A275732 One-based positions of 1-digits in the factorial base representation of n are converted to primes with those indices, then multiplied together.

1, 2, 3, 6, 1, 2, 5, 10, 15, 30, 5, 10, 1, 2, 3, 6, 1, 2, 1, 2, 3, 6, 1, 2, 7, 14, 21, 42, 7, 14, 35, 70, 105, 210, 35, 70, 7, 14, 21, 42, 7, 14, 7, 14, 21, 42, 7, 14, 1, 2, 3, 6, 1, 2, 5, 10, 15, 30, 5, 10, 1, 2, 3, 6, 1, 2, 1, 2, 3, 6, 1, 2, 1, 2, 3, 6, 1, 2, 5, 10, 15, 30, 5, 10, 1, 2, 3, 6, 1, 2, 1, 2, 3, 6, 1, 2, 1, 2, 3, 6, 1, 2, 5, 10, 15, 30

Offset: 0

Author

Antti Karttunen, Aug 08 2016

Keywords

• nonn
• base

Comments

All terms are squarefree (A005117), and each squarefree number occurs an infinitely many times.

Examples

			22 has factorial base representation "320" (= A007623(22)), which does not contain any "1". Thus a(22) = 1, as the empty product is 1.
35 has factorial base representation "1121" (= A007623(35)). 1's occur in the following positions, when counted from right, starting with 1: 1, 3 and 4. Thus a(35) = prime(1)*prime(3)*prime(4) = 2*5*7 = 70.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..40320
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A000040, A001221, A001222, A002110, A005117, A007623, A007489, A048675, A257261, A257511, A275730.

Cf. A255411 (indices of ones).

Can be used to compute A275733 and A275734.

Cf. also to A275736.

Programs

Mathematica

nn = 105; m = 1; While[Factorial@ m < nn, m++]; m; Map[Times @@ Map[Prime, Flatten@ Position[#, 1]] &@ Reverse@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] &, Range[0, nn]] (* Michael De Vlieger, Aug 11 2016, Version 10.2 *)

Python

from operator import mul
from sympy import prime
def a007623(n, p=2): return n if n
Indranil Ghosh, Jun 19 2017

Scheme

;; Recursive definition using memoizing definec-macro:
(definec (A275732 n) (cond ((zero? (A257261 n)) 1) (else (* (A000040 (A257261 n)) (A275732 (A275730bi n (- (A257261 n) 1)))))))
(define (A275732 n) (let loop ((z 1) (n n)) (let ((y (A257261 n))) (cond ((zero? y) z) (else (loop (* z (A000040 y)) (A275730bi n (- y 1))))))))
;; Code for A275730bi given in A275730.

Formula

If A257261(n) = 0, then a(n) = 1, otherwise a(n) = A000040(A257261(n)) * a(A275730(n, A257261(n)-1)). [Here A275730(n,p) is a bivariate function that "clears" the digit at zero-based position p in the factorial base representation of n].

Other identities and observations. For all n >= 0:

a(A007489(n)) = A002110(n).

a(A255411(n)) = 1.

A001221(a(n)) = A001222(a(n)) = A257511(n).

A048675(a(n)) = A275736(n).