A001015 -id:A001015 - cp's OEIS frontend

A009641 a(n) = Product_{i=0..6} floor((n+i)/7).

0, 0, 0, 0, 0, 0, 0, 1, 2, 4, 8, 16, 32, 64, 128, 192, 288, 432, 648, 972, 1458, 2187, 2916, 3888, 5184, 6912, 9216, 12288, 16384, 20480, 25600, 32000, 40000, 50000, 62500, 78125, 93750, 112500, 135000, 162000, 194400, 233280, 279936, 326592, 381024

Offset: 0

N. J. A. Sloane

For n >= 7, a(n) is the maximal product of seven positive integers with sum n. - Wesley Ivan Hurt, Jun 29 2022

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. El-Mikkawy and T. Sogabe, A new family of k-Fibonacci numbers, Appl. Math. Comput. 215 (2010) 4456-4461, Table 1 k=7.
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 6, -12, 6, 0, 0, 0, 0, -15, 30, -15, 0, 0, 0, 0, 20, -40, 20, 0, 0, 0, 0, -15, 30, -15, 0, 0, 0, 0, 6, -12, 6, 0, 0, 0, 0, -1, 2, -1).

Maximal product of k positive integers with sum n, for k = 2..10: A002620 (k=2), A006501 (k=3), A008233 (k=4), A008382 (k=5), A008881 (k=6), this sequence (k=7), A009694 (k=8), A009714 (k=9), A354600 (k=10).

Cf. A001015, A013665.

A009641[n_] := Product[Floor[(n + i)/7], {i, 0, 6}]; Array[A009641, 50, 0] (* Paolo Xausa, Jan 17 2025 *)

a(n) = prod(k=0, 6, (n+k)\7); \\ Georg Fischer, Nov 07 2019

a(n) = 2*a(n-1) - a(n-2) + 6*a(n-7) - 12*a(n-8) + 6*a(n-9) - 15*a(n-14) + 30*a(n-15) - 15*a(n-16) + 20*a(n-21) - 40*a(n-22) + 20*a(n-23) - 15*a(n-28) + 30*a(n-29) - 15*a(n-30) + 6*a(n-35) - 12*a(n-36) + 6*a(n-37) - a(n-42) + 2*a(n-43) - a(n-44). - Wesley Ivan Hurt, Jun 29 2022
a(7*n) = n^7 (A001015). - Bernard Schott, Nov 04 2022
Sum_{n>=7} 1/a(n) = 1 + zeta(7). - Amiram Eldar, Jan 10 2023

a(40)-a(44) from Georg Fischer, Nov 07 2019

A017503 a(n) = (11*n + 9)^7.

Original entry on oeis.org

4782969, 1280000000, 27512614111, 230539333248, 1174711139837, 4398046511104, 13348388671875, 34792782221696, 80798284478113, 171382426877952, 337931541778439, 627485170000000, 1107984764452581, 1874584905187328, 3057125241215467, 4828861374436224

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Powers of the form (11*n+9)^m: A017497 (m=1), A017498 (m=2), A017499 (m=3), A017500 (m=4), A017501 (m=5), A017502 (m=6), this sequence (m=7), A017504 (m=8), A017505 (m=9), A017506 (m=10), A017607 (m=11), A017508 (m=12).

Subsequence of A001015.

List([0..20], n-> (11*n+9)^7); # G. C. Greubel, Oct 28 2019

[(11*n+9)^7: n in [0..20]]; // G. C. Greubel, Oct 28 2019

A017503:=n->(11*n+9)^7; seq(A017503(n), n=0..50); # Wesley Ivan Hurt, Nov 20 2013

Table[(11n+9)^7, {n,0,50}] (* Wesley Ivan Hurt, Nov 20 2013 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{4782969,1280000000,27512614111,230539333248,1174711139837,4398046511104,13348388671875,34792782221696},20] (* Harvey P. Dale, Nov 12 2022 *)

vector(21, n, (11*n-2)^7) \\ G. C. Greubel, Oct 28 2019

[(11*n+9)^7 for n in (0..20)] # G. C. Greubel, Oct 28 2019

a(n) = A001015(A017497(n)). - Michel Marcus, Nov 21 2013
From G. C. Greubel, Oct 28 2019: (Start)
G.f.: (4782969 + 1241736248*x + 17406537243*x^2 + 46010574096*x^3 + 29404476791*x^4 + 4084486872*x^5 + 62747493*x^6 + 128*x^7)/(1-x)^8.
E.g.f.: (4782969 + 1275217031*x + 12478698540*x^2 + 25306117991*x^3 + 17188094770*x^4 + 4676276836*x^5 + 520838934*x^6 + 19487171*x^7)*exp(x). (End)

A300785 Triangle read by rows: T(n,k) = 140k^3(n-k)^3 - 14k(n-k) + 1; n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 127, 1, 1, 1093, 1093, 1, 1, 3739, 8905, 3739, 1, 1, 8905, 30157, 30157, 8905, 1, 1, 17431, 71569, 101935, 71569, 17431, 1, 1, 30157, 139861, 241753, 241753, 139861, 30157, 1, 1, 47923, 241753, 472291, 573217, 472291, 241753, 47923, 1, 1, 71569, 383965, 816229, 1119721, 1119721, 816229, 383965, 71569, 1

Offset: 0

Kolosov Petro, Mar 12 2018

From Kolosov Petro, Apr 12 2020: (Start)
Let A(m, r) = A302971(m, r) / A304042(m, r).
Let L(m, n, k) = Sum_{r=0..m} A(m, r) * k^r * (n - k)^r.
Then T(n, k) = L(3, n, k).
T(n, k) is symmetric: T(n, k) = T(n, n-k). (End)

			Triangle begins:
--------------------------------------------------------------------
k=   0      1       2       3       4       5       6      7     8
--------------------------------------------------------------------
n=0: 1;
n=1: 1,     1;
n=2: 1,   127,      1;
n=3: 1,  1093,   1093,      1;
n=4: 1,  3739,   8905,   3739,      1;
n=5: 1,  8905,  30157,  30157,   8905,      1;
n=6: 1, 17431,  71569, 101935,  71569,  17431,      1;
n=7: 1, 30157, 139861, 241753, 241753, 139861,  30157,     1;
n=8: 1, 47923, 241753, 472291, 573217, 472291, 241753, 47923,    1;

Muniru A Asiru, Rows n=0..100 of triangle, flattened.
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
Petro Kolosov, Polynomial identity involving binomial theorem and Faulhaber's formula, 2023.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.

Various cases of L(m, n, k): A287326 (m=1), A300656 (m=2), This sequence (m=3). See comments for L(m, n, k).
Row sums give A258806.
Cf. A000584, A287326, A007318, A077028, A294317, A068236, A300656, A302971, A304042, A001015, A094053, A258808, A024005, A316387.

T:=Flat(List([0..9], n->List([0..n], k->140*k^3*(n-k)^3 - 14*k*(n-k)+1))); # G. C. Greubel, Dec 14 2018

/* As triangle */ [[140*k^3*(n-k)^3-14*k*(n-k)+1: k in [0..n]]: n in [0..10]]; // Bruno Berselli, Mar 21 2018

T:=(n,k)->140*k^3*(n-k)^3-14*k*(n-k)+1: seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Dec 14 2018

T[n_, k_] := 140*k^3*(n - k)^3 - 14*k*(n - k) + 1; Column[
Table[T[n, k], {n, 0, 10}, {k, 0, n}], Center] (* From Kolosov Petro, Apr 12 2020 *)

t(n, k) = 140*k^3*(n-k)^3-14*k*(n-k)+1
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
/* Print initial 9 rows of triangle as follows */ trianglerows(9)

[[140*k^3*(n-k)^3 - 14*k*(n-k)+1 for k in range(n+1)] for n in range(12)] # G. C. Greubel, Dec 14 2018

From Kolosov Petro, Apr 12 2020: (Start)
T(n, k) = 140*k^3*(n-k)^3 - 14*k*(n-k) + 1.
T(n, k) = 140*A094053(n, k)^3 + 0*A094053(n, k)^2 - 14*A094053(n, k)^1 + 1.
T(n+3, k) = 4*T(n+2, k) - 6*T(n+1, k) + 4*T(n, k) - T(n-1, k), for n >= k.
Sum_{k=1..n} T(n, k) = A001015(n).
Sum_{k=0..n} T(n, k) = A258806(n).
Sum_{k=0..n-1} T(n, k) = A001015(n).
Sum_{k=1..n-1} T(n, k) = A258808(n).
Sum_{k=1..n-1} T(n, k) = -A024005(n).
Sum_{k=1..r} T(n, k) = -A316387(3, r, 0)*n^0 + A316387(3, r, 1)*n^1 - A316387(3, r, 2)*n^2 + A316387(3, r, 3)*n^3. (End)
G.f.: (1 + 127*x^6*y^3 - 3*x*(1 + y) + 585*x^5*y^2*(1 + y) + 129*x^4*y*(1 + 17*y + y^2) + 3*x^2*(1 + 45*y + y^2) - x^3*(1 - 579*y - 579*y^2 + y^3))/((1 - x)^4*(1 - x*y)^4). - Stefano Spezia, Sep 14 2024

A016783 a(n) = (3*n+1)^7.

Original entry on oeis.org

1, 16384, 823543, 10000000, 62748517, 268435456, 893871739, 2494357888, 6103515625, 13492928512, 27512614111, 52523350144, 94931877133, 163840000000, 271818611107, 435817657216, 678223072849, 1028071702528

Offset: 0

N. J. A. Sloane

The inverse binomial transform is 1, 16383, 790776, 7578522, 27624240, 46539360, 36741600, 11022480, 0, 0,... (0 continued). - R. J. Mathar, May 07 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A001015, A016777.

[(3*n+1)^7: n in [0..30]]; // Vincenzo Librandi, Sep 21 2011

Table[(3n+1)^7,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)

G.f.: (1+16376x+692499x^2+3870352x^3+4890287x^4+1475736x^5+77101x^6 +128x^7)/ (1-x)^8. - R. J. Mathar, May 07 2008
E.g.f.: exp(x)*(2187*x^7+51030*x^6+387828*x^5+1151010*x^4 +1263087*x^3 +395388*x^2 +16383*x+1). - Robert Israel, Jun 15 2016
a(n) = A001015(A016777(n)). - Michel Marcus, Jun 16 2016
Sum_{n>=0} 1/a(n) = (147555*zeta(7) + 28*sqrt(3)*Pi^7)/295245. - Ilya Gutkovskiy, Jun 16 2016

A036085 Centered cube numbers: (n+1)^7 + n^7.

Original entry on oeis.org

1, 129, 2315, 18571, 94509, 358061, 1103479, 2920695, 6880121, 14782969, 29487171, 55318979, 98580325, 168162021, 276272879, 439294831, 678774129, 1022558705, 1506091771, 2173871739, 3081088541, 4295446429, 5899183335, 7991296871, 10689987049, 14135325801

Offset: 0

N. J. A. Sloane

Never prime, as a(n) = (2n+1)*(n^6 + 3n^5 + 9n^4 + 13n^3 + 11n^2 + 5n + 1). Semiprimes in the sequence begin for n = 1, 2, 8, 9, 21, 30, 33, 53, 65, 81, 83. - Jonathan Vos Post, Aug 26 2011

B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A100267, A154535, A100266, A152913, A194155, A194185, A194216.

[(n+1)^7+n^7: n in [0..30]]; // Vincenzo Librandi, Aug 27 2011

a(n)=(n+1)^7+n^7 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A001015(n+1) + A001015(n).

G.f.: (1+x)*(x^6 + 120*x^5 + 1191*x^4 + 2416*x^3 + 1191*x^2 + 120*x + 1) / (x-1)^8. - R. J. Mathar, Aug 27 2011

A101104 a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4.

Original entry on oeis.org

1, 12, 23, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24

Offset: 1

Cecilia Rossiter, Dec 15 2004

Original name: The first summation of row 4 of Euler's triangle - a row that will recursively accumulate to the power of 4.

D. J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference [pdf], Kristiansand, 2002, (ed. Otto Bekken et al), National Center for Mathematics Education, University of Gothenburg, Sweden, in press.
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Dead link]
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]
Eric Weisstein, Link to section of MathWorld: Worpitzky's Identity of 1883
Eric Weisstein, Link to section of MathWorld: Eulerian Number
Eric Weisstein, Link to section of MathWorld: Nexus number
Eric Weisstein, Link to section of MathWorld: Finite Differences
Index entries for linear recurrences with constant coefficients, signature (1).

For other sequences based upon MagicNKZ(n,k,z):
..... |  n = 1  |  n = 2  |  n = 3  |  n = 4  |  n = 5  |  n = 6  |  n = 7
---------------------------------------------------------------------------
z = 0 | A000007 | A019590 | .......MagicNKZ(n,k,0) = A008292(n,k+1) .......
z = 1 | A000012 | A040000 | A101101 | thisSeq | A101100 | ....... | .......
z = 2 | A000027 | A005408 | A008458 | A101103 | A101095 | ....... | .......
z = 3 | A000217 | A000290 | A003215 | A005914 | A101096 | ....... | .......
z = 4 | A000292 | A000330 | A000578 | A005917 | A101098 | ....... | .......
z = 5 | A000332 | A002415 | A000537 | A000583 | A022521 | ....... | A255181
z = 6 | A000389 | A005585 | A024166 | A000538 | A000584 | A022522 | A255177
z = 7 | A000579 | A040977 | A101094 | A101089 | A000539 | A001014 | A022523
z = 8 | A000580 | A050486 | A101097 | A101090 | A101092 | A000540 | A001015
z = 9 | A000581 | A053347 | A101102 | A101091 | A101099 | A101093 | A000541
Cf. A101095 for an expanded table and more about MagicNKZ.

MagicNKZ = Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}];Table[MagicNKZ, {n, 4, 4}, {z, 1, 1}, {k, 0, 34}]
Join[{1, 12, 23},LinearRecurrence[{1},{24},56]] (* Ray Chandler, Sep 23 2015 *)

a(k) = MagicNKZ(4,k,1) where MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n+1-z,j)*(k-j+1)^n (cf. A101095). That is, a(k) = Sum_{j=0..k+1} (-1)^j*binomial(4, j)*(k-j+1)^4.
a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4. - Joerg Arndt, Nov 30 2014
G.f.: x*(1+11*x+11*x^2+x^3)/(1-x). - Colin Barker, Apr 16 2012

New name from Joerg Arndt, Nov 30 2014

Original Formula edited and Crossrefs table added by Danny Rorabaugh, Apr 22 2015

A010803 15th powers: a(n) = n^15.

Original entry on oeis.org

0, 1, 32768, 14348907, 1073741824, 30517578125, 470184984576, 4747561509943, 35184372088832, 205891132094649, 1000000000000000, 4177248169415651, 15407021574586368, 51185893014090757, 155568095557812224

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16, -120, 560, -1820, 4368, -8008, 11440, -12870, 11440, -8008, 4368, -1820, 560, -120, 16, -1).

Cf. A013673 (zeta(15)).

Cf. A000290 (squares), A000578 (cubes), A000583 (4th powers), A000584 (5th powers), A001015 (7th powers), A008455 (11th powers).

[n^15: n in [0..15]]; // Vincenzo Librandi, Jun 19 2011

Table[n^15,{n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2010 *)

for(n=0,15,print1(n^15,", ")) \\ Derek Orr, Feb 27 2017

A010803(n)=n^15 \\ M. F. Hasler, Jul 03 2025

A010803 = lambda n: n**15 # M. F. Hasler, Jul 03 2025

Totally multiplicative with a(p) = p^15 for prime p. Multiplicative with a(p^e) = p^(15e). - Jaroslav Krizek, Nov 01 2009
From Ilya Gutkovskiy, Feb 27 2017: (Start)
Dirichlet g.f.: zeta(s-15).
Sum_{n>=1} 1/a(n) = zeta(15) = A013673. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 16383*zeta(15)/16384. - Amiram Eldar, Oct 08 2020

A155014 a(n) = floor(sqrt(n^7)).

Original entry on oeis.org

0, 1, 11, 46, 128, 279, 529, 907, 1448, 2187, 3162, 4414, 5985, 7921, 10267, 13071, 16384, 20256, 24743, 29897, 35777, 42439, 49943, 58350, 67723, 78125, 89620, 102275, 116159, 131338, 147885, 165869, 185363, 206442, 229179, 253651, 279936

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jan 19 2009

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A001015 (n^7).

[Floor(Sqrt(n^7)): n in [0..30]]; // G. C. Greubel, Dec 30 2017

a={};Do[AppendTo[a,IntegerPart[(n^7)^(1/2)]],{n,0,5!}];a
IntegerPart[(Sqrt[Range[0,40]])^7] (* Harvey P. Dale, Jan 19 2016 *)
Table[Floor[Sqrt[n^7]], {n,0,30}] (* G. C. Greubel, Dec 30 2017 *)

for(n=1,30, print1(floor(sqrt(n^7)), ", ")) \\ G. C. Greubel, Dec 30 2017

Offset corrected by Alois P. Heinz, Sep 27 2014

A330486 Earliest start of a run of n numbers divisible by a seventh power larger than one.

Original entry on oeis.org

128, 76544, 2372890624, 390491792890623, 2083234733888734218749, 18962123650219836035505781245

Offset: 1

Jud McCranie, Dec 15 2019

In De Koninck's book, a(4) was probable but not certain.  It also gives probable terms for a(5) and a(6).
From David A. Corneth, Dec 18 2019: (Start)
a(6) <= 18962123650219836035505781245
a(7) <= 105574810222799317012520534891328125
a(8) <= 352061173826332779662070232460962002804453118
a(9) <= 74322635218313783849790472504951802188239215127109371
If any a(k) of these values are less than presented values, one of a(k) + m is divisible only by at least one p^7 for some p > 31, 0 <= m < k.
For a(6) this minimum prime is some prime p > 59. (End)

			2372890624 is divisible by 2^7, 2372890625 is divisible by 5^7, 2372890626 is divisible by 3^7.  This is the smallest number with this property, so a(3)=2372890624.

J.-M. De Koninck, Those Fascinating Numbers, Entry 242, p. 63, Amer. Math. Soc., 2009.

David A. Corneth, PARI program for finding candidates
Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., (2009), page 63.

Cf. A045882, A271443, A001015.

a(5) from Giovanni Resta, Dec 17 2019

a(6) from Giovanni Resta, Dec 19 2019

A016759 a(n) = (2*n + 1)^7.

Original entry on oeis.org

1, 2187, 78125, 823543, 4782969, 19487171, 62748517, 170859375, 410338673, 893871739, 1801088541, 3404825447, 6103515625, 10460353203, 17249876309, 27512614111, 42618442977, 64339296875, 94931877133, 137231006679, 194754273881, 271818611107, 373669453125, 506623120463

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Philippe A. J. G. Chevalier, On the discrete geometry of physical quantities, Preprint, 2012.
Philippe A. J. G. Chevalier, A "table of Mendeleev" for physical quantities?, Slides from a talk, May 14 2014, Leuven, Belgium.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A001015, A005408, A215960, A258814.

[(2*n+1)^7: n in [0..30]]; // Vincenzo Librandi, Sep 07 2011

Table[(2*n+1)^7, {n,0,30}] (* G. C. Greubel, Sep 15 2018 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,2187,78125,823543,4782969,19487171,62748517,170859375},20] (* Harvey P. Dale, Jul 09 2019 *)

a(n) = (2*n+1)^7; \\ Michel Marcus, Mar 07 2016

a(n) = A001015(A005408(n)). - Michel Marcus, Mar 07 2016
G.f.: (1+x)*(x^6 + 2178*x^5 + 58479*x^4 + 201244*x^3 + 58479*x^2 + 2178*x + 1)/(x-1)^8. - R. J. Mathar, Jul 07 2017
From Amiram Eldar, Oct 10 2020: (Start)
Sum_{n>=0} 1/a(n) = 127*zeta(7)/128.
Sum_{n>=0} (-1)^n/a(n) = 61*Pi^7/184320 (A258814). (End)

cp's OEIS Frontend

A009641 a(n) = Product_{i=0..6} floor((n+i)/7).

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A017503 a(n) = (11*n + 9)^7.

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A300785 Triangle read by rows: T(n,k) = 140*k^3*(n-k)^3 - 14*k*(n-k) + 1; n >= 0, 0 <= k <= n.

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A016783 a(n) = (3*n+1)^7.

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Magma

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A036085 Centered cube numbers: (n+1)^7 + n^7.

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Magma

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A101104 a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4.

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A010803 15th powers: a(n) = n^15.

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A300785 Triangle read by rows: T(n,k) = 140k^3(n-k)^3 - 14k(n-k) + 1; n >= 0, 0 <= k <= n.