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).

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, 29, 1, 31, 256, 1, 17, 1, 1, 37, 19, 1, 625, 41, 1, 43, 121, 1, 23, 47, 6561, 1, 1, 1, 169, 53, 1, 1, 2401, 1, 29, 59, 1, 61, 31, 1, 65536, 1, 1, 67, 289, 1, 1, 71

Offset: 0

Author

Peter Luschny and Arie Groeneveld, Mar 30 2013

Keywords

• nonn

Comments

a(n) is 1 or a prime or an even power of a prime (A084400, A050376).

If n > 0 then a(n) = 1 if and only if n is an element of A110473.

Examples

			a(20) = (7/(5*7))^2*((3*5)/3)^4 = 25.
a(22) = ((13*17*19)/(11*13*17*19))*((7*11)/7)^2 = 11.
		

Links

• Peter Luschny, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A220027, the partial products of a(n).

Programs

J

genSeq=: 3 :0
p=. x: i.&.(_1&p:) y1=.y+1
i=.(#~y1>])&.> <:@((i.@>.&.(2&^.)y1)*])&.> p
y{.(;p(^2x^0,i.@<:@#)&.>i) (;i) } y1$1
)

Maple

A219964 := proc(n) local l, m, z;
if isprime(n) then RETURN(n) fi;
z := 1; l := n - 1; m := n;
do l := iquo(l, 2); m := iquo(m, 2);
   if l = 0 then break fi;
   if l < m then if isprime(l+1) then RETURN((l+1)^z) fi fi;
   z := z + z;
od; 1 end:  seq(A219964(k), k=0..71);

Mathematica

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 *)

Sage

def A219964(n):
    if is_prime(n): return n
    z = 1; l = max(0,n-1); m = n
    while true:
        l = l // 2
        m = m // 2
        if l == 0: break
        if l < m:
            if is_prime(l+1): return (l+1)^z
        z = z + z
    return 1
[A219964(n) for n in (0..71)]

Formula

a(n) = A220027(n) / A220027(n-1).