A062965 Positive numbers which are one less than a perfect square that is also another power.

15, 63, 80, 255, 624, 728, 1023, 1295, 2400, 4095, 6560, 9999, 14640, 15624, 16383, 20735, 28560, 38415, 46655, 50624, 59048, 65535, 83520, 104975, 117648, 130320, 159999, 194480, 234255, 262143, 279840, 331775, 390624, 456975, 531440, 614655

Offset

1,1

References

William Dunham, Euler: The Master of Us All, The Mathematical Association of America, Washington D.C., 1999, p. 65.

Leonhard Euler, "Variae observationes circa series infinitas," Opera Omnia, Ser. 1, Vol. 14, pp. 216-244.

Nicolao Fvss, "Demonstratio Theorematvm Qvorvndam Analyticorvm," Nova Acta Academiae Scientiarum Imperialis Petropolitanae, 8 (1794) 223-226.

Links

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

Leonard Euler, Variae observationes circa series infinitas.

Leonard Euler, Variae observationes circa series infinitas.

Eric Weisstein's World of Mathematics, Perfect Power.

Formula

From Terry D. Grant, Oct 25 2020: (Start)

a(n) = A001597(n+1)^2 - 1.

Sum_{k>=1} 1/a(k) = 7/4 - Pi^2/6 = 7/4 - zeta(2).

Sum_{k>=1} 1/(a(k)+1) = Sum_{k>=2} mu(k)*(1-zeta(2*k)).

(End)

Example

a(2) = 63 because the perfect square 64 = 8^2 = 4^3.

Mathematica

Take[ Select[ Range[ 2, 150 ], GCD@@(Last/@FactorInteger[ # ])>1& ]^2-1] (* corrected by Jon Maiga, Sep 28 2019 *)

Crossrefs

Cf. A037450, A062834, A062757, A001597.

Cf. A131605.

Sequence in context: A067572 A066584 A065915 * A157968 A141759 A335574

Adjacent sequences: A062962 A062963 A062964 * A062966 A062967 A062968

Keyword

nonn

Author

Jason Earls, Jul 16 2001

Extensions

More terms from Dean Hickerson, Jul 24 2001

Status

approved