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

0,2

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

Links

Table of n, a(n) for n=0..55.

Index entries for sequences related to binary expansion of n

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

Example

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

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}]

Prog

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

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.

Sequence in context: A100181 A180620 A336882 * A277323 A340194 A353340

Adjacent sequences: A332379 A332380 A332381 * A332383 A332384 A332385

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 10 2020

Status

approved