A072413 -id:A072413 - cp's OEIS frontend

A072412 Numbers k such that the LCM of exponents in the prime factorization of k does not equal the largest exponent.

72, 108, 200, 288, 360, 392, 432, 500, 504, 540, 600, 648, 675, 756, 792, 800, 864, 936, 968, 972, 1125, 1152, 1176, 1188, 1224, 1323, 1350, 1352, 1368, 1372, 1400, 1404, 1440, 1500, 1568, 1656, 1800, 1836, 1944, 1960, 2000, 2016, 2052, 2088, 2160

Offset: 1

Labos Elemer, Jun 17 2002

nonn

This sequence differs from the Achilles numbers (A052486).

			k = 360 = 2*2*2*3*3*5, exponent set = {3,2,1}; LCM=6, max=3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005361, A051903, A051904, A052409, A052486, A072411, A072413, A072414.

Select[Range[2000], LCM @@ (e = FactorInteger[#][[;; , 2]]) != Max[e] &] (* Amiram Eldar, Jul 30 2022 *)

is(n)=my(f=factor(n)[,2]); n>9 && vecmax(f)!=lcm(f) \\ Charles R Greathouse IV, Oct 16 2015

A051903(a(n)) != A072411(a(n)).

A377817 Numbers that have more than one even exponent in their prime factorization.

Original entry on oeis.org

36, 100, 144, 180, 196, 225, 252, 300, 324, 396, 400, 441, 450, 468, 484, 576, 588, 612, 676, 684, 700, 720, 784, 828, 882, 900, 980, 1008, 1044, 1089, 1100, 1116, 1156, 1200, 1225, 1260, 1296, 1300, 1332, 1444, 1452, 1476, 1521, 1548, 1575, 1584, 1600, 1620, 1692, 1700, 1764, 1800

Offset: 1

Amiram Eldar, Nov 09 2024

Subsequence of A072413 and differs from it by not having the terms 216, 1000, 1080, 1512, ... .
Each term can be represented in a unique way as m * k^2, where m is an exponentially odd number (A268335) and k is a composite number that is coprime to m.
Numbers k such that A350388(k) is a square of a composite number (A062312 \ {1}).
The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p*(p+1))) * (1 + Sum_{p prime} 1/(p^2+p-1)) = 0.032993560887093165933... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of the union of A268335 and A377816.
Subsequence of A072413.
Cf. A062312, A162645, A350388.

Select[Range[1800], Count[FactorInteger[#][[;; , 2]], _?EvenQ] > 1 &]

is(k) = if(k == 1, 0, my(e = factor(k)[, 2]); #select(x -> !(x%2), e) > 1);

A273058 Numbers having pairwise coprime exponents in their canonical prime factorization.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Giuseppe Coppoletta, May 14 2016

nonn

The complement of A072413.

			36 is not a term because 36 = 2^2 * 3^2 and gcd(2,2) = 2 > 1.
360 is a term because 360 = 2^3 * 3^2 * 5 and gcd(3,2) = gcd(2,1) = 1.
10800 is not a term because 10800 = 2^4 * 3^3 * 5^2 and gcd(4,2) > 1

Giuseppe Coppoletta, Table of n, a(n) for n = 1..10000

Cf. A005361, A072411, A130091, A072413.

Select[Range@ 120, LCM @@ # == Times @@ # &@ Map[Last, FactorInteger@ #] &] (* Michael De Vlieger, May 15 2016 *)

is(n)=my(f=factor(n)[,2]); factorback(f)==lcm(f) \\ Charles R Greathouse IV, Jan 14 2017

def d(n):
    v=factor(n)[:]; L=len(v); diff=prod(v[j][1] for j in range(L)) - lcm([v[j][1] for j in range(L)])
    return diff
[k for k in (1..100) if d(k)==0]

A005361(a(n)) = A072411(a(n)).

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A072412 Numbers k such that the LCM of exponents in the prime factorization of k does not equal the largest exponent.

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A377817 Numbers that have more than one even exponent in their prime factorization.

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A273058 Numbers having pairwise coprime exponents in their canonical prime factorization.

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