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.

A256450 Numbers that have at least one 1-digit in their factorial base representation (A007623).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93, 95, 97, 98, 99, 101
Offset: 0

Views

Author

Antti Karttunen, Apr 27 2015

Keywords

Comments

Numbers n for which A257679(n) = 1, i.e., numbers n such that the least nonzero digit in their factorial base representation (A007623) is 1.
Involution A225901 maps each term of this sequence to a unique term of A273670, and vice versa.
Starting offset is zero (with a(0) = 1) because it is the most natural offset for the given fast recurrence.

Links

Crossrefs

Cf. A007623, A257679.
Complement of A255411.
Cf. A257680 (characteristic function), A273662 (left inverse).
First row of A257503, first column of A257505.
Subsequences: A059590 (apart from its zero-term), A255341, A255342, A255343, A257262, A257263, A258198, A258199.
Cf. also A227187 (numbers with at least one nonleading zero) and A273670, A225901.

Programs

  • Mathematica
    Select[Range@ 101, MemberQ[IntegerDigits[#, MixedRadix[Reverse@ Range@ 12]], 1] &] (* Michael De Vlieger, May 30 2016, Version 10.2 *)
r = MixedRadix[Reverse@ Range[2, 12]]; Select[Range@ 101, Min[IntegerDigits[#, r] /. 0 -> Nothing] == 1 &]  (* Michael De Vlieger, Aug 14 2016, Version 10.2 *)
  • Python
    def A(n, p=2): return n if n
    =1]) # Indranil Ghosh, Jun 19 2017

Formula

a(0) = 1, and for n >= 1, if A257511(1+a(n-1)) > 0, then a(n) = a(n-1) + 1, otherwise a(n-1) + 2. [In particular, if the previous term is 2k, then the next term is 2k+1, because all odd numbers are members.]
Other identities:
For all n >= 0, A273662(a(n)) = n. [A273662 works as the left inverse for this sequence.]
From Antti Karttunen, May 26 2015: (Start)
Alternative recurrence for the same sequence:
Set k = A258198(n), d = n - A258199(n) and f = A000142(k+1) = (k+1)! If d < f then b(n) = f+d, otherwise b(n) = ((2+floor((d-f)/A258199(n))) * f) + b((d-f) mod A258199(n)). For offset=1 sequence, define a(n) = b(n-1).
(End)

Extensions

Starting offset changed from 1 to 0 by Antti Karttunen, May 30 2016

A227157 Numbers k whose factorial base representation A007623(k) does not contain any nonleading zeros.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 17, 21, 23, 33, 35, 39, 41, 45, 47, 57, 59, 63, 65, 69, 71, 81, 83, 87, 89, 93, 95, 105, 107, 111, 113, 117, 119, 153, 155, 159, 161, 165, 167, 177, 179, 183, 185, 189, 191, 201, 203, 207, 209, 213, 215, 225, 227, 231, 233, 237, 239, 273
Offset: 1

Views

Author

Antti Karttunen, Jul 04 2013

Keywords

Comments

a(A003422(n)) = A007489(n).
a(A007489(n)) = (n+1)!-1 thus A007489(n) gives the number of terms less than (n+1)! in this sequence.
Equivalently, there are n! terms in the sequence with their magnitude in range n!..(n+1)!.
Also numbers k such that A304036(k) = 1 for k > 0. - Seiichi Manyama, May 06 2018

Links

Crossrefs

The sequence gives all n for which A208575(n) is not zero. Complement of A227187. Subsets: A071156 (apart from zero), A231716, A231720.
Cf. also A003422, A007489, A007623, A153880, A227130, A227132, A304036.

Programs

  • Mathematica
    q[n_] := Module[{k = n, m = 2, c = 0, r}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r == 0, c++]; m++]; c == 0]; Select[Range[300], q] (* Amiram Eldar, Jan 23 2024 *)

A257510 Number of nonleading zeros in factorial base representation of n (A007623).

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 3, 2, 2, 1, 2, 1, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 3, 2, 2, 1, 2, 1, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 3, 2, 2, 1, 2, 1, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 3, 2, 2, 1, 2, 1, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 4
Offset: 1

Views

Author

Antti Karttunen, Apr 27 2015

Keywords

Comments

Sequence starts from n=1, because 0 is an ambiguous case.

Links

Crossrefs

Cf. A227157 (numbers n such that a(n) = 0), A227187 (n for which a(n) > 0).
Cf. A007623, A060130, A084558, A230403.
Cf. also A257511.
Cf. also A023416, A080791 (analogous sequences for base-2), A055641 (for base-10).

Programs

  • Mathematica
    factBaseIntDs[n_] := Module[{m, i, len, dList, currDigit}, i = 1; While[n > i!, i++]; m = n; len = i; dList = Table[0, {len}]; Do[currDigit = 0; While[m >= j!, m = m - j!; currDigit++]; dList[[len - j + 1]] = currDigit, {j, i, 1, -1}]; If[dList[[1]] == 0, dList = Drop[dList, 1]]; dList]; s = Table[FromDigits[factBaseIntDs[n]], {n, 120}]; Last@ DigitCount[#] & /@ s (* Michael De Vlieger, Apr 27 2015, after Alonso del Arte at A007623 *)
  • Scheme
    (define (A257510 n) (let loop ((n n) (i 2) (s 0)) (cond ((zero? n) s) (else (loop (floor->exact (/ n i)) (+ 1 i) (+ s (if (zero? (modulo n i)) 1 0)))))))

Formula

a(n) = A084558(n) - A060130(n).
Other identities and observations:
For all n >= 0, a(A000142(n+1)) = n. [(n+1)! gives the position where n first appears.]
For all n, a(n) >= A230403(n).
Showing 1-3 of 3 results.

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