A085231 -id:A085231 - cp's OEIS frontend

A057715 Numbers m = Product p_i^{e_i}, not a power of a prime, such that p_j^{e_j} > p_k^{e_k} for all p_j < p_k.

12, 24, 40, 45, 48, 56, 63, 80, 96, 112, 135, 144, 160, 175, 176, 189, 192, 208, 224, 275, 288, 297, 320, 325, 351, 352, 384, 405, 416, 425, 448, 459, 475, 513, 539, 544, 567, 575, 576, 608, 621, 637, 640, 675, 704, 720, 736, 768, 800, 832, 833, 864, 875

Offset: 1

Leroy Quet, Oct 24 2000

			720 is included because 720 = 2^4 * 3^2 * 5^1 and 2^4 > 3^2 > 5^1.

Lei Zhou, Table of n, a(n) for n = 1..10000

Subsequence of A085231, A097320, A126706.

Cf. A363063.

Select[Range[575], Greater @@ Power @@@ (fi = FactorInteger[#]) && Length[fi] > 1 &] (* Ray Chandler, Nov 06 2008 *)

Title clarified by Sean A. Irvine, Jun 24 2022 and Peter Munn, May 26 2025

A085232 In canonical prime factorization: power of smallest prime factor is less than power of greatest prime factor.

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)) < A053585(a(n));
p*a(n) is a term for all primes p with A020639(a(n))A006530(a(n));
a(n)=A057714(n-1) for n<28: a(28)=60, A057714(28-1)=62.

			60 = 2^2 * 3 * 5 with 2^2=4 < 5, therefore 60 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..2500

Cf. A085231.

spfQ[n_]:=Module[{fi=FactorInteger[n]},Length[fi]>1&&fi[[1,1]]^fi[[1,2]] < fi[[-1,1]]^fi[[-1,2]]]; Select[Range[120],spfQ] (* Harvey P. Dale, Jul 30 2018 *)

A085233 Numbers k such that k is divisible by a power of its smallest prime factor that exceeds its square root.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 24, 25, 27, 29, 31, 32, 37, 40, 41, 43, 45, 47, 48, 49, 53, 56, 59, 61, 63, 64, 67, 71, 73, 79, 80, 81, 83, 89, 96, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 135, 137, 139, 144, 149, 151, 157, 160

Offset: 1

Reinhard Zumkeller, Jun 22 2003

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A028233, A116882\{1} (even terms).

Subsequence of A085231.

Select[Range[160], #[[1]]^(2 #[[2]]) &@FactorInteger[#][[1]] > # &] (* Ivan Neretin, Dec 30 2015 *)

A028233(a(n))^2 > a(n).

Edited by Peter Munn, Jun 01 2025

cp's OEIS Frontend

A057715 Numbers m = Product p_i^{e_i}, not a power of a prime, such that p_j^{e_j} > p_k^{e_k} for all p_j < p_k.

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A085232 In canonical prime factorization: power of smallest prime factor is less than power of greatest prime factor.

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A085233 Numbers k such that k is divisible by a power of its smallest prime factor that exceeds its square root.

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