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Numerator of sum of first p reciprocals of p-simplex numbers divided by p^4, where p = prime(n) for n > 2.
1

%I #15 Nov 26 2017 09:47:48

%S 1,5,53,789,237493,2576561,338350897,616410400171,2603853251291,

%T 5745400286707685,3081677433937346539,41741941495866750557,

%U 7829195555633964779233,21066131970056662377432067

%N Numerator of sum of first p reciprocals of p-simplex numbers divided by p^4, where p = prime(n) for n > 2.

%H G. C. Greubel, <a href="/A118583/b118583.txt">Table of n, a(n) for n = 3..250</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Composition.html">Composition</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TetrahedralNumber.html">Tetrahedral Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>.

%F a(n) = numerator( Sum_{k=1..prime(n)} ( 1/binomial( k + prime(n) - 1, prime(n) ) ))/prime(n)^4 for n > 2.

%e Prime(3) = 5.

%e a(3) = 1 because A118431(5)/5^4 = 1, where A118431(5) = Numerator[ 1/C(4+1,5) + 1/C(4+2,5) + 1/C(4+3,5) + 1/C(4+4,5) +1/C(4+5,5) ] = Numerator[ 1/1 + 1/6 + 1/21 + 1/56 + 1/126 ] = 625.

%t Table[Numerator[Sum[1 /Binomial[ n+Prime[k]-1, Prime[k]], {n,1,Prime[k]} ]]/ Prime[k]^4, {k,3,25}]

%o (PARI) for(n=3,10, print1(numerator(sum(k=1,prime(n), 1/(binomial(k+ prime(n)-1, prime(n)))))/prime(n)^4, ", ")) \\ _G. C. Greubel_, Nov 25 2017

%Y Cf. A022998 = Numerator of sum of reciprocals of first n triangular numbers

%Y Cf. A118391 = Numerator of sum of reciprocals of first n tetrahedral numbers A000292.

%Y Cf. A118431 = Numerator of sum of reciprocals of first n 5-simplex numbers A000389.

%K nonn

%O 3,2

%A _Alexander Adamchuk_, May 09 2007