A008606 -id:A008606 - cp's OEIS frontend

A288777 Triangle read by rows in which column k lists the positive multiples of the factorial of k, with 1 <= k <= n.

1, 2, 2, 3, 4, 6, 4, 6, 12, 24, 5, 8, 18, 48, 120, 6, 10, 24, 72, 240, 720, 7, 12, 30, 96, 360, 1440, 5040, 8, 14, 36, 120, 480, 2160, 10080, 40320, 9, 16, 42, 144, 600, 2880, 15120, 80640, 362880, 10, 18, 48, 168, 720, 3600, 20160, 120960, 725760, 3628800, 11, 20, 54, 192, 840, 4320, 25200, 161280, 1088640

Offset: 1

Omar E. Pol, Jun 15 2017

T(n,k) is the number of k-digit numbers in base n+1 with distinct positive digits that form an integer interval when sorted.
T(9,k) is also the number of numbers with k digits in A288528.
The number of terms in A288528 is also A014145(9) = 462331, the same as the sum of the 9th row of this triangle.
Removing the left column of A137267 and of A137948 then this triangle appears in both cases.

			Triangle begins:
   1;
   2,  2;
   3,  4,  6;
   4,  6, 12,  24;
   5,  8, 18,  48, 120;
   6, 10, 24,  72, 240,  720;
   7, 12, 30,  96, 360, 1440,  5040;
   8, 14, 36, 120, 480, 2160, 10080,  40320;
   9, 16, 42, 144, 600, 2880, 15120,  80640,  362880;
  10, 18, 48, 168, 720, 3600, 20160, 120960,  725760, 3628800;
  11, 20, 54, 192, 840, 4320, 25200, 161280, 1088640, 7257600, 39916800;
  ...
For n = 9 and k = 2: T(9,2) is the number of numbers with two digits in A288528.
For n = 9 the row sum is 9 + 16 + 42 + 144 + 600 + 2880 + 15120 + 80640 + 362880 = 462331, the same as A014145(9) and also the same as the number of terms in A288528.

Right border gives A000142, n>=1.
Middle diagonal gives A001563, n>=1.
Row sums give A014145, n>=1.
Column 1..4: A000027, A005843, A008588, A008606.
Cf. A007489, A000142, A004736, A137267, A137948, A166350, A288528, A288778, A288780.

Table[(n - k + 1) k!, {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Jun 15 2017 *)

T(n,k) = (n-k+1)*k! = (n-k+1)*A000142(k) = A004736(n,k)*A166350(n,k).
T(n,k) = Sum_{j=1..n} A166350(j,k).
T(n,k) =  A288778(n,k) + A000142(k-1).

A291900 Sum of the divisors of 24*n - 1, divided by 24, minus n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 0, 0, 2, 0, 0, 2, 0, 3, 0, 0, 2, 0, 9, 0, 0, 2, 2, 7, 0, 4, 0, 3, 6, 0, 0, 3, 5, 7, 0, 0, 0, 0, 15, 6, 0, 3, 0, 9, 4, 0, 10, 0, 13, 5, 0, 3, 3, 22, 0, 4, 0, 5, 12, 0, 19, 0, 0, 13, 0, 0, 0, 10, 14, 4, 6, 7, 5, 19, 11, 0, 0, 0, 16, 5, 4, 12, 8, 28, 0, 0, 0, 0, 35, 6, 4, 0, 5, 32, 4, 18, 8, 0, 31, 0

Offset: 1

Omar E. Pol, Nov 02 2017

The indices of the zeros give A131210.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A008606, A086463, A131210, A134517, A183010, A183011, A280097, A280098, A294614.

a[n_] := DivisorSigma[1, 24 n - 1]/24 - n; Array[a, 90] (* Robert G. Wilson v, Nov 04 2017 *)

a(n) = sigma(24*n-1)/24 - n; \\ Michel Marcus, Nov 04 2017

a(n) = sigma(24*n-1)/24 - n = A000203(A183010(n))/24 - n = A280097(n)/24 - n = A280098(n) - n.

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/18 - 1/2 = 0.048311... . - Amiram Eldar, Mar 28 2024

A348845 Part two of the trisection of A017101: a(n) = 11 + 24*n.

Original entry on oeis.org

11, 35, 59, 83, 107, 131, 155, 179, 203, 227, 251, 275, 299, 323, 347, 371, 395, 419, 443, 467, 491, 515, 539, 563, 587, 611, 635, 659, 683, 707, 731, 755, 779, 803, 827, 851, 875, 899, 923, 947, 971, 995, 1019, 1043, 1067

Offset: 0

Wolfdieter Lang, Dec 11 2021

The trisection of A017101 = {3 + 8*k}A017077%20=%20%7B3*(1%20+%2012*n)%7D">{k>=0} gives 3*A017077 = {3*(1 + 12*n)}{n>=0}, {a(n)}A350051%20=%20%7B19%20+%2024*n%7D">{n >= 0} and A350051 = {19 + 24*n}{n>=0}. These three sequences are congruent to 3 modulo 8 and to 3, 5, and 1 modulo 6, respectively.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008606, 3*A017077, A017101, A350051.

24 * Range[0, 44] + 11 (* Amiram Eldar, Dec 18 2021 *)

a(n) = 11 + 24*n = 11 + A008606(n), for n >= 0
a(n) = 2*a(n-1) - a(n-2), for n >= 1, with a(-1) = -13, a(0) = 11.
G.f.: (11 + 13*x)/(1-x)^2.
E.g.f.: (11 + 24*x)*exp(x).

A350051 Part three of the trisection of A017101: a(n) = 19 + 24*n.

Original entry on oeis.org

19, 43, 67, 91, 115, 139, 163, 187, 211, 235, 259, 283, 307, 331, 355, 379, 403, 427, 451, 475, 499, 523, 547, 571, 595, 619, 643, 667, 691, 715, 739, 763, 787, 811, 835, 859, 883, 907, 931, 955, 979, 1003, 1027, 1051, 1075

Offset: 0

Wolfdieter Lang, Dec 11 2021

The trisection of A017101 = {3 + 8*k}A017077%20=%20%7B3*(1%20+%2012*n)%7D">{k>=0} gives 3*A017077 = {3*(1 + 12*n)}{n>=0}, {A348845(n)}{n >= 0} and {a(n)}{n>=0}. These three sequences are congruent to 3 modulo 8 and to 3, 5, and 1 modulo 6, respectively.

Winston de Greef, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008606, 3*A017077, A017101, A348845.

24 * Range[0, 44] + 19 (* Amiram Eldar, Dec 18 2021 *)

a(n) = 19 + 24*n \\ Winston de Greef, Jan 28 2024

a(n) = 19 + 24*n = 19 + A008606(n), for n >= 0
a(n) = 2*a(n-1) - a(n-2), for n >= 1, with a(-1) = -5, a(0) = 19.
G.f.: (19 + 5*x)/(1-x)^2.
E.g.f.: (19 + 24*x)*exp(x).

A383488 Numbers k that have at least one divisor d_i(k) for which a divisor d_j(k) exists such that d_i(k) < d_j(k) < sigma(d_i(k)).

Original entry on oeis.org

12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 130, 132, 138, 140, 144, 150, 154, 156, 160, 162, 168, 170, 174, 176, 180, 186, 189, 192, 196, 198, 200, 204, 208, 210, 216

Offset: 1

Felix Huber, May 03 2025

nonn

Numbers k (without multiplicity) that are multiples of lcm(c,i), where c is any composite and i is any integer from [c + 1, sigma(c) - 1].

			All multiples of 12 (A008594) are terms because 12 has the divisors 4 and 6 where sigma(4) = 7 > 6.
All multiples of 18 (A008600) are terms because 18 has the divisors 6 and 9 where sigma(6) = 12 > 9.
All multiples of 20 (A008602) are terms because 20 has the divisors 4 and 5 where sigma(4) = 7 > 5.

Felix Huber, Table of n, a(n) for n = 1..10000

Supersequence of A008594, A008600, A008602, A008606, A044102, A135628.

Cf. A000203, A002808, A027750, A109042, A383360, A383489.

with(NumberTheory):
A383488:=proc(n)
    option remember;
    local k,i,L;
    if n=1 then
        12
    else
        for k from procname(n-1)+1 do
            L:=Divisors(k);
            for i to nops(L)-1 do
                if sigma(L[i])>L[i+1] then
                    return k
                fi
            od
        od
    fi;
end proc;
seq(A383488(n),n=1..57);

A157517 a(n) = 7 + 12n - 6n^2.

Original entry on oeis.org

7, 13, 7, -11, -41, -83, -137, -203, -281, -371, -473, -587, -713, -851, -1001, -1163, -1337, -1523, -1721, -1931, -2153, -2387, -2633, -2891, -3161, -3443, -3737, -4043, -4361, -4691, -5033, -5387, -5753, -6131, -6521, -6923, -7337, -7763, -8201, -8651

Offset: 0

Paul Curtz, Mar 02 2009

From John Couch Adams multisteps integration of differential equations, 1855.

P. Curtz Integration numerique des systemes differentiels, C.C.S.A., Arcueil, 1969, p. 36.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

[7+12*n-6*n^2: n in [0..50]]; // Vincenzo Librandi, Aug 07 2011

a(n)=7+12*n-6*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 12*n + 6 - A140811(n) = A017593(n) - A140811(n).
Recurrences: a(n) = 2*a(n-1) - a(n-2) - 12 = 3*a(n-1) - 3*a(n-2) + a(n-3).
First differences: a(n+1) - a(n) = -A017593(n-1), n > 0. Second differences are all -12.
a(n+2) - a(n) = -A008606(n).
G.f.: (-7 + 8*x + 11*x^2)/(x-1)^3. - R. J. Mathar, Mar 15 2009

Edited and extended by R. J. Mathar, Mar 15 2009

A183008 a(n) = 24p(n) = 24A000041(n).

Original entry on oeis.org

24, 24, 48, 72, 120, 168, 264, 360, 528, 720, 1008, 1344, 1848, 2424, 3240, 4224, 5544, 7128, 9240, 11760, 15048, 19008, 24048, 30120, 37800, 46992, 58464, 72240, 89232, 109560, 134496, 164208, 200376, 243432, 295440, 357192, 431448, 519288, 624360, 748440

Offset: 0

Omar E. Pol, Jan 22 2011

a(n) is also the area of one of the faces of the rectangular cuboid which is a three-dimensional version of the shell model of partitions of n. The areas of the other faces are A008606(n)=24*n and A066186(n)=n*p(n), where p(n) is the number of partitions of n. See A135010 for more information.

			The number of partitions of 6 is p(6) = A000041(6) = 11, so a(6) = 24*11 = 264.

Cf. A000041, A008606, A018253, A066186, A135010, A138121, A138011.

24 PartitionsP[Range[0,40]] (* Harvey P. Dale, Sep 17 2024 *)

A207036 Numbers k such that the sum of divisors of k is a multiple of 24.

Original entry on oeis.org

14, 15, 23, 30, 33, 35, 42, 46, 47, 51, 54, 55, 56, 60, 62, 66, 69, 70, 71, 77, 78, 87, 92, 94, 95, 102, 105, 110, 114, 115, 119, 120, 123, 126, 132, 135, 138, 140, 141, 142, 143, 154, 155, 158, 159, 161, 165, 167, 168, 174, 177, 182, 184, 186, 187, 188

Offset: 1

Omar E. Pol, Feb 20 2012

nonn

			33 is in the sequence because the divisors of 33 are 1, 3, 11, 33 and the sum of divisors of 33 is 1+3+11+33 = A000203(33) = 48 and 48 is a multiple de 24 because 48 divided by 24 gives a positive integer.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A000203, A008606, A183010, A207374.

Select[Range[188], Mod[DivisorSigma[1, #], 24] == 0 &] (* T. D. Noe, Feb 27 2012 *)

A207374 Composites of the form 24n - 1.

Original entry on oeis.org

95, 119, 143, 215, 287, 335, 407, 455, 527, 551, 575, 623, 671, 695, 767, 791, 815, 935, 959, 1007, 1055, 1079, 1127, 1175, 1199, 1247, 1271, 1295, 1343, 1391, 1415, 1463, 1535, 1631, 1655, 1679, 1703, 1727, 1751, 1775, 1799, 1895, 1919, 1943, 1967, 1991, 2015

Offset: 1

Omar E. Pol, Feb 18 2012

nonn

Also denominators that are composite numbers A002808 in the Bruinier-Ono formula for the partition function (see A183010 and A183011).

The union of A134517 and this sequence gives A183010.

Cf. A008606, A134517, A183010, A183011.

Select[24*Range[250] - 1, ! PrimeQ[#] &] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2012 *)

A002808 INTERSECT A183010.

A350052 Third part of the trisection of A017077: a(n) = 17 + 24*n.

Original entry on oeis.org

17, 41, 65, 89, 113, 137, 161, 185, 209, 233, 257, 281, 305, 329, 353, 377, 401, 425, 449, 473, 497, 521, 545, 569, 593, 617, 641, 665, 689, 713, 737, 761, 785, 809, 833, 857, 881, 905, 929, 953, 977, 1001, 1025, 1049, 1073

Offset: 0

Wolfdieter Lang, Dec 11 2021

The trisection of A017077 = {1 + 8*k}A103214%20=%20%7B1%20+%2024*n%7D">{k>=0} gives A103214 = {1 + 24*n}{n>=0}, 3*A017101 = {3*(3 + 8*n)}{n >= 0} and {a(n)}{n>=0}. These three sequences are congruent to 1 modulo 8 and to 1, 3, and 5 modulo 6, respectively.

Winston de Greef, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008606, A017077, 3*A017101, A103214.

24 * Range[0, 44] + 17 (* Amiram Eldar, Dec 18 2021 *)

a(n) = 17 + 24*n \\ Winston de Greef, Jan 28 2024

a(n) = 17 + 24*n = 17 + A008606(n), for n >= 0
a(n) = 2*a(n-1) - a(n-2), for n >= 1, with a(-1) = -7, a(0) = 17.
G.f.: (17 + 7*x)/(1-x)^2.
E.g.f.: (17 + 24*x)*exp(x).

cp's OEIS Frontend

A288777 Triangle read by rows in which column k lists the positive multiples of the factorial of k, with 1 <= k <= n.

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A291900 Sum of the divisors of 24*n - 1, divided by 24, minus n.

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A348845 Part two of the trisection of A017101: a(n) = 11 + 24*n.

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A350051 Part three of the trisection of A017101: a(n) = 19 + 24*n.

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A383488 Numbers k that have at least one divisor d_i(k) for which a divisor d_j(k) exists such that d_i(k) < d_j(k) < sigma(d_i(k)).

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A157517 a(n) = 7 + 12*n - 6*n^2.

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Magma

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A183008 a(n) = 24*p(n) = 24*A000041(n).

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A207036 Numbers k such that the sum of divisors of k is a multiple of 24.

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A157517 a(n) = 7 + 12n - 6n^2.

A183008 a(n) = 24p(n) = 24A000041(n).