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).

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

Author

Kyle Schalm (kschalm(AT)math.utexas.edu), Jan 07 2007

Keywords

• nonn

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.

Examples

			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.
		

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.

Links

• 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), II.

Crossrefs

Cf. A000522, A127015, A138761.

Programs

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 *)

Formula

A(a(n)) = A138761(n) = Sum_{k=0..a(n)} a(n)!/k! for n > 0. - Jonathan Sondow, Jun 12 2009