OFFSET
1,1
COMMENTS
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
Euler (1744, written in 1737) published a proof that the sum of reciprocals of this sequence is 1 and stated that the proof was communicated to him by Goldbach. He also mentioned this result in his 1837 paper (written in 1732). Goldbach's letter was lost, but a related letter with the result was sent to Bernoulli in 1729 and was published in 1843. - Amiram Eldar, Oct 14 2025
REFERENCES
Bruce Burdick and Edward Sandifer, Fooling with an Euler Series, International Journal of Mathematics and Computer Science, Vol. 4, No. 1 (2009).
Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, 2nd edition, Addison-Wesley, Reading, MA, 1994, p. 66, exercise 35.
Gabriel Klambauer, Mathematical Analysis, M. Dekker Inc., New York, 1975, p. 120.
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Lluís Bibiloni, Pelegrí Viader, and Jaume Paradís, On a Series of Goldbach and Euler, Amer. Math. Monthly, Vol. 113, No. 3 (2006), pp. 206-220.
Eugène Catalan, Note sur la sommation de quelques séries, Journal de Mathématiques Pures et Appliquées, Serie 1, Volume 7 (1842), pp. 1-12.
Leonhard Euler, Methodus generalis summandi progressiones, Commentarii academiae scientiarum Petropolitanae, Vol. 6 (1738), pp. 68-97.
Leonhard Euler, Variae observationes circa series infinitas, Commentarii academiae scientiarum Petropolitanae, Vol. 9 (1744), pp. 160-188; reprinted in Opera Omnia, Series 1, Vol. 14, pp. 217-244.
Christian Goldbach, Lettre XLII, letter to Daniel Bernoulli, May 26 1729, in P. H. Fuss, Correspondance mathématique et physique de quelques céleèbres géomètres du XVIIIème siècle, Vol. II, St.-Pétersbourg, 1843.
Adolf P. Juškevič and Judith Kh. Kopelevič, Christian Goldbach 1690-1764, Birkhäuser Basel, 2012, p. 154.
Ed Sandifer, Goldbach's series, How Euler Did It, February 2005. [Wayback Machine link]
J. D. Shallit and Karel Zikan, Problem E 2999, The American Mathematical Monthly, Vol. 90, No. 5 (1983), p. 335; A Theorem of Goldbach, solution to Problem E 2999 by University of South Alabama Problem Group, ibid., Vol. 93, No. 5 (1986), pp. 402-403.
Wikipedia, Goldbach-Euler theorem.
Doron Zeilberger, A Proof of the Celebrated Goldbach's Theorem, 2006.
FORMULA
a(n) = A001597(n + 1) - 1.
a(n) = A216765(n) - 2. - Amiram Eldar, Oct 14 2025
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
(Python)
from sympy import mobius, integer_nthroot
from oeis_sequences.OEISsequences import bisection
def A045542(n): return bisection(lambda x:int(n+x+sum(mobius(k)*(integer_nthroot(x+1, k)[0]-1) for k in range(2, (x+1).bit_length()))), n, n) # Chai Wah Wu, Oct 21 2025
CROSSREFS
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