A062739 Odd powerful numbers.

1, 9, 25, 27, 49, 81, 121, 125, 169, 225, 243, 289, 343, 361, 441, 529, 625, 675, 729, 841, 961, 1089, 1125, 1225, 1323, 1331, 1369, 1521, 1681, 1849, 2025, 2187, 2197, 2209, 2401, 2601, 2809, 3025, 3087, 3125, 3249, 3267, 3375, 3481, 3721, 3969, 4225

Offset

1,2

Comments

Smallest term of this sequence not also in A075109 is 675, followed by 1125. - Alonso del Arte, Nov 22 2011

References

R. K. Guy, Unsolved Problems in Number Theory, B16

Links

Zak Seidov, Table of n, a(n) for n = 1..10000

W. A. Sentance, Occurrences of consecutive odd powerful numbers, Amer. Math. Monthly, 88 (1981), 272-274.

Formula

It is not true that a(n) = A001694(2n-1).

Sum_{n>=1} 1/a(n) = (2/3) * Sum_{n>=1} 1/A001694(n) = 2*zeta(2)*zeta(3)/(3*zeta(6)) = (2/3) * A082695 = 1.2957309... - Amiram Eldar, Jun 23 2020

Example

Consecutive-odd examples from Sentance: {25,27},{70225,70227},{189750625,189750627}

Mathematica

Powerful[n_Integer] := (n ==1) || Min[Transpose[FactorInteger[n]][[2]]]>=2; Select[Range[5000], OddQ[ # ]&&Powerful[ # ]&] (* T. D. Noe, May 04 2006 *)
Join[{1}, Select[Range[3, 4301, 2], Min[FactorInteger[#][[All, 2]]]>1&]] (* Harvey P. Dale, Jan 08 2021 *)

Crossrefs

Cf. A001694, A060355, A060859, A060860, A082695.

Cf. A076445 (consecutive odd powerful numbers).

Sequence in context: A020210 A275196 A325373 * A075109 A319165 A319152

Adjacent sequences: A062736 A062737 A062738 * A062740 A062741 A062742

Keyword

nonn

Author

Labos Elemer, Jul 12 2001

Extensions

Checked by T. D. Noe, May 04 2006

Status

approved