A051683 -id:A051683 - cp's OEIS frontend

A276001 Numbers n for which A060502(n) <= 1; numbers with at most one distinct slope in their factorial representation.

0, 1, 2, 4, 5, 6, 12, 14, 18, 19, 22, 23, 24, 48, 54, 72, 74, 84, 86, 96, 97, 100, 101, 114, 115, 118, 119, 120, 240, 264, 360, 366, 408, 414, 480, 482, 492, 494, 552, 554, 564, 566, 600, 601, 604, 605, 618, 619, 622, 623, 696, 697, 700, 701, 714, 715, 718, 719, 720, 1440, 1560, 2160, 2184, 2400, 2424, 2880, 2886, 2928, 2934, 3240, 3246, 3288, 3294

Offset: 0

Antti Karttunen, Aug 16 2016

Indexing starts from zero, because a(0)=0 is a special case in this sequence. To get those n for which A060502(n) = 1, start listing terms from a(1) = 1 onward.
From n=1 onward numbers in whose factorial base representation (A007623) the difference i_x - d_x is the same for all nonzero digits d_x present. Here i_x is the position of digit d_x from the least significant end.
From n=1 onward also n such that A060498(n) is a one-ball juggling pattern.

			4 ("20" in factorial base) is present, because all nonzero digits are on the same slope as there is only one nonzero digit.
14 ("210" in factorial base) is present, because all nonzero digits are on the same slope, as 3-2 = 2-1.
19 ("301" in factorial base) is present, because all nonzero digits are on the same slope, as 3-3 = 1-1.
21 ("311" in factorial base) is NOT present, because not all of its nonzero digits are on the same slope, as 3-3 <> 2-1.

Index entries for sequences related to factorial base representation

Cf. A007623, A060498, A060502, A276002, A276003.

Cf. A000142, A033312, A051683 (subsequences).

A377020 Numbers whose prime factorization has exponents that are all numbers of the form m*k!, where 1 <= m <= k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76

Offset: 1

Amiram Eldar, Oct 13 2024

First differs from A138302 and A270428 at n = 57: a(57) = 64 is not a term of A138302 and A270428.
First differs from A337052 at n = 193: A337052(193) = 216 is not a term of this sequence.
First differs from A335275 at n = 227: A335275(227) = 256 is not a term of this sequence.
First differs from A220218 at n = 903: A220218(903) = 1024 is not a term of this sequence.
Numbers k such that A376886(k) = A001221(k).
The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^3 + (1 - 1/p) * (Sum_{k>=3} 1/p^A051683(k))) = 0.87902453718626485582... .
a(n) = A096432(n-1) for 2<=n<380, but then the sequences start to differ: A096432 contains 432, 648, 1024, 1728, 2000, 2160,... which are not in this sequence. - R. J. Mathar, Oct 15 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001221, A051683, A376886, A377021, A377022.
Subsequences: A005117, A004709, A377019.
Cf. A138302, A220218, A270428, A335275, A337052.

expQ[n_] := expQ[n] = Module[{m = n, k = 2}, While[Divisible[m, k], m /= k; k++]; m < k]; q[n_] := AllTrue[FactorInteger[n][[;;, 2]], expQ]; Select[Range[100], q]

isf(n) = {my(k = 2); while(!(n % k), n /= k; k++); n < k;}
is(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!isf(e[i]), return(0))); 1;}

A257686 a(0) = 0, for n >= 1: a(n) = A099563(n) * A048764(n).

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 6, 6, 6, 6, 6, 6, 12, 12, 12, 12, 12, 12, 18, 18, 18, 18, 18, 18, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 72

Offset: 0

Antti Karttunen, May 04 2015

For n >= 1, a(n) = the smallest term of A051683 >= n.
Can also be obtained by replacing with zeros all other digits except the first (the most significant) in the factorial base representation of n (A007623), then converting back to decimal.
Useful when computing A257687.

			Factorial base representation (A007623) of 2 is "10", zeroing all except the most significant digit does not change anything, thus a(2) = 2.
Factorial base representation (A007623) of 3 is "11", zeroing all except the most significant digit gives "10", thus a(3) = 2.
Factorial base representation of 23 is "321", zeroing all except the most significant digit gives "300" which is factorial base representation of 18, thus a(23) = 18.

Antti Karttunen, Table of n, a(n) for n = 0..5040

Cf. A007623, A048764, A051683, A099563, A257687.

Cf. also A053644 (analogous sequence for base-2).

from sympy import factorial as f
def a007623(n, p=2): return n if nIndranil Ghosh, Jun 21 2017

(define (A257686 n) (if (zero? n) n (* (A099563 n) (A048764 n))))

a(0) = 0, and for n >= 1: a(n) = A099563(n) * A048764(n).
Other identities:
For all n >= 0, a(n) = n - A257687(n).
a(n) = A000030(A007623(n))*(A055642(A007623(n)))! - Indranil Ghosh, Jun 21 2017

A213168 a(n) = n!/2 - (n-1)! - n + 2.

Original entry on oeis.org

0, 0, 4, 33, 236, 1795, 15114, 141113, 1451512, 16329591, 199583990, 2634508789, 37362124788, 566658892787, 9153720575986, 156920924159985, 2845499424767984, 54420176498687983, 1094805903679487982, 23112569077678079981, 510909421717094399980

Offset: 2

Olivier Gérard, Nov 02 2012

Row sums of A142706 for k=1..n-1.

Vincenzo Librandi, Table of n, a(n) for n = 2..200

Cf. A001286.
Cf. A142706, A051683, A152883.
Cf. A200748 (considered as a triangular array).

[Factorial(n)/2-Factorial(n-1)-n+2: n in [2..25]]; // Vincenzo Librandi, Sep 09 2016

f:=gfun:-rectoproc({2*(n-3)*a(n) - (2*n^2-6*n+4)*a(n-1)- 2*(n-3)*(n-2)^2, a(2)=0,a(3)=0},a(n),remember): map(f, [$2..22]); # Georg Fischer, Aug 25 2021

Table[n!/2 - (n - 1)! - n + 2, {n, 2, 20}]

A213168(n):=n!/2-(n-1)!-n+2$
makelist(A213168(n),n,2,30); /* Martin Ettl, Nov 03 2012 */

a(n) = A001286(n-1) - n + 2. - Anton Zakharov, Sep 08 2016
D-finite with recurrence: 2*(n-3)*a(n) - (2*n^2-6*n+4)*a(n-1)- 2*(n-3)*(n-2)^2 = 0. - Georg Fischer, Aug 25 2021
E.g.f.: 1/(2-2*x)+log(1-x)+(2-x)*exp(x). - Alois P. Heinz, Aug 25 2021

A237450 Triangle read by rows, T(n,k) = !n + (k-1)*(n-1)!, with n>=1, 1<=k<=n; Position of the first n-letter permutation beginning with number k in the list of lexicographically sorted permutations A030299.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 16, 22, 28, 34, 58, 82, 106, 130, 154, 274, 394, 514, 634, 754, 874, 1594, 2314, 3034, 3754, 4474, 5194, 5914, 10954, 15994, 21034, 26074, 31114, 36154, 41194, 46234, 86554, 126874, 167194, 207514, 247834, 288154, 328474, 368794, 409114, 771994, 1134874, 1497754, 1860634, 2223514, 2586394, 2949274, 3312154, 3675034

Offset: 1

Antti Karttunen, Feb 08 2014

When organized as a triangular table
1;
2, 3;
4, 6, 8;
10, 16, 22, 28;
34, 58, 82, 106, 130;
...
the k-th term of row n gives the position of the first n-letter permutation beginning with number k among all the lexicographically ordered permutations A030299. Thus the terms give the positions of rows of irregular table A237265 among the rows of A030298.
Note: the notation !n stands for the left factorial, A003422(n).

Antti Karttunen, Rows 1..45 of the triangular table, flattened

Left edge: A003422.

Cf. also A002024, A002262, A000142, A030298, A030299, A051683, A237265.

lf[n_] := lf[n] = (-1)^n n! Subfactorial[-n - 1] - Subfactorial[-1] // FullSimplify;
T[n_, k_] := lf[n] + (k - 1)(n - 1)!;
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten

(define (A237450 n) (+ (A003422 (A002024 n)) (* (A002262 (- n 1)) (A000142 (- (A002024 n) 1)))))

a(n) = A003422(A002024(n)) + (A002262(n-1)*A000142(A002024(n)-1)).

A355678 For any nonnegative number n with factorial base expansion Sum_{k > 0} d_k * k!, a(n) = Sum_{k > 0} d_k * k! * (-1)^(Sum_{i < k} sign(d_i)).

Original entry on oeis.org

0, 1, 2, -1, 4, -3, 6, -5, -4, 5, -2, 3, 12, -11, -10, 11, -8, 9, 18, -17, -16, 17, -14, 15, 24, -23, -22, 23, -20, 21, -18, 19, 20, -19, 22, -21, -12, 13, 14, -13, 16, -15, -6, 7, 8, -7, 10, -9, 48, -47, -46, 47, -44, 45, -42, 43, 44, -43, 46, -45, -36, 37

Offset: 0

Rémy Sigrist, Jul 14 2022

This sequence establishes a bijection from the nonnegative integers (N) to the integers (Z).
This sequence is to factorial base what A065620 is to base 2.
To compute a(n): write n as a minimal sum of terms of A051683 and take the alternating sum.

			For n = 28:
  28 = 4! + 2*2!,
  so a(28) = -4! + 2*2! = -20.

Cf. A051683, A065620, A355679.

a(n) = { my (v=0, f=1, s=1, d); for (r=2, oo, if (n==0, return (v), d=n%r; if (d, v+=d*f*s; s=-s); n\=r; f*=r)) }

a(n) = n iff n = 0 or n belongs to A051683.

A123316 Triangle read by rows: T(n,k)=(k+1)*n!/2 (1<=k<=n).

Original entry on oeis.org

1, 2, 3, 6, 9, 12, 24, 36, 48, 60, 120, 180, 240, 300, 360, 720, 1080, 1440, 1800, 2160, 2520, 5040, 7560, 10080, 12600, 15120, 17640, 20160, 40320, 60480, 80640, 100800, 120960, 141120, 161280, 181440, 362880, 544320, 725760, 907200, 1088640

Offset: 1

Roger L. Bagula, Nov 09 2006

			Triangle begins:
{1},
{2, 3},
{6, 9, 12},
{24, 36, 48, 60},
{120, 180, 240, 300, 360},
{720, 1080, 1440, 1800, 2160, 2520},
{5040, 7560, 10080, 12600, 15120, 17640, 20160}

Cf. A051683.

T:=(n,k)->(k+1)*n!/2: for n from 1 to 10 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

t[n_, m_] := (m + 1)*n!/2; a = Table[Table[t[n, m], {m, 1, n}], {n, 1, 10}]; Flatten[a]

Edited by N. J. A. Sloane, Dec 03 2006

A181416 Irregular table T(n,k) = n*A178883(n,k) read by rows.

Original entry on oeis.org

1, 2, 4, 6, 12, 18, 24, 32, 16, 72, 96, 120, 120, 120, 180, 180, 480, 600, 720, 576, 576, 288, 648, 1296, 216, 1152, 1728, 3600, 4320, 5040, 3360, 3360, 3360, 3024, 6048, 3024, 3024, 4032, 12096, 4032, 8400, 16800, 30240, 35280, 40320

Offset: 1

Alford Arnold, Oct 17 2010

The row sum of row n is A001286(n).

			In row n=3 the products are (3,3,3) times (2,4,6) yielding (6,12,18) which adds to 36, the third Lah number.
The table starts in row n=1 with row lengths A000041(n) as:
1;
2,4;
6,12,18;
24,32,16,72,96;
120,120,120,180,180,480,600;

Cf. A051683.

T(n,k) = A036042(n,k)*A178883(n,k), 1<=k<= A000041(n).

cp's OEIS Frontend

A276001 Numbers n for which A060502(n) <= 1; numbers with at most one distinct slope in their factorial representation.

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A377020 Numbers whose prime factorization has exponents that are all numbers of the form m*k!, where 1 <= m <= k.

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A257686 a(0) = 0, for n >= 1: a(n) = A099563(n) * A048764(n).

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A213168 a(n) = n!/2 - (n-1)! - n + 2.

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A237450 Triangle read by rows, T(n,k) = !n + (k-1)*(n-1)!, with n>=1, 1<=k<=n; Position of the first n-letter permutation beginning with number k in the list of lexicographically sorted permutations A030299.

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A355678 For any nonnegative number n with factorial base expansion Sum_{k > 0} d_k * k!, a(n) = Sum_{k > 0} d_k * k! * (-1)^(Sum_{i < k} sign(d_i)).

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A123316 Triangle read by rows: T(n,k)=(k+1)*n!/2 (1<=k<=n).

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A181416 Irregular table T(n,k) = n*A178883(n,k) read by rows.

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