factor is greater than the power of the greatest prime factor.

12, 24, 40, 45, 48, 56, 63, 80, 96, 112, 120, 135, 144, 160, 168, 175, 176, 189, 192, 208, 224, 240, 275, 280, 288, 297, 315, 320, 325, 336, 351, 352, 360, 384, 405, 416, 425, 448, 459, 475, 480, 504, 513, 528, 539, 544, 560, 567, 575, 576, 608, 621, 624

Offset: 1

Reinhard Zumkeller, Jun 22 2003

p*a(n) is a term for all primes p with A020639(a(n)) < p < A006530(a(n)).

			The canonical factorization of 240 is 2^4 * 3 * 5. 2^4 = 16 > 5, therefore 240 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A006530, A020639, A028233, A053585, A085232, A363122.
A085233 is a subsequence.
Subsequence of A102749.

pfgQ[n_]:=Module[{fe=#[[1]]^#[[2]]&/@FactorInteger[n]},fe[[1]]>fe[[-1]]]; Select[Range[700],pfgQ] (* Harvey P. Dale, Dec 11 2017 *)

A028233(a(n)) > A053585(a(n)).

Edited by Peter Munn, Jun 01 2025

A379094 Numbers whose factors in the canonical prime factorization neither increase weakly nor decrease weakly.

Original entry on oeis.org

60, 84, 90, 120, 126, 132, 156, 168, 180, 204, 228, 240, 252, 264, 270, 276, 280, 300, 312, 315, 336, 348, 350, 360, 372, 378, 408, 420, 440, 444, 456, 480, 492, 495, 504, 516, 520, 525, 528, 540, 550, 552, 560, 564, 585, 588, 594, 600, 616, 624, 630, 636, 650

Offset: 1

Peter Luschny, Dec 17 2024

nonn

A379097 is a subsequence.
From Michael De Vlieger, Dec 18 2024: (Start)
Proper subset of A126706.
Smallest powerful number is a(314) = 2700. (End)

			60 is a term because the factors in the canonical prime factorization are [4, 3, 5], a list that is neither increasing nor decreasing.
Primorials (A002110) are not terms of this sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A053585, A057714, A085232, A140831, A379097.

with(ArrayTools):
fact := n -> local p; [seq(p[1]^p[2], p in ifactors(n)[2])]:
isA379094 := proc(n) local f; f := fact(n);
is(not IsMonotonic(f, direction=decreasing, strict=false) and not IsMonotonic(f, direction=increasing, strict=false)) end:
select(isA379094, [seq(1..650)]);

Select[Range[650], Function[f, NoneTrue[{Sort[f], ReverseSort[f]}, # == f &]][Power @@@ FactorInteger[#]] &] (* Michael De Vlieger, Dec 18 2024 *)

is_a379094(n) = my(C=apply(x->x[1]^x[2], Vec(factor(n)~))); vecsort(C)!=C && vecsort(C,,4)!=C \\ Hugo Pfoertner, Dec 18 2024

A085234 (Greatest power of smallest prime factor of n) < square root(n).

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 26, 28, 30, 33, 34, 35, 36, 38, 39, 42, 44, 46, 50, 51, 52, 54, 55, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 114, 115

Offset: 1

Reinhard Zumkeller, Jun 22 2003

nonn

A028233(a(n))^2 < a(n);

a(n)=A085232(n) for n<69: a(69)=120, A085232(69)=122=a(70).

isA085234 := proc(n)
    if A028233(n)^2 < n then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 115 do
    if isA085234(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Jul 09 2016

okQ[n_] := Power @@ FactorInteger[n][[1]] < Sqrt[n]; Select[Range[120], okQ] (* Jean-François Alcover, Feb 13 2018 *)

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A085231 Numbers k in whose canonical factorization the power of the smallest prime factor is greater than the power of the greatest prime factor.

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A379094 Numbers whose factors in the canonical prime factorization neither increase weakly nor decrease weakly.

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A085234 (Greatest power of smallest prime factor of n) < square root(n).

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