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A219964 a(n) = product(i >= 0, (P(n, i)/P(n-1, i))^(2^i)) where P(n, i) = product(p prime, n/2^(i+1) < p <= n/2^i).

%I A219964 #22 Jul 08 2018 02:08:37
%S A219964 1,1,2,3,2,5,3,7,4,1,5,11,9,13,7,1,16,17,1,19,25,1,11,23,81,1,13,1,49,
%T A219964 29,1,31,256,1,17,1,1,37,19,1,625,41,1,43,121,1,23,47,6561,1,1,1,169,
%U A219964 53,1,1,2401,1,29,59,1,61,31,1,65536,1,1,67,289,1,1,71
%N A219964 a(n) = product(i >= 0, (P(n, i)/P(n-1, i))^(2^i)) where P(n, i) = product(p prime, n/2^(i+1) < p <= n/2^i).
%C A219964 a(n) is 1 or a prime or an even power of a prime (A084400, A050376).
%C A219964 If n > 0 then a(n) = 1 if and only if n is an element of A110473.
%H A219964 Peter Luschny, <a href="/A219964/b219964.txt">Table of n, a(n) for n = 0..1000</a>
%F A219964 a(n) = A220027(n) / A220027(n-1).
%e A219964 a(20) = (7/(5*7))^2*((3*5)/3)^4 = 25.
%e A219964 a(22) = ((13*17*19)/(11*13*17*19))*((7*11)/7)^2 = 11.
%p A219964 A219964 := proc(n) local l, m, z;
%p A219964 if isprime(n) then RETURN(n) fi;
%p A219964 z := 1; l := n - 1; m := n;
%p A219964 do l := iquo(l, 2); m := iquo(m, 2);
%p A219964    if l = 0 then break fi;
%p A219964    if l < m then if isprime(l+1) then RETURN((l+1)^z) fi fi;
%p A219964    z := z + z;
%p A219964 od; 1 end:  seq(A219964(k), k=0..71);
%t A219964 a[n_] := Module[{l, m, z}, If[PrimeQ[n] , Return[n] ]; z = 1; l = Max[0, n - 1]; m = n; While[True, l = Quotient[l, 2]; m = Quotient[m, 2]; If[l == 0 , Break[]]; If[l < m , If[ PrimeQ[l+1], Return[(l+1)^z]]]; z = z+z]; 1]; Table[a[k], {k, 0, 71}] (* _Jean-François Alcover_, Jan 15 2014, after Maple *)
%o A219964 (J)
%o A219964 genSeq=: 3 :0
%o A219964 p=. x: i.&.(_1&p:) y1=.y+1
%o A219964 i=.(#~y1>])&.> <:@((i.@>.&.(2&^.)y1)*])&.> p
%o A219964 y{.(;p(^2x^0,i.@<:@#)&.>i) (;i) } y1$1
%o A219964 )
%o A219964 (Sage)
%o A219964 def A219964(n):
%o A219964     if is_prime(n): return n
%o A219964     z = 1; l = max(0,n-1); m = n
%o A219964     while true:
%o A219964         l = l // 2
%o A219964         m = m // 2
%o A219964         if l == 0: break
%o A219964         if l < m:
%o A219964             if is_prime(l+1): return (l+1)^z
%o A219964         z = z + z
%o A219964     return 1
%o A219964 [A219964(n) for n in (0..71)]
%Y A219964 Cf. A220027, the partial products of a(n).
%K A219964 nonn
%O A219964 0,3
%A A219964 _Peter Luschny_ and _Arie Groeneveld_, Mar 30 2013