A062838 -id:A062838 - cp's OEIS frontend

A376173 Numbers whose prime factorization has an odd minimum exponent that is larger than 1.

8, 27, 32, 125, 128, 216, 243, 343, 432, 512, 648, 864, 1000, 1331, 1728, 1944, 2000, 2048, 2187, 2197, 2744, 3125, 3375, 3456, 4000, 4913, 5000, 5488, 5832, 6859, 6912, 7776, 8000, 8192, 9261, 10125, 10648, 10976, 12167, 13824, 15552, 16000, 16807, 16875, 17496

Offset: 1

Amiram Eldar, Sep 13 2024

Numbers k such that A051904(k) is odd and larger than 1.

The minimum exponent in the prime factorization of 1 is considered to be A051904(1) = 0, and therefore 1 is not a term of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Subsequence of A036966.
Complement of A376172 within A001694.
Subsequences: A030078, A062838 \ {1}.
Cf. A051904.

seq[lim_] := Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}], # > 1 && OddQ[Min[FactorInteger[#][[;; , 2]]]] &]; seq[10000]

is(k) = {my(f = factor(k), e = f[,2]); #e && (ispowerful(f) && vecmin(e) % 2);}

Sum_{n>=1} 1/a(n) = -1 + Sum_{k>=2} (-1)^k * s(k) = 0.2379998147971880759099..., where s(k) = Product_{p prime} (1 + 1/(p^k*(p-1))).

A079712 Numbers m such that bigomega(m) = 3*omega(m).

Original entry on oeis.org

1, 8, 27, 96, 125, 144, 160, 216, 224, 324, 343, 352, 400, 416, 486, 544, 608, 736, 784, 928, 992, 1000, 1184, 1215, 1312, 1331, 1376, 1504, 1696, 1701, 1888, 1920, 1936, 1952, 2025, 2144, 2197, 2272, 2336, 2500, 2528, 2656, 2673, 2688, 2704, 2744, 2848

Offset: 1

Benoit Cloitre, Jan 31 2003

nonn

A cube k is a term iff k belongs to A062838; in this case, k = p_1^3 * p_2^3 *...* p_r^3 and bigomega(k) = 3*omega(k) = 3*r. - Bernard Schott, May 09 2022

Enrique Pérez Herrero, Table of n, a(n) for n = 1..2000

Cf. A067801, A062838 (subsequence of cubes).

Select[Range[3000], PrimeOmega[#] == 3*PrimeNu[#] &] (* Amiram Eldar, Jun 29 2022 *)

is(n)=my(f=factor(n)[,2]); vecsum(f)==3*#f \\ Charles R Greathouse IV, Oct 16 2015

A228532 a(n) = order of the torsion subgroup of the elliptic curve y^2 = x^3 + n if n is a sixth-power-free integer, otherwise -1.

Original entry on oeis.org

6, 0, 0, 3, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0

Offset: 1

Arkadiusz Wesolowski, Aug 24 2013

Dale Husemoller, Elliptic Curves, Graduate Texts in Mathematics 111, Springer-Verlag, 2004, pp. 35-37.

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
Wikipedia, Torsion subgroup

Cf. A000290, A062838, A179126.

a(A062838(n)) = 2 for n > 1.

a(A179126(n)) = 3.

cp's OEIS Frontend

A376173 Numbers whose prime factorization has an odd minimum exponent that is larger than 1.

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A079712 Numbers m such that bigomega(m) = 3*omega(m).

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A228532 a(n) = order of the torsion subgroup of the elliptic curve y^2 = x^3 + n if n is a sixth-power-free integer, otherwise -1.

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