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A102647 a(n) = product of the remainders when the n-th prime is divided by primes up to the (n-1)-st prime.

%I A102647 #18 May 28 2025 12:45:52
%S A102647 1,1,2,2,8,36,288,1920,2880,120960,362880,6386688,34836480,217728000,
%T A102647 3881779200,275904921600,1785411403776,28217548800000,608662978560000,
%U A102647 3492203839488000,964122158039040000,2224367550332928000,1314079960596480000000,3758268687305932800000
%N A102647 a(n) = product of the remainders when the n-th prime is divided by primes up to the (n-1)-st prime.
%H A102647 Robert Israel, <a href="/A102647/b102647.txt">Table of n, a(n) for n = 1..413</a>
%e A102647 Prime(6) = 13, 13 mod 2 = 1, 13 mod 3 = 1, 13 mod 5 = 3, 13 mod 7 = 6, 13 mod 11 = 2 so a(6) = 1*1*3*6*2 = 36.
%p A102647 f:= proc(n) local p,i;
%p A102647   p:= ithprime(n);
%p A102647   mul(p mod ithprime(i),i=1..n-1)
%p A102647 end proc:
%p A102647 map(f, [$1..25]); # _Robert Israel_, Jan 12 2021
%t A102647 f[n_] := Times @@ Mod[ Prime[n], Table[ Prime[i], {i, n - 1}]]; Table[ f[n], {n, 22}] (* _Robert G. Wilson v_, Feb 04 2005 *)
%t A102647 Join[{0},Table[Times@@Mod[Prime[n],Prime[Range[n-1]]],{n,2,30}]] (* _Harvey P. Dale_, May 16 2019 *)
%o A102647 (PARI) a(n) = my(pr = 1, pn = prime(n)); forprime (q=1, precprime(pn-1), pr *= (pn % q)); pr; \\ _Michel Marcus_, Jan 12 2021
%Y A102647 Cf. A033955, A062347.
%K A102647 nonn
%O A102647 1,3
%A A102647 Hans Boelens (h.p.m.boelens(AT)pl.hanze.nl), Feb 02 2005
%E A102647 More terms from _Robert G. Wilson v_, Feb 04 2005
%E A102647 a(1) (an empty product, therefore 1 by standard convention) corrected by _N. J. A. Sloane_, Jan 11 2021