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A125294 Numerator of Sum[ k^2, {k,1,n} ] / Product[ k^2, {k,1,n} ]. 1
1, 5, 7, 5, 11, 91, 1, 17, 19, 11, 23, 13, 1, 29, 31, 17, 1, 703, 1, 41, 43, 23, 47, 1, 1, 53, 1, 29, 59, 1891, 1, 1, 67, 1, 71, 2701, 1, 1, 79, 41, 83, 43, 1, 89, 1, 47, 1, 97, 1, 101, 103, 53, 107, 109, 1, 113, 1, 59, 1, 61, 1, 1, 127, 1, 131, 67, 1, 137, 139, 71, 1, 73, 1, 149 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

All a(n) are either 1, semiprime or prime. a(n) = 1 for n = 1 and n = {7,13,17,19,24,25,27,31,32,34,37,38,43,45,47,49,...} = A067656 Numbers n such that n!*B(2n) is an integer, where B(2n) are the Bernoulli numbers. p divides a(p-1) for prime p>3. p divides a((p-1)/2) for prime p>3.

a(p-1) = p*(2p-1) is a semiprime hexagonal number for prime p = {7,19,31,37,79,97,139,157,199,211,229,271,307,331,337,367,379,439,499,...} = A005382(n) for n>2, where A005382(n) are the numbers n such that n and 2n-1 are primes.

a(p-1) = p for prime p = {5, 11, 13, 17, 23, 29, 41, 43, 47, 53, 59, 61, 67, 71, 73, 83, 89, ...} = Primes that do not belong to A005382(n). a((p-1)/2) = p for prime p = {5, 7, 11, 17, 19, 23, 29, 31, 41, 43, 47, 53, 59, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 259, 271, 281, 283, 293, 307, 311, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 401, ...} that is apparently a union of {5} and A034849(n).

LINKS

Table of n, a(n) for n=1..74.

FORMULA

a(n) = Numerator[ Sum[ k^2, {k,1,n} ] / Product[ k^2, {k,1,n} ] ]. a(n) = Numerator[ n(n+1)(2n+1)/6/(n!)^2 ].

MATHEMATICA

Table[Numerator[n(n+1)(2n+1)/6/(n!)^2], {n, 1, 500}]

CROSSREFS

Cf. A067656, A005382, A034849.

Sequence in context: A205694 A121595 A226660 * A177735 A139428 A303574

Adjacent sequences:  A125291 A125292 A125293 * A125295 A125296 A125297

KEYWORD

nonn,frac

AUTHOR

Alexander Adamchuk, Jan 17 2007

STATUS

approved

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Last modified February 17 10:59 EST 2019. Contains 320219 sequences. (Running on oeis4.)