A045542 Sub-perfect powers: perfect powers (squares, cubes etc.) minus 1.

3, 7, 8, 15, 24, 26, 31, 35, 48, 63, 80, 99, 120, 124, 127, 143, 168, 195, 215, 224, 242, 255, 288, 323, 342, 360, 399, 440, 483, 511, 528, 575, 624, 675, 728, 783, 840, 899, 960, 999, 1023, 1088, 1155, 1224, 1295, 1330, 1368, 1443, 1520, 1599, 1680, 1727

Offset

1,1

Comments

Goldbach showed that Sum 1/a(n) = 1, see A214390, A214391.

The only primes in the sequence are 3,7,31,127,... the Mersenne primes (A000668). - Zak Seidov, Dec 08 2011

Repdigits of two or more digits, interpreted in the smallest possible base. E.g., the smallest base for 222 is 3, 222 in base 3 is 26, and 26 is in the sequence. - Franklin T. Adams-Watters, Aug 11 2014

References

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd edition, p. 66.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

L. Bibiloni, P. Viader, and J. Paradis, On a Series of Goldbach and Euler, Amer. Math. Monthly, 113 (2006), pp. 206-220.

Formula

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

Mathematica

f[upto_] := Union[Flatten[Table[n^pwr - 1, {pwr, 2, Log[2, upto+1]}, {n, 2, (upto+1)^(1/pwr)}]]]; f[1763] (* Zak Seidov, Dec 08 2011 *)
Select[Range[2000], GCD@@FactorInteger[#][[All, 2]]>1&]-1 (* Harvey P. Dale, Jan 31 2023 *)

Prog

(Haskell)
a045542 n = a045542_list !! (n-1)
a045542_list = map (subtract 1) $ tail a001597_list
-- Reinhard Zumkeller, Jul 15 2012
(PARI) list(lim)=my(v=List()); for(e=2, logint(lim\=1, 2), for(k=2, sqrtnint(lim, e), listput(v, k^e-1))); Set(v) \\ Charles R Greathouse IV, Aug 26 2015

Crossrefs

Cf. A000668, A001597.

Sequence in context: A105173 A309197 A076683 * A249435 A192120 A031404

Adjacent sequences: A045539 A045540 A045541 * A045543 A045544 A045545

Keyword

easy,nice,nonn

Author

William M. Glasgow (billg(AT)wakely.com)

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jun 11 2002

Status

approved