login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A127014
a(n) = smallest k such that A(k) == 0 (mod 2^n), where A(0) = 1 and A(k) = k*A(k-1) + 1 = A000522(k).
2
1, 3, 3, 3, 19, 51, 115, 115, 115, 627, 627, 2675, 2675, 2675, 2675, 35443, 35443, 166515, 166515, 166515, 1215091, 3312243, 3312243, 3312243, 3312243, 36866675
OFFSET
1,2
COMMENTS
a(n+1) - a(n) = 2^n or 0; see A127015.
In the 2-adic integers, lim_{n->oo} a(n) = 11001110010100010100110001...; see A127015.
REFERENCES
N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta-Functions, 2nd ed., Springer, New York, 1996.
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
FORMULA
A(a(n)) = A138761(n) = Sum_{k=0..a(n)} a(n)!/k! for n > 0. - Jonathan Sondow, Jun 12 2009
EXAMPLE
A(0) = 1, A(1) = 2, A(2) = 5 and A(3) = 16 = 2^4, so a(1) = 1 and a(2) = a(3) = a(4) = 3. Also, A(19) = 330665665962404000 is the first A(k) divisible by 2^5, so a(5) = 19.
MATHEMATICA
a522[n_] := E Gamma[n + 1, 1];
a[1] = 1; a[n_] := a[n] = For[k = a[n - 1], True, k++, If[Mod[a522[k], 2^n] == 0, Print[n, " ", k]; Return[k]]];
Table[a[n], {n, 1, 17}] (* Jean-François Alcover, Feb 20 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Kyle Schalm (kschalm(AT)math.utexas.edu), Jan 07 2007
STATUS
approved