A045542 - OEIS

%I #36 Jan 31 2023 16:25:22

%S 3,7,8,15,24,26,31,35,48,63,80,99,120,124,127,143,168,195,215,224,242,

%T 255,288,323,342,360,399,440,483,511,528,575,624,675,728,783,840,899,

%U 960,999,1023,1088,1155,1224,1295,1330,1368,1443,1520,1599,1680,1727

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

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

%C The only primes in the sequence are 3,7,31,127,... the Mersenne primes (A000668). - _Zak Seidov_, Dec 08 2011

%C 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

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

%H Reinhard Zumkeller, <a href="/A045542/b045542.txt">Table of n, a(n) for n = 1..10000</a>

%H L. Bibiloni, P. Viader, and J. Paradis, <a href="https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/bibiloni206.pdf">On a Series of Goldbach and Euler</a>, Amer. Math. Monthly, 113 (2006), pp. 206-220.

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

%t 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 *)

%t Select[Range[2000],GCD@@FactorInteger[#][[All,2]]>1&]-1 (* _Harvey P. Dale_, Jan 31 2023 *)

%o (Haskell)

%o a045542 n = a045542_list !! (n-1)

%o a045542_list = map (subtract 1) $ tail a001597_list

%o -- _Reinhard Zumkeller_, Jul 15 2012

%o (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

%Y Cf. A000668, A001597.

%K easy,nice,nonn

%O 1,1

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

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