cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A275731 Transpose of array A275730.

Original entry on oeis.org

0, 0, 0, 2, 1, 0, 2, 0, 1, 0, 4, 1, 2, 1, 0, 4, 0, 3, 2, 1, 0, 6, 1, 4, 3, 2, 1, 0, 6, 6, 5, 4, 3, 2, 1, 0, 8, 7, 0, 5, 4, 3, 2, 1, 0, 8, 6, 1, 6, 5, 4, 3, 2, 1, 0, 10, 7, 2, 7, 6, 5, 4, 3, 2, 1, 0, 10, 6, 3, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 7, 4, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 12, 5, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0
Offset: 0

Views

Author

Antti Karttunen, Aug 08 2016

Keywords

Comments

See comments and links in A275730.

Links

Crossrefs

Transpose: A275730.

Programs

A275736 a(n) has base-2 representation with ones in those digit-positions where n contains ones in its factorial base representation, and zeros in all the other positions.

Original entry on oeis.org

0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 8, 9, 10, 11, 8, 9, 12, 13, 14, 15, 12, 13, 8, 9, 10, 11, 8, 9, 8, 9, 10, 11, 8, 9, 0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0
Offset: 0

Views

Author

Antti Karttunen, Aug 09 2016

Keywords

Comments

Each natural numbers occurs an infinite number of times.
Can be used when computing A275727.

Examples

			22 has factorial base representation "320" (= A007623(22)), which does not contain any "1". Thus a(22) = 0, as the empty sum is 0.
35 has factorial base representation "1121" (= A007623(35)). Here 1's occur in the following positions, when counted from right (starting with 0 for the least significant position): 0, 2 and 3. Thus a(35) = 2^0 + 2^2 + 2^3 = 1*4*8 = 13.

Links

Crossrefs

Cf. A000079, A000120, A000225, A007489, A048675, A257261, A257511, A275727, A275730.
Left inverse of A059590.
Cf. A255411 (indices of zeros).
Cf. also A275732.

Programs

  • Mathematica
    nn = 120; m = 1; While[Factorial@ m < nn, m++]; m; Map[FromDigits[#, 2] &[IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] /. k_ /; k != 1 -> 0] &, Range[0, nn]] (* Michael De Vlieger, Aug 11 2016, Version 10.2 *)

Formula

If A257261(n) = 0, then a(n) = 0, otherwise a(n) = A000079(A257261(n)-1) + 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(n) = A048675(A275732(n)).
A000120(a(n)) = A257511(n).
a(A007489(n)) = A000225(n).
a(A059590(n)) = n.
a(A255411(n)) = 0.

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 08 2016

Keywords

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

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).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.