A121663 -id:A121663 - cp's OEIS frontend

A332306 a(n) is the least k such that A121663(k) = n.

0, 1, 2, 4, 8, 3, 32, 5, 128, 9, 512, 6, 2048, 33, 10, 65, 32768, 18, 131072, 12, 34, 513, 2097152, 7, 8388608, 2049, 130, 36, 134217728, 11, 536870912, 68, 514, 32769, 40, 19, 34359738368, 131073, 2050, 13, 549755813888, 35, 2199023255552, 516, 136, 2097153

Offset: 1

Rémy Sigrist, Feb 09 2020

The binary representation of a(n) encodes the colexicographically earliest factorization of n into distinct factors greater than 1.

			The first terms, alongside their binary representations and factorizations, are:
  n   a(n)    bin(a(n))           Factorization
  --  ------  ------------------  -------------
   1       0                   0
   2       1                   1              2
   3       2                  10              3
   4       4                 100              4
   5       8                1000              5
   6       3                  11            2*3
   7      32              100000              7
   8       5                 101            2*4
   9     128            10000000              9
  10       9                1001            2*5
  11     512          1000000000             11
  12       6                 110            3*4
  13    2048        100000000000             13
  14      33              100001            2*7
  15      10                1010            3*5
  16      65             1000001            2*8
  17   32768    1000000000000000             17
  18      18               10010            3*6
  19  131072  100000000000000000             19
  20      12                1100            4*5

Rémy Sigrist, PARI program for A332306

Cf. A000430, A045778, A121663.

PARI
```
See Links section.
```

a(n) = 2^(n-2) iff n is a prime number of the square of a prime number (A000430).
a(n!) = 2^(n-1)-1 for any n > 0.
a(p_1*...*p_k) = 2^(p_1-2)+...+2^(p_k-2) for distinct prime numbers p_1, ..., p_k.

A096111 If n = 2^k - 1, then a(n) = k+1, otherwise a(n) = (A000523(n)+1)*a(A053645(n)).

Original entry on oeis.org

1, 2, 2, 3, 3, 6, 6, 4, 4, 8, 8, 12, 12, 24, 24, 5, 5, 10, 10, 15, 15, 30, 30, 20, 20, 40, 40, 60, 60, 120, 120, 6, 6, 12, 12, 18, 18, 36, 36, 24, 24, 48, 48, 72, 72, 144, 144, 30, 30, 60, 60, 90, 90, 180, 180, 120, 120, 240, 240, 360, 360, 720, 720, 7, 7, 14, 14, 21, 21

Offset: 0

Amarnath Murthy, Jun 29 2004

nonn

Each n > 1 occurs 2*A045778(n) times in the sequence.
f(n+2^k) = (k+1)*f(n) if 2^k > n+1. - Robert Israel, Apr 25 2016
If the binary indices of n (row n of A048793) are the positions 1's in its reversed binary expansion, then a(n) is the product of all binary indices of n + 1. The number of binary indices of n is A000120(n), their sum is A029931(n), and their average is A326699(n)/A326700(n). - Gus Wiseman, Jul 27 2019

Peter Kagey, Table of n, a(n) for n = 0..10000

Permutation of A096115, i.e. a(n) = A096115(A122198(n+1)) [Note the different starting offsets]. Bisection: A121663. Cf. A096113, A052330.
Cf. A029931.
Cf. A000120, A034797, A048793, A070939, A291166, A326031, A326672, A326673.

f:= proc(n) local L;
    L:= convert(2*n+2,base,2);
    convert(subs(0=NULL,zip(`*`,L, [$0..nops(L)-1])),`*`);
end proc:
map(f, [$0..100]); # Robert Israel, Apr 25 2016

CoefficientList[(Product[1 + k x^(2^(k - 1)), {k, 7}] - 1)/x, x] (* Michael De Vlieger, Apr 08 2016 *)
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];Table[Times@@bpe[n+1],{n,0,100}] (* Gus Wiseman, Jul 26 2019 *)

N=166; q='q+O('q^N);
gf= (prod(n=1,1+ceil(log(N)/log(2)), 1+n*q^(2^(n-1)) ) - 1) / q;
Vec(gf)
/* Joerg Arndt, Oct 06 2012 */

(define (A096111 n) (cond ((pow2? (+ n 1)) (+ 2 (A000523 n))) (else (* (+ 1 (A000523 n)) (A096111 (A053645 n))))))
(define (pow2? n) (and (> n 0) (zero? (A004198bi n (- n 1)))))

G.f.: ( prod(k>=1, 1+k*x^(2^(k-1)) )- 1 ) / x. - Vladeta Jovovic, Nov 08 2005

a(n) is the product of the exponents in the binary expansion of 2*n + 2. - Peter Kagey, Apr 24 2016

Edited, extended and Scheme code added by Antti Karttunen, Aug 25 2006

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

Original entry on oeis.org

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

Ilya Gutkovskiy, Feb 10 2020

nonn

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

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

Index entries for sequences related to binary expansion of n

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.

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

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}]

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

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

A309840 If n = Sum (2^e_k) then a(n) = Product (Fibonacci(e_k + 3)).

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 8, 16, 24, 48, 40, 80, 120, 240, 13, 26, 39, 78, 65, 130, 195, 390, 104, 208, 312, 624, 520, 1040, 1560, 3120, 21, 42, 63, 126, 105, 210, 315, 630, 168, 336, 504, 1008, 840, 1680, 2520, 5040, 273, 546, 819, 1638, 1365, 2730, 4095, 8190

Offset: 0

Ilya Gutkovskiy, Aug 19 2019

nonn

			23 = 2^0 + 2^1 + 2^2 + 2^4 so a(23) = Fibonacci(3) * Fibonacci(4) * Fibonacci(5) * Fibonacci(7) = 390.

Michael De Vlieger, Table of n, a(n) for n = 0..16384
Michael De Vlieger, Fan style binary tree showing a(n) for n = 0..2^14, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and magenta, where magenta additionally represents powerful numbers that are not perfect prime powers and bright green represents primorials.

Cf. A000045, A003266, A019565, A022290, A029930, A121663, A160009.

nmax = 55; CoefficientList[Series[Product[(1 + Fibonacci[k + 3] x^(2^k)), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := Fibonacci[Floor[Log[2, n]] + 3] a[n - 2^Floor[Log[2, n]]]; Table[a[n], {n, 0, 55}]

a(n)={vecprod([fibonacci(k+2) | k<-Vec(select(b->b, Vecrev(digits(n, 2)), 1))])} \\ Andrew Howroyd, Aug 19 2019

G.f.: Product_{k>=0} (1 + Fibonacci(k + 3) * x^(2^k)).
a(0) = 1; a(n) = Fibonacci(floor(log_2(n)) + 3) * a(n - 2^floor(log_2(n))).
a(2^(k-2)-1) = A003266(k).

A309841 If n = Sum (2^e_k) then a(n) = Product ((e_k + 2)!).

Original entry on oeis.org

1, 2, 6, 12, 24, 48, 144, 288, 120, 240, 720, 1440, 2880, 5760, 17280, 34560, 720, 1440, 4320, 8640, 17280, 34560, 103680, 207360, 86400, 172800, 518400, 1036800, 2073600, 4147200, 12441600, 24883200, 5040, 10080, 30240, 60480, 120960, 241920, 725760, 1451520, 604800

Offset: 0

Ilya Gutkovskiy, Aug 19 2019

			21 = 2^0 + 2^2 + 2^4 so a(21) = 2! * 4! * 6! = 34560.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000142, A000178, A019565, A029930, A058295, A059590, A121663, A283477.

a:= n-> (l-> mul((i+1)!^l[i], i=1..nops(l)))(convert(n, base, 2)):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 10 2020

nmax = 40; CoefficientList[Series[Product[(1 + (k + 2)! x^(2^k)), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := (Floor[Log[2, n]] + 2)! a[n - 2^Floor[Log[2, n]]]; Table[a[n], {n, 0, 40}]

a(n)={vecprod([(k+1)! | k<-Vec(select(b->b, Vecrev(digits(n, 2)), 1))])} \\ Andrew Howroyd, Aug 19 2019

G.f.: Product_{k>=0} (1 + (k + 2)! * x^(2^k)).
a(0) = 1; a(n) = (floor(log_2(n)) + 2)! * a(n - 2^floor(log_2(n))).
a(2^(k-1)-1) = A000178(k).

cp's OEIS Frontend

A332306 a(n) is the least k such that A121663(k) = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A096111 If n = 2^k - 1, then a(n) = k+1, otherwise a(n) = (A000523(n)+1)*a(A053645(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Scheme

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A309840 If n = Sum (2^e_k) then a(n) = Product (Fibonacci(e_k + 3)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A309841 If n = Sum (2^e_k) then a(n) = Product ((e_k + 2)!).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula