A000142 -id:A000142 - cp's OEIS frontend

A152656 Triangle read by rows: denominators of polynomials from A000142: P(0,x) = 1, P(n,x) = 1/n! + x*Sum_{i=0..n-1} P(n-i-1)/i!. Numerators are A152650.

1, 1, 1, 2, 1, 1, 6, 1, 1, 1, 24, 3, 2, 1, 1, 120, 3, 2, 1, 1, 1, 720, 15, 8, 3, 2, 1, 1, 5040, 45, 40, 3, 6, 1, 1, 1, 40320, 315, 80, 15, 24, 1, 2, 1, 1, 362880, 315, 560, 45, 24, 1, 6, 1, 1, 1, 3628800, 2835, 4480, 315, 144, 5, 24, 3, 2, 1, 1

Offset: 0

Paul Curtz, Dec 10 2008

a(n) is the last sequence of a trio with, first, A141412 and, second, A142048 (denominators).

			Contribution from _Vincenzo Librandi_, Dec 16 2012: (Start)
Triangle begins:
        1,
        1,    1,
        2,    1,    1,
        6,    1,    1,   1,
       24,    3,    2,   1,   1,
      120,    3,    2,   1,   1, 1,
      720,   15,    8,   3,   2, 1,  1,
     5040,   45,   40,   3,   6, 1,  1, 1,
    40320,  315,   80,  15,  24, 1,  2, 1, 1,
   362880,  315,  560,  45,  24, 1,  6, 1, 1, 1,
  3628800, 2835, 4480, 315, 144, 5, 24, 3, 2, 1, 1,
  ...
First column: A000142; second column: A049606. (End)

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A152650, A142048, A142412,

ClearAll[u, p]; u[n_] := 1/n!; p[0][x_] := u[0]; p[n_][x_] := p[n][x] = u[n] + x*Sum[u[i]*p[n-i-1][x] , {i, 0, n-1}] // Expand; row[n_] := CoefficientList[p[n][x], x]; Table[row[n], {n, 0, 10}] // Flatten // Denominator (* Jean-François Alcover, Oct 02 2012 *)

More terms from Jean-François Alcover, Oct 02 2012

A081401 Pseudologarithm (A056239) of n!: a(n) = A056239(A000142(n)).

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 15, 18, 22, 26, 31, 35, 41, 46, 51, 55, 62, 67, 75, 80, 86, 92, 101, 106, 112, 119, 125, 131, 141, 147, 158, 163, 170, 178, 185, 191, 203, 212, 220, 226, 239, 246, 260, 267, 274, 284, 299, 305, 313, 320, 329, 337, 353, 360, 368, 375, 385, 396, 413

Offset: 1

Labos Elemer, Mar 31 2003

nonn

As A056239 is fully additive sequence, this sequence gives its partial sums. - Antti Karttunen, Jun 28 2020

			n=8: 8! = 40320 = 2*2*2*2*2*2*2*3*3*5*7, p-indices = {1,2,3,4}, exponents = {7,2,1,1}; a(8) = 1*7 + 2*2 + 3*1 + 4*1 = 7 + 4 + 3 + 4 = 18.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000142, A056239.

Cf. also A335860.

a:= n-> add (numtheory[pi](i[1])*i[2], i=ifactors(n!)[2]):
seq (a(n), n=1..100);  # Alois P. Heinz, Aug 09 2012

Array[Total[FactorInteger[#!] /. {p_, c_} /; p > 0 :> PrimePi[p] c] &, 59] (* Michael De Vlieger, Jun 26 2020 *)

a(n) = Sum(k*e(k)) where k runs through indices of prime factors of n!, while e(k) is the exponent of the corresponding prime factor.

A230961 Boustrophedon transform of factorials beginning with 1, cf. A000142.

Original entry on oeis.org

1, 3, 11, 50, 273, 1746, 12823, 106462, 986689, 10103074, 113309991, 1381835454, 18209834849, 257911743506, 3907538236631, 63066584719982, 1080340925760129, 19577690297352258, 374214932301757255, 7524626434657416286, 158783753482817132065

Offset: 0

Reinhard Zumkeller, Nov 05 2013

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Peter Luschny, An old operation on sequences: the Seidel transform
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).
Wikipedia, Boustrophedon_transform
Index entries for sequences related to boustrophedon transform

Cf. A230960.

a230961 n = sum $ zipWith (*) (a109449_row n) $ tail a000142_list

T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}];
a[n_] := Sum[T[n, k] (k + 1)!, {k, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 23 2019 *)

from itertools import accumulate, count, islice
def A230961_gen(): # generator of terms
    blist, m = tuple(), 1
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=(m := m*i))))[-1]
A230961_list = list(islice(A230961_gen(),40)) # Chai Wah Wu, Jun 12 2022

a(n) = sum(A109449(n,k)*A000142(k+1): k=0..n).
E.g.f.: conjecture: (tan(x)+sec(x))/(1-2*x+x^2) = (1- 12*x/ (Q(0)+6*x+3*x^2))/(1-x)^2, where Q(k) = 2*(4*k+1)*(32*k^2+16*k - x^2-6) - x^4*(4*k-1)*(4*k+7)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013
a(n) ~ n! * n * (1+sin(1))/cos(1). - Vaclav Kotesovec, Jun 12 2015

A082083 a(n)=A082081[n! ]=A082081[A000142[n]] Fixed points of iterated A008475 function initiated at factorials as initial values.

Original entry on oeis.org

0, 2, 5, 11, 16, 7, 37, 149, 7, 27, 11, 11, 23, 2389, 49, 11, 31, 19, 67, 109, 13, 8, 25, 8, 461, 179, 1319, 9, 193, 16, 7, 4931, 121, 7, 9, 8, 7, 8, 2895630887, 25, 19, 13, 19, 41, 2209493509721, 32, 5939, 23, 43, 11

Offset: 1

Labos Elemer, Apr 08 2003

nonn

			Fixed point is always a prime or a true power of prime:
a term from A000961.
n=20!=2432902008176640000, a(20)=109 because
fixed point list={2432902008176640000,269439,214,109}}

Cf. A001414, A056239, A008475, A082081-A082084, A000961, A000142.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] sex[x_] := Apply[Plus, ba[x]^ep[x]] Table[FixedPoint[sex, w! ], {w, 1, 128}]

A108552 Integer values of (12...*k)/(1+2+...+k) = k!/T(k) = A000142(k)/A000217(k), k>=1.

Original entry on oeis.org

1, 1, 8, 180, 1120, 8064, 604800, 68428800, 830269440, 10897286400, 2324754432000, 640237370572800, 11585247657984000, 221172909834240000, 93666727314800640000, 2068161339110798131200, 47726800133326110720000, 1148978521728221184000000, 28806532937614688256000000

Offset: 1

Rick L. Shepherd, Jun 09 2005

nonn

A000142(n)/A000217(n) = n!/(n*(n+1)/2) = 2*(n-1)!/(n+1) is an integer iff n = 1 or n + 1 is composite; i.e., iff n is a term of A060462.

Cf. A060462 (corresponding k), A000142 (factorials), A000217 (triangular numbers).

select(x-> denom(x)=1, [k!/(k*(k+1)/2)$k=1..30])[];  # Alois P. Heinz, Dec 11 2020

Select[Table[(n - 1)!/((n (n - 1))/2), {n, 2, 50}], IntegerQ[#] &] (* Geoffrey Critzer, May 02 2015 *)

for(n=1,50, r=2*(n-1)!/(n+1); if(denominator(r)==1, print1(r,",")))

a(m) = 2*(A060462(m)-1)!/(A060462(m)+1) = A000142(A060462(m))/A000217(A060462(m)).

Offset corrected by Alois P. Heinz, Dec 11 2020

A139004 Number of operations A000142 (i.e., x!) or A000196 (i.e., floor(sqrt(x))) needed to get n, starting with 4.

Original entry on oeis.org

2, 1, 10, 0, 7, 11, 24, 27, 29, 9, 36, 40, 36, 17, 37, 31, 22, 31, 37, 42, 19, 37, 21, 1, 26, 13, 51, 41, 36, 6, 30, 41, 44, 33, 16, 33, 31, 64, 35, 50, 25, 43, 12, 18, 41, 18, 42, 55, 39, 23, 71, 65, 45, 43, 52, 39, 49, 44, 51, 60, 57, 59, 24, 66, 26, 36, 46, 51, 46, 26, 48, 76

Offset: 1

M. F. Hasler, Apr 09 2008

nonn

Knuth conjectured that any number can be obtained in this way, starting from 4.
This sequence gives the minimal number of operations needed to do so.
To ensure the sequence is well-defined, define a(n)=-1 if it is not possible to get n in the given way.
If we are allowed to use tan(x) just once, then a single 2 is enough to get any positive integer, if Knuth's conjecture that one 4 is enough is true. From 2, (((-tan(2.))!)!)! = 5.592..., then floor, factorial gets 120, then sqrt, sqrt gives 3.162..., and floor gives 3, or negate, floor, negate gives 4. - N. J. A. Sloane, Feb 26 2025
The article by Bendersky is relevant because it gives an explicit formula for n using four 2's (and some logs). Good illustration of techniques. - N. J. A. Sloane, Feb 26 2025

			Representing the operation x -> floor(sqrt(x)) by "s" and x -> x! by "f", we have:
a(1) = 2 since 1 = ss4 is clearly the shortest way to obtain 1, starting with 4.
a(2) = 1 since 2 = s4 is clearly the shortest way to obtain 2, starting with 4.
a(4) = 0 since no operation is required to get 4.
a(3) = 10 = 3+a(5) since 3 = ssf5 and it cannot be obtained from 4 with fewer operations.
a(5) = 7 since 5 = sssssff4.
a(6) = 11 = 1+a(3) since 6 = f3. a(10) = 9 since 10 = sfsssssff4 is the shortest way to obtain 9, starting with 4.

Jon E. Schoenfield, Table of n, a(n) for n = 1..1000
Eli Bendersky, Making any integer with four 2s, Blog Post, Feb 22 2025
Eli Bendersky, Making any integer with four 2s, Blog Post, Feb 22 2025 [Local copy, with the author's permission]
D. E. Knuth, Representing numbers using only one 4, Mathematics Magazine, Vol. 37, No. 5 (Nov. 1964), pp. 308-310.
John E. Maxfield, A Note on N!, Mathematics Magazine, Vol. 43, No. 2 (March 1970), pp. 64-67.
Perlmonks.org, Chasing Knuth's Conjecture.
Jon E. Schoenfield, Table of n, a(n), and shortest path for n = 1..1000

Cf. A139003, A055226.

A139004( n, S=Set(4), LIM=10^4 )={ for( i=0,LIM, setsearch( S, n) & return(i); S=setunion( S, setunion( Set( vector( #S, j, sqrtint(eval(S[j])))), Set( vector( #S, j, if( LIM > j=eval(S[j]), j!))))))}

{ search(x,r,l=0) = local(ll,xx); ll=l; xx=x; while(llMax Alekseyev, Nov 01 2008

a(4) = 0, a(n) = min { a(k)+1 ; n^2 <= k < (n+1)^2 or k! = n }

a(7)-a(9) from Max Alekseyev, Oct 17, Nov 01 2008

More terms from Jon E. Schoenfield, Nov 10 2008

A240993 A000142 (n+1) * A002109(n), a product of factorials and hyperfactorials.

Original entry on oeis.org

1, 2, 24, 2592, 3317760, 62208000000, 20316635136000000, 133852981198454784000000, 20211123400293732996612096000000, 78302033109811407811828935756349440000000, 8613223642079254859301182933198438400000000000000000

Offset: 0

Reinhard Zumkeller, Aug 31 2014

nonn

a(n+1) / a(n) = A055897(n+2);

row products of the triangle A245334.

Reinhard Zumkeller, Table of n, a(n) for n = 0..36

Cf. A000142, A002109, A245334, A055897, A245334, A111063, A074962.

Haskell
```
a240993 n = a000142 (n + 1) * a002109 n
```

Table[(n+1)!*Hyperfactorial[n], {n, 0, 10}] (* Vaclav Kotesovec, Nov 14 2014 *)
Table[(n+1)*(n!)^(n+1)/BarnesG[n+1], {n, 0, 10}] (* Vaclav Kotesovec, Nov 14 2014 *)

a(n) ~ A * sqrt(2*Pi) * n^(n^2/2+3*n/2+19/12) / exp(n*(n+4)/4), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 14 2014

A353969 Lexicographically earliest sequence of distinct positive integers with no finite subset summing to a factorial number (A000142).

Original entry on oeis.org

3, 4, 5, 7, 11, 18, 22, 23, 25, 26, 96, 103, 107, 114, 118, 158, 600, 696, 703, 707, 714, 718, 782, 4320, 4920, 5016, 5023, 5027, 5034, 5038, 5222, 35280, 39600, 40200, 40296, 40303, 40307, 40314, 40318, 41222, 322560, 357840, 362160, 362760, 362856, 362863

Offset: 1

Rémy Sigrist, May 12 2022

nonn

The sequence is well defined:
- a(1) = 3,
- for n > 0, let k be such that A000142(k) + 1 + a(1) + ... + a(n) < A000142(k+1),
- then a(n+1) <= A000142(k) + 1.

Rémy Sigrist, C++ program

See A353889 for similar sequences.

Cf. A000142, A353970, A353971.

A072480 Shadow transform of factorials A000142.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 3, 0, 4, 3, 5, 0, 8, 0, 7, 10, 10, 0, 12, 0, 15, 14, 11, 0, 20, 15, 13, 18, 21, 0, 25, 0, 24, 22, 17, 28, 30, 0, 19, 26, 35, 0, 35, 0, 33, 39, 23, 0, 42, 35, 40, 34, 39, 0, 45, 44, 49, 38, 29, 0, 55, 0, 31, 56, 56, 52, 55, 0, 51, 46, 63, 0, 66, 0, 37, 65, 57, 66, 65

Offset: 0

N. J. A. Sloane and Vladeta Jovovic, Aug 02 2002

For n > 1, a(n) is the number of solutions (n,k) of k! = n! (mod n) where 1 <= k < n. - Clark Kimberling, Feb 11 2012

For n > 1, a(n) is the smallest number k such that n divides (n - k)! but not (n - k - 1)!. - Jianing Song, Aug 29 2018

Antti Karttunen, Table of n, a(n) for n = 0..65537
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150. (ps, pdf); see Definition 7 for the shadow transform.
OEIS Wiki, Shadow transform.
N. J. A. Sloane, Transforms.

Cf. A000142, A002034, A072458.

a:= n-> add(`if`(modp(j!, n)=0, 1, 0), j=0..n-1):
seq(a(n), n=0..120);  # Alois P. Heinz, Sep 16 2019

s[k_] := k!;
f[n_, k_] := If[Mod[s[n] - s[k], n] == 0, 1, 0];
t[n_] := Flatten[Table[f[n, k], {k, 1, n - 1}]]
a[n_] := Count[Flatten[t[n]], 1]
Table[a[n], {n, 2, 420}] (* A072480 *)
Flatten[Position[%, 0]]  (* A006093, primes-1 *)
(* Agrees with A072480 for n > 1, from Clark Kimberling, Feb 12 2012 *)

A002034(n) = if(1==n,n,my(s=factor(n)[, 1], k=s[#s], f=Mod(k!, n)); while(f, f*=k++); (k)); \\ From A002034
A072480(n) = if(n<2,n,(n-A002034(n))); \\ Antti Karttunen, Oct 01 2018

For n > 1, a(n) = n - A002034(n).

A101751 Table (read by rows) giving the coefficients of sum formulas of n-th Factorials (A000142). The k-th row (k>=1, n>=2) contains T(i,k) for i=1 to k+1, where k=[2n+1+(-1)^(n-1)]/4 and T(i,k) satisfies Fact(n) = Sum_{i=1..k+1} T(i,k) (n-1)^(k-i+1) / (2*k-2)!.

Original entry on oeis.org

1, 0, 1, 3, -6, 32, 264, -2024, 2400, 3420, 55800, -666540, 909720, 2570400, 90440, 13101144, 72406040, -3757930680, 13117344800, 72965762016, -261763004160

Offset: 1

André F. Labossière, Dec 17 2004

			Fact(8) = 5040; substituting n=8 in the formula of the k-th row we obtain k=4 and the coefficients
T(i,4) will be the following: 3420,55800,-666540,909720,2570400, => Fact(8) = [ 3420*7^4 +55800*7^3 -666540*7^2 +909720*7 +2570400 ]/6! = 7! =5040.

Cf. A008276, A094216, A000142, A094638, A101752, A003422, A101559, A101032, A099731.

cp's OEIS Frontend

A152656 Triangle read by rows: denominators of polynomials from A000142: P(0,x) = 1, P(n,x) = 1/n! + x*Sum_{i=0..n-1} P(n-i-1)/i!. Numerators are A152650.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A081401 Pseudologarithm (A056239) of n!: a(n) = A056239(A000142(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A230961 Boustrophedon transform of factorials beginning with 1, cf. A000142.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Python

Formula

A082083 a(n)=A082081[n! ]=A082081[A000142[n]] Fixed points of iterated A008475 function initiated at factorials as initial values.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A108552 Integer values of (1*2*...*k)/(1+2+...+k) = k!/T(k) = A000142(k)/A000217(k), k>=1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A139004 Number of operations A000142 (i.e., x!) or A000196 (i.e., floor(sqrt(x))) needed to get n, starting with 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A240993 A000142 (n+1) * A002109(n), a product of factorials and hyperfactorials.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A353969 Lexicographically earliest sequence of distinct positive integers with no finite subset summing to a factorial number (A000142).

A108552 Integer values of (12...*k)/(1+2+...+k) = k!/T(k) = A000142(k)/A000217(k), k>=1.

A101751 Table (read by rows) giving the coefficients of sum formulas of n-th Factorials (A000142). The k-th row (k>=1, n>=2) contains T(i,k) for i=1 to k+1, where k=[2n+1+(-1)^(n-1)]/4 and T(i,k) satisfies Fact(n) = Sum_{i=1..k+1} T(i,k) (n-1)^(k-i+1) / (2*k-2)!.