A332382 If n = Sum (2^e_k) then a(n) = Product (prime(e_k + 2)).

1, 3, 5, 15, 7, 21, 35, 105, 11, 33, 55, 165, 77, 231, 385, 1155, 13, 39, 65, 195, 91, 273, 455, 1365, 143, 429, 715, 2145, 1001, 3003, 5005, 15015, 17, 51, 85, 255, 119, 357, 595, 1785, 187, 561, 935, 2805, 1309, 3927, 6545, 19635, 221, 663, 1105, 3315, 1547, 4641, 7735, 23205

Offset: 0

Author

Ilya Gutkovskiy, Feb 10 2020

Keywords

• nonn

Comments

Permutation of odd squarefree numbers (A056911).

a(n) is the n-th power of 3 in the monoid defined in A331590. - Peter Munn, May 02 2020

Examples

			21 = 2^0 + 2^2 + 2^4 so a(21) = prime(2) * prime(4) * prime(6) = 3 * 7 * 13 = 273.
		

Links

• Index entries for sequences related to binary expansion of n

Crossrefs

Bisection of A019565.

Cf. A002110, A029930, A048675, A056911, A121663, A225546.

A003961, A003987, A059897, A331590, A334748 are used to express relationship between terms of this sequence.

Programs

Maple

a:= n-> (l-> mul(ithprime(i+1)^l[i], i=1..nops(l)))(convert(n, base, 2)):
seq(a(n), n=0..55);  # Alois P. Heinz, Feb 10 2020

Mathematica

nmax = 55; CoefficientList[Series[Product[(1 + Prime[k + 2] x^(2^k)), {k, 0, Floor[Log[2, nmax]]}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := Prime[Floor[Log[2, n]] + 2] a[n - 2^Floor[Log[2, n]]]; Table[a[n], {n, 0, 55}]

PARI

a(n) = my(b=Vecrev(binary(n))); prod(k=1, #b, if (b[k], prime(k+1), 1)); \\ Michel Marcus, Feb 10 2020

Formula

G.f.: Product_{k>=0} (1 + prime(k+2) * x^(2^k)).

a(0) = 1; a(n) = prime(floor(log_2(n)) + 2) * a(n - 2^floor(log_2(n))).

a(2^(k-1)-1) = A002110(k)/2 for k > 0.

From Peter Munn, May 02 2020: (Start)

a(2n) = A003961(a(n)).

a(2n+1) = 3 * a(2n).

a(n) = A225546(4^n).

a(n+k) = A331590(a(n), a(k)).

a(n XOR k) = A059897(a(n), a(k)), where XOR denotes bitwise exclusive-or, A003987.

A048675(a(n)) = 2n.

(End)

a(n+1) = A334748(a(n)). - Peter Munn, Mar 04 2022