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.

A265334 Numbers that are >= k! but < 2*k! for some k; numbers whose factorial base representation (A007623) begins with digit "1".

Original entry on oeis.org

1, 2, 3, 6, 7, 8, 9, 10, 11, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150
Offset: 1

Views

Author

Antti Karttunen, Dec 25 2015

Keywords

Comments

Numbers k for which A099563(k) = 1.
Numbers k for which A265333(k) = 1.

Examples

			1 is present as 1 >= 1! but < 2*1!
2 is present as 2 >= 2! but < 2*2!
3 is present as 3 >= 2! but < 2*2!
4 is not present as 4 >= 2! but not < 2*2! (and not >= 3!).

Links

Crossrefs

Cf. A007623, A099563, A265333 (characteristic function).
Cf. A059590, A060112 (subsequences).
Cf. also A265897, A265898.

A265891 a(n) = A099563(A000407(n)); the most significant digit in factorial base representation of (2n+1)! / n!.

Original entry on oeis.org

1, 1, 2, 1, 3, 8, 2, 6, 1, 3, 10, 1, 5, 14, 1, 5, 16, 1, 5, 15, 1, 4, 12, 1, 3, 9, 28, 2, 6, 19, 1, 3, 11, 35, 2, 6, 19, 1, 3, 10, 30, 1, 4, 14, 44, 2, 6, 20, 61, 2, 8, 25, 1, 3, 10, 31, 1, 3, 11, 35, 1, 4, 12, 38, 1, 4, 12, 39, 1, 4, 12, 39, 1, 3, 11, 36, 1, 3, 10, 33, 102, 3, 9, 28, 89, 2, 7, 23, 74, 1, 6, 19, 59
Offset: 0

Views

Author

Antti Karttunen, Dec 20 2015

Keywords

Examples

			The terms A000407(0) .. A000407(8) in factorial base representation (A007623) look as:
  1, 100, 2200, 110000, 3000000, 82000000, 2374000000, 65500000000, 1550000000000, ...
Taking the first digit (actually: a place holder value) of each gives the terms a(0) .. a(8) of this sequence: 1, 1, 2, 1, 3, 8, 2, 6, 1, ...

Links

Crossrefs

Cf. A000407, A007623, A099563.
Main diagonal of A265890 (apart from the corner term).
Cf. A265897 (positions of ones).
Cf. also A265894.

Programs

  • Mathematica
    a[n_] := Module[{k = (2*n+1)!/n!, m = 2, r, d=0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r > 0, d = r]; m++]; d]; Array[a, 100, 0] (* Amiram Eldar, Feb 14 2024 *)
  • PARI
    allocatemem((2^31)); \\ Enough?
A099563(n) = { my(i=2,dig=0); until(0==n, dig = n % i; n = (n - dig)/i; i++); return(dig); };
A265891 = n -> A099563(((2*n)+1)! / n!);
  • Scheme
    (define (A265891 n) (A099563 (A000407 n)))
  • Scheme
    (define (A265891 n) (A265890bi (+ 1 n) (+ 1 n))) ;; Code for A265890bi given in A265890.

Formula

a(n) = A099563(A000407(n)).
a(n) = A265890(n+1, n+1).

A265898 Numbers n for which (2n)! / n! [= A001813(n)] is >= k! but < 2*k! for some k; positions of ones in A265894.

Original entry on oeis.org

0, 1, 3, 6, 8, 11, 14, 17, 20, 27, 34, 41, 45, 49, 81, 85, 89, 102, 106, 115, 124, 128, 137, 142, 146, 151, 160, 169, 174, 188, 193, 202, 207, 212, 231, 236, 241, 246, 251, 256, 306, 311, 316, 321, 326, 331, 336, 357, 362, 367, 383, 388, 393, 409, 414, 425, 430, 446, 462, 478, 489, 494, 505, 516, 521, 532, 543, 554, 565
Offset: 0

Views

Author

Antti Karttunen, Dec 24 2015

Keywords

Comments

a(0) = 0 is a special case in this sequence, thus the indexing starts from zero.
Numbers n such that A001813(n) is in A265334.

Links

Crossrefs

After zero-term, a subsequence of A265899.
Cf. A001813, A265894, A265334.
Cf. also A265897.
Showing 1-3 of 3 results.

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