A068424 -id:A068424 - cp's OEIS frontend

A345708 a(n) is the least positive number starting an interval of consecutive integers whose product of elements is n.

1, 1, 3, 4, 5, 1, 7, 8, 9, 10, 11, 3, 13, 14, 15, 16, 17, 18, 19, 4, 21, 22, 23, 1, 25, 26, 27, 28, 29, 5, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 6, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 7, 57, 58, 59, 3, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 1

Rémy Sigrist, Jun 24 2021

nonn

This sequence is similar to A118235; here we multiply, there we add.
a(n) is the index of the first row of A068424 (interpreted as a square array) containing n.
If n is the product of k consecutive integers, then k! divides n.

			The square array A068424(n, k) begins:
  n\k|   1    2     3      4       5        6
  ---+---------------------------------------
    1|   1    2     6     24     120      720
    2|   2    6    24    120     720     5040
    3|   3   12    60    360    2520    20160
    4|   4   20   120    840    6720    60480
- so a(1) = a(2) = a(6) = a(24) = a(120) = a(720) = 1,
     a(3) = a(12) = a(60) = a(360) = 3,
     a(4) = a(20) = 4.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A000142, A001710, A045619, A068424, A118235, A246655.

a(n) = { fordiv (n, d, my (r=n); for (k=d, oo, if (r==1, return (d), r%k, break, r/=k))) }

a(n) = { for (i=2, oo, if (n%i!, forstep (j=i-1, 2, -1, my (d=sqrtnint(n,j)); while (d-1 && n%(d-1)==0, d--); while (n%d==0, my (r=n); for
(k=d, oo, if (r==1, return (if (d==2, 1, d)), r%k, break, r/=k)); d++)); break)); return (n) }

from sympy import divisors
def a(n):
    if n%2 == 0: return n
    divs = divisors(n)
    for i, d in enumerate(divs[:len(divs)//2]):
        p = lastj = d
        for j in divs[i+1:]:
            if p >= n or j - lastj > 1: break
            p, lastj = p*j, j
        if p == n: return d
    return n
print([a(n) for n in range(1, 70)]) # Michael S. Branicky, Jun 29 2021

a(n) = 1 iff n is a factorial number (A000142).
a(n) <> 2.
a(n) = 3 iff n >= 3 and n belongs to A001710.
a(n) <= n.
a(p! / (n-1)!) = n for any n >= 3 and any prime number p >= n.
a(q) = q for any prime power q > 2.
a(n) = n for any odd number n.
a(n) < n iff n belongs to A045619.

A346928 Irregular triangle read by rows; the n-th row contains, in ascending order, the distinct integers of the form n! / m! (with 1 <= m <= n) that do not appear in former rows.

Original entry on oeis.org

1, 2, 3, 6, 4, 12, 24, 5, 20, 60, 120, 30, 360, 720, 7, 42, 210, 840, 2520, 5040, 8, 56, 336, 1680, 6720, 20160, 40320, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 10, 90, 30240, 151200, 604800, 1814400, 3628800, 11, 110, 990, 7920, 55440, 332640, 1663200, 6652800, 19958400, 39916800

Offset: 1

Rémy Sigrist, Oct 16 2021

This sequence has similarities with A209260; here we consider quotients of factorial numbers, there differences of triangular numbers.
The n-th row is included in the n-th row of A068424 and has greatest term n!.
As a flat sequence, we have a permutation of the positive integers (any n > 0 appears among the first n rows, see A348401).
For any prime number p, the p-th row contains p-1 terms.

			Triangle begins:
    1;
    2;
    3, 6;
    4, 12, 24;
    5, 20, 60, 120;
    30, 360, 720;
    7, 42, 210, 840, 2520, 5040;
    8, 56, 336, 1680, 6720, 20160, 40320;
    9, 72, 504, 3024, 15120, 60480, 181440, 362880;
    10, 90, 30240, 151200, 604800, 1814400, 3628800;
    11, 110, 990, 7920, 55440, 332640, 1663200, 6652800, 19958400, 39916800;
    ...

Cf. A000142, A068424, A209260, A348401.

s=[]; for (n=1, 11, p=1; forstep (m=n, 1, -1, if (!setsearch(s, p*=m), s=setunion(s, [p]); print1 (p", "))))

from math import factorial
def auptor(rows):
    alst, aset = [1], {1}
    for n in range(2, rows+1):
        fn = factorial(n)
        for m in range(n-1, 0, -1):
            fm = factorial(m)
            q, r = divmod(fn, factorial(m))
            if r == 0 and q not in aset:
                alst.append(q); aset.add(q)
    return alst
print(auptor(11)) # Michael S. Branicky, Oct 17 2021

A347034 Triangle read by columns: T(n,k) is the number of functions from an n-element set to a k-element set that are not one-to-one, k>=n>=1.

Original entry on oeis.org

0, 0, 2, 0, 3, 21, 0, 4, 40, 232, 0, 5, 65, 505, 3005, 0, 6, 96, 936, 7056, 45936, 0, 7, 133, 1561, 14287, 112609, 818503, 0, 8, 176, 2416, 26048, 241984, 2056832, 16736896, 0, 9, 225, 3537, 43929, 470961, 4601529, 42683841, 387057609, 0, 10, 280, 4960, 69760, 848800

Offset: 1

Mohammad K. Azarian, Aug 28 2021

The formula for this sequence is Theorem 2.2(iv) of the author's paper, p. 131 (see the link).

			For T(2,3): the number of functions is 3^2 and the number of one-to-one functions is 6, so 3^2 - 6 = 3 and thus T(2,3) = 3.
Triangle T(n,k) begins:
       k=1  k=2   k=3   k=4    k=5     k=6
  n=1:  0    0    0     0      0       0
  n=2:       2    3     4      5       6
  n=3:            21    40     65      96
  n=4:                  232    505     936
  n=5:                         3005    7056
  n=6:                                 45936

Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000312, A002416, A036679, A068424, A089072, A101030, A199656, A344110, A344112, A344113, A344114, A344115, A344116.

A347034 := proc(n,k)
    k^n-k!/(k-n)! ;
end proc:
seq(seq(A347034(n,k),n=1..k),k=1..12) ; # R. J. Mathar, Jan 12 2023

Table[k^n - k!/(k - n)!, {k, 12}, {n, k}] // Flatten

T(n,k) = k^n - k!/(k - n)!;
row(k) = vector(k, i, T(i, k)); \\ Michel Marcus, Oct 01 2021

T(n,k) = k^n - k!/(k - n)!, k>=n.

T(n,n) = A036679(n).

A121922 The result of the integration Integral_{t=0..oo} -rhoexp(-rhost)t^jslog(1+t) dt can be written as (F(u,j)exp(u)Ei(1,u) + G(u,j))/u^j, where rho>0, s>0, and u=rho*s. Sequence is the regular triangle corresponding to G(u,j).

Original entry on oeis.org

-1, 1, -3, -1, 4, -11, 1, -5, 18, -50, -1, 6, -27, 96, -274, 1, -7, 38, -168, 600, -1764, -1, 8, -51, 272, -1200, 4320, -13068, 1, -9, 66, -414, 2200, -9720, 35280, -109584, -1, 10, -83, 600, -3750, 19920, -88200, 322560, -1026576, 1, -11, 102, -836, 6024, -37620, 199920, -887040, 3265920, -10628640

Offset: 0

Arie Harel, Sep 09 2006

			At j=7, the result of the integration Integral_{t=0..oo} -rho*exp(-rho*s*t)*t^j*s*log(1+t) dt
can be written as (F(u,7)*exp(u)*Ei(1,u) + G(u,7))/u^7, where
F(u,7) = u^7 - 7*u^6 + 42*u^5 - 210*u^4 + 840*u^3 -2520*u^2 + 5040*u - 5040,
G(u,7) = - u^6 + 8*u^5 - 51*u^4 + 272*u^3 - 1200*u^2 + 4320*u - 13068,
and u=rho*s.
The coefficients of F(u,7), i.e., (1, -7, 42, -210, 840, 2520, 5040, -5040), comprise the 7th row of A008279 (see also A068424). The coefficients of G(u,7), i.e., (-1, 8, -51, 272, -1200, 4320, -13068) give the 7th row of the triangle below.
Triangle begins:
  -1
  1, -3
  -1, 4, -11
  1, -5, 18, -50
  -1, 6, -27, 96, -274
  1, -7, 38, -168, 600, -1764
  -1, 8, -51, 272, -1200, 4320, -13068

The right-hand diagonal is A000254, the one before that is A001563.

Cf. A008279, A068424.

Edited by Jon E. Schoenfield, Oct 20 2013

A145836 Coefficients of a symmetric matrix representation of the 9th falling factorial power, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10080, 0, 0, 0, 15120, 544320, 544320, 15120, 0, 0, 40320, 1958040, 6108480, 1958040, 40320, 0, 0, 24192, 1796760, 12267360, 12267360, 1796760, 24192, 0, 1, 4608, 588168, 7988904, 18329850, 7988904, 588168, 4608, 1, 255, 74124, 2066232, 9874746, 9874746, 2066232, 74124, 255, 3025, 218484, 2229402, 4690350, 2229402, 218484, 3025, 7770, 212436, 965790, 965790, 212436, 7770, 6951, 85680, 185766, 85680, 6951, 2646, 15624, 15624, 2646, 462, 1260, 462, 36, 36, 1

Offset: 0

Jonathan Vos Post, Oct 21 2008

Osgood and Wu abstract: We investigate the coefficients generated by expressing the falling factorial (xy)_k as a linear combination of falling factorial products (x)_l (y)_m for l,m = 1,...,k. Algebraic and combinatoric properties of these coefficients are discussed, including recurrence relations, closed-form formulas, relations with Stirling numbers and a combinatorial characterization in terms of conjoint ranking tables.

			Full array of coefficients:
[0,     0,       0,        0,        0,       0,      0,       0,    1],
[0,     0,       0,        0,    15120,   40320,   24192,   4608,  255],
[0,     0,   10080,   544320,  1958040, 1796760,  588168,  74124, 3025],
[0,     0,  544320,  6108480, 12267360, 7988904, 2066232, 218484, 7770],
[0, 15120, 1958040, 12267360, 18329850, 9874746, 2229402, 212436, 6951],
[0, 40320, 1796760,  7988904,  9874746, 4690350,  965790,  85680, 2646],
[0, 24192,  588168,  2066232,  2229402,  965790,  185766,  15624,  462],
[0,  4608,   74124,   218484,   212436,   85680,   15624,   1260,   36],
[1,   255,    3025,     7770,     6951,    2646,     462,     36,    1]

Brad Osgood, William Wu, Falling Factorials, Generating Functions and Conjoint Ranking Tables, arXiv:0810.3327 [math.CO], 2008.

Cf. A008277, A068424.

rows = 9;
c[k_, l_ /; l <= rows, m_ /; m <= rows] := Sum[(-1)^(k-p) Abs[StirlingS1[k, p]] StirlingS2[p, l] StirlingS2[p, m], {p, 1, k}];
c[rows, , ] = Nothing;
Table[Table[c[rows, l-m+1, m], {m, 1, l}], {l, 1, 2rows-1}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)

Corrected by Michel Marcus, Dec 15 2014

A385577 Array read by ascending antidiagonals: A(n,m) = n*Pochhammer(n+1,m+1)/(m+2).

Original entry on oeis.org

0, 1, 0, 3, 2, 0, 6, 8, 6, 0, 10, 20, 30, 24, 0, 15, 40, 90, 144, 120, 0, 21, 70, 210, 504, 840, 720, 0, 28, 112, 420, 1344, 3360, 5760, 5040, 0, 36, 168, 756, 3024, 10080, 25920, 45360, 40320, 0, 45, 240, 1260, 6048, 25200, 86400, 226800, 403200, 362880, 0

Offset: 0

Stefano Spezia, Jul 03 2025

			Array begins as:
   0,  0,   0,    0,     0,     0,      0, ...
   1,  2,   6,   24,   120,   720,   5040, ...
   3,  8,  30,  144,   840,  5760,  45360, ...
   6, 20,  90,  504,  3360, 25920, 226800, ...
  10, 40, 210, 1344, 10080, 86400, 831600, ...
  ...

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 20.

Paul W. Haggard and Bonnie L. Sadler, A Generalization of Triangular Numbers, International Journal of Mathematical Education in Science and Technology 25 (2): 195-202, (1994).

Cf. A000142, A001048, A002740, A006231, A007290, A068424, A238474.
Cf. A000217 (m=0), A033487 (m=2), A158874 (m=3).
Cf. A000004 (n=0).

A[n_,m_]:=n*Pochhammer[n+1,m+1]/(m+2); Table[A[n-m,m],{n,0,9},{m,0,n}]//Flatten

Sum_{m=0..n} A(n-m,m) = A006231(n+1).
A(n,1) = A007290(n+2).
A(1,n) = A000142(n+1).
A(2,n) = A001048(n+2).
A(3,n) = abs(A238474(n+1)).
A(n,n) = n!*A002740(n+2)

cp's OEIS Frontend

A345708 a(n) is the least positive number starting an interval of consecutive integers whose product of elements is n.

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A346928 Irregular triangle read by rows; the n-th row contains, in ascending order, the distinct integers of the form n! / m! (with 1 <= m <= n) that do not appear in former rows.

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A347034 Triangle read by columns: T(n,k) is the number of functions from an n-element set to a k-element set that are not one-to-one, k>=n>=1.

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A121922 The result of the integration Integral_{t=0..oo} -rho*exp(-rho*s*t)*t^j*s*log(1+t) dt can be written as (F(u,j)*exp(u)*Ei(1,u) + G(u,j))/u^j, where rho>0, s>0, and u=rho*s. Sequence is the regular triangle corresponding to G(u,j).

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A145836 Coefficients of a symmetric matrix representation of the 9th falling factorial power, read by antidiagonals.

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Mathematica

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A385577 Array read by ascending antidiagonals: A(n,m) = n*Pochhammer(n+1,m+1)/(m+2).

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A121922 The result of the integration Integral_{t=0..oo} -rhoexp(-rhost)t^jslog(1+t) dt can be written as (F(u,j)exp(u)Ei(1,u) + G(u,j))/u^j, where rho>0, s>0, and u=rho*s. Sequence is the regular triangle corresponding to G(u,j).