A177378 a(n) is the smallest prime p>2 such that there are 2*n or 2*n+1 positive integers m for which the exponents of 2 and p in the prime power factorization of m! are both powers of 2.

11, 13, 3, 29, 31, 251, 127, 509, 1021, 4091, 4093, 65519, 8191, 131063, 262133, 262139, 131071, 1048571, 524287, 8388593, 4194301, 67108837, 16777213, 67108861, 1073741789, 2147483587, 2147483629, 536870909

Offset

1,1

Links

Table of n, a(n) for n=1..28.

V. Shevelev, Compact integers and factorials, Acta Arithmetica 126 (2007), no. 3, 195-236.

Formula

For sufficiently large n, 2^n - 1 <= a(n) <= 2^ceiling(40*n/19). Let k >= n. Put g = g(n,k) = min{odd j >= 2^(k-n): 2^k - j is prime} and h(n) = min{k: k - n = floor(log_2(g))}. Then a(n) = 2^h(n) - g(n,h(n)).

Example

By the formula, for n=6, consider k >= 6. If k=6, then g(6,6) = 3, but 6 does not equal to 6 - floor(log_2(3)); if k=7, then g=15, but 6 does not equal to 7 - floor(log_2(15)); if k=8, then g=5 and we see that 6 = 8 - floor(log_2(5)). Therefore a(6) = 2^8 - 5 = 251.

Crossrefs

Cf. A000142, A050376, A169655, A169661, A177349, A177355, A177436, A177458, A177459, A177498.

Sequence in context: A262282 A034080 A107805 * A110059 A004476 A347522

Adjacent sequences: A177375 A177376 A177377 * A177379 A177380 A177381

Keyword

nonn

Author

Vladimir Shevelev, May 07 2010

Status

approved