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

Author

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

Keywords

• easy
• nice
• nonn

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.

Crossrefs

Cf. A000668, A001597.

Programs

Haskell

a045542 n = a045542_list !! (n-1)
a045542_list = map (subtract 1) $ tail a001597_list
-- Reinhard Zumkeller, Jul 15 2012

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 *)

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

Formula

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

Extensions

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