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

0,2

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 0..40320

Index entries for sequences related to factorial base representation

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).

Example

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.

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 *)

Prog

(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.
(Python)
from operator import mul
from sympy import prime
def a007623(n, p=2): return n if n<p else a007623(n//p, p+1)*10 + n%p
def a(n):
    x=str(a007623(n))[::-1]
    return 1 if n==0 or "1" not in x else reduce(mul, [prime(i + 1) for i in range(len(x)) if x[i]=='1'])
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 19 2017

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.

Sequence in context: A226871 A178483 A133031 * A200594 A319191 A124795

Adjacent sequences: A275729 A275730 A275731 * A275733 A275734 A275735

Keyword

nonn,base

Author

Antti Karttunen, Aug 08 2016

Status

approved