

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.


5



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
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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, 195236.


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^(kn): 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 A112133
Adjacent sequences: A177375 A177376 A177377 * A177379 A177380 A177381


KEYWORD

nonn


AUTHOR

Vladimir Shevelev, May 07 2010


STATUS

approved



