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

Author

Labos Elemer, Jul 12 2001

Keywords

• nonn

Comments

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

Examples

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

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.

Crossrefs

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

Cf. A076445 (consecutive odd powerful numbers).

Programs

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 *)

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

Extensions

Checked by T. D. Noe, May 04 2006