A001015 -id:A001015 - cp's OEIS frontend

A092759 a(n) = prime(n)^7.

128, 2187, 78125, 823543, 19487171, 62748517, 410338673, 893871739, 3404825447, 17249876309, 27512614111, 94931877133, 194754273881, 271818611107, 506623120463, 1174711139837, 2488651484819, 3142742836021, 6060711605323

Offset: 1

Jorge Coveiro, Apr 13 2004

Seventh powers of prime numbers. - Wesley Ivan Hurt, Mar 27 2014

			a(1) = 128 since the seventh power of the first prime is 2^7 = 128. - _Wesley Ivan Hurt_, Mar 27 2014

Subsequence of A030626.

Cf. A085967, A013665, A013672.

[p^7: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

A092759:=n->ithprime(n)^7; seq(A092759(n), n=1..30); # Wesley Ivan Hurt, Mar 27 2014

Array[Prime[ # ]^7 &, 30] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
Table[Prime[n]^7, {n, 30}] (* Wesley Ivan Hurt, Mar 27 2014 *)

a(n)=prime(n)^7 \\ Charles R Greathouse IV, Jul 20 2011

a(n) = A086874(n-1), n>1. - R. J. Mathar, Sep 08 2008
a(n) = A000040(n)^7 = A001015(A000040(n)). - Wesley Ivan Hurt, Mar 27 2014
Sum_{n>=1} 1/a(n) = P(7) = 0.0082838328... (A085967). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 24 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(7)/zeta(14) = A013665/A013672.
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(7) = 1/A013665. (End)

A003370 Numbers that are the sum of 3 positive 7th powers.

Original entry on oeis.org

3, 130, 257, 384, 2189, 2316, 2443, 4375, 4502, 6561, 16386, 16513, 16640, 18572, 18699, 20758, 32769, 32896, 34955, 49152, 78127, 78254, 78381, 80313, 80440, 82499, 94510, 94637, 96696, 110893, 156251, 156378, 158437, 172634, 234375, 279938, 280065, 280192, 282124, 282251

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 04 2020: (Start)
1320614777 is in the sequence as 1320614777 = 9^7 + 12^7 + 20^7.
5868568576 is in the sequence as 5868568576 = 8^7 + 20^7 + 24^7.
17285710304 is in the sequence as 17285710304 = 3^7 + 12^7 + 29^7. (End)

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A001015.

list(lim)=my(v=List(),x7,t); lim\=1; for(x=1,sqrtnint(lim-2,7), x7=x^7; for(y=1,min(sqrtnint(lim-x7,7),x), t=x7+y^7; for(z=1,min(sqrtnint(lim-t,7),y), listput(v,t+z^7)))); Set(v) \\ Charles R Greathouse IV, Nov 05 2017

A003374 Numbers that are the sum of 7 positive 7th powers.

Original entry on oeis.org

7, 134, 261, 388, 515, 642, 769, 896, 2193, 2320, 2447, 2574, 2701, 2828, 2955, 4379, 4506, 4633, 4760, 4887, 5014, 6565, 6692, 6819, 6946, 7073, 8751, 8878, 9005, 9132, 10937, 11064, 11191, 13123, 13250, 15309, 16390, 16517, 16644, 16771, 16898, 17025, 17152, 18576

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
2200409 is in the sequence as 2200409 = 3^7 + 3^7 + 3^7 + 3^7 + 4^7 + 5^7 + 8^7.
4122089 is in the sequence as 4122089 = 3^7 + 3^7 + 7^7 + 7^7 + 7^7 + 7^7 + 7^7.
6356992 is in the sequence as 6356992 = 4^7 + 4^7 + 4^7 + 4^7 + 8^7 + 8^7 + 8^7. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001015 (seventh powers).

Column k=7 of A336725.

M = 21027478; m = M^(1/7) // Ceiling; Reap[
For[a = 1, a <= m, a++, For[b = a, b <= m, b++, For[c = b, c <= m, c++,
For[d = c, d <= m, d++, For[e = d, e <= m, e++, For[f = e, f <= m, f++,
For[g = f, g <= m, g++,
s = a^7 + b^7 + c^7 + d^7 + e^7 + f^7 + g^7;
If[s <= M, Sow[s]]]]]]]]]][[2, 1]] // Union (* Jean-François Alcover, Dec 01 2020 *)

A003378 Numbers that are the sum of 11 positive 7th powers.

Original entry on oeis.org

11, 138, 265, 392, 519, 646, 773, 900, 1027, 1154, 1281, 1408, 2197, 2324, 2451, 2578, 2705, 2832, 2959, 3086, 3213, 3340, 3467, 4383, 4510, 4637, 4764, 4891, 5018, 5145, 5272, 5399, 5526, 6569, 6696, 6823, 6950, 7077, 7204, 7331, 7458, 7585, 8755, 8882, 9009, 9136

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
340152 is in the sequence as 340152 = 1^7 + 2^7 + 3^7 + 3^7 + 3^7 + 3^7 + 3^7 + 4^7 + 4^7 + 4^7 + 6^7.
595136 is in the sequence as 595136 = 1^7 + 2^7 + 2^7 + 2^7 + 2^7 + 3^7 + 5^7 + 5^7 + 5^7 + 5^7 + 6^7.
978139 is in the sequence as 978139 = 3^7 + 3^7 + 3^7 + 3^7 + 3^7 + 4^7 + 4^7 + 4^7 + 4^7 + 5^7 + 7^7. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001015 (seventh powers).

With[{upto=10000},Select[Total/@Tuples[Range[Floor[Surd[upto,7]]]^7,11]//Union,#<=upto&]] (* Harvey P. Dale, May 13 2018 *)

A003372 Numbers that are the sum of 5 positive 7th powers.

Original entry on oeis.org

5, 132, 259, 386, 513, 640, 2191, 2318, 2445, 2572, 2699, 4377, 4504, 4631, 4758, 6563, 6690, 6817, 8749, 8876, 10935, 16388, 16515, 16642, 16769, 16896, 18574, 18701, 18828, 18955, 20760, 20887, 21014, 22946, 23073, 25132, 32771, 32898, 33025, 33152, 34957, 35084, 35211

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 04 2020: (Start)
16898692 is in the sequence as 16898692 = 3^7 + 4^7 + 8^7 + 9^7 + 10^7.
34270142 is in the sequence as 34270142 = 1^7 + 1^7 + 9^7 + 10^7 + 11^7.
43759626 is in the sequence as 43759626 = 2^7 + 3^7 + 9^7 + 11^7 + 11^7. (End)

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A001015 (seventh powers).

list(lim)=my(v=List(), A, B, C, D); lim\=1; for(a=1, sqrtnint(lim-4, 7), A=a^7; for(b=1, min(sqrtnint(lim-A-3, 7), a), B=A+b^7; for(c=1, min(sqrtnint(lim-B-2, 7), b), C=B+c^7; for(d=1,min(sqrtnint(lim-C-1,7),c), D=C+d^7; for(e=1,min(sqrtnint(lim-D,7),d), listput(v,D+e^7)))))); Set(v) \\ Charles R Greathouse IV, Nov 05 2017

A003375 Numbers that are the sum of 8 positive 7th powers.

Original entry on oeis.org

8, 135, 262, 389, 516, 643, 770, 897, 1024, 2194, 2321, 2448, 2575, 2702, 2829, 2956, 3083, 4380, 4507, 4634, 4761, 4888, 5015, 5142, 6566, 6693, 6820, 6947, 7074, 7201, 8752, 8879, 9006, 9133, 9260, 10938, 11065, 11192, 11319, 13124, 13251, 13378, 15310, 15437, 16391

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
858636 is in the sequence as 858636 = 1^7 + 2^7 + 2^7 + 3^7 + 4^7 + 6^7 + 6^7 + 6^7.
2146435 is in the sequence as 2146435 = 1^7 + 1^7 + 1^7 + 2^7 + 4^7 + 4^7 + 4^7 + 8^7.
2846170 is in the sequence as 2846170 = 2^7 + 4^7 + 4^7 + 5^7 + 5^7 + 6^7 + 6^7 + 8^7. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001015.

A003376 Numbers that are the sum of 9 positive 7th powers.

Original entry on oeis.org

9, 136, 263, 390, 517, 644, 771, 898, 1025, 1152, 2195, 2322, 2449, 2576, 2703, 2830, 2957, 3084, 3211, 4381, 4508, 4635, 4762, 4889, 5016, 5143, 5270, 6567, 6694, 6821, 6948, 7075, 7202, 7329, 8753, 8880, 9007, 9134, 9261, 9388, 10939, 11066, 11193, 11320, 11447, 13125

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
867374 is in the sequence as 867374 = 2^7 + 3^7 + 3^7 + 3^7 + 3^7 + 3^7 + 4^7 + 4^7 + 7^7.
1388047 is in the sequence as 1388047 = 1^7 + 1^7 + 2^7 + 2^7 + 3^7 + 3^7 + 6^7 + 6^7 + 7^7.
1688730 is in the sequence as 1688730 = 2^7 + 3^7 + 3^7 + 3^7 + 3^7 + 4^7 + 4^7 + 7^7 + 7^7. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001015 (seventh powers).

A003377 Numbers that are the sum of 10 positive 7th powers.

Original entry on oeis.org

10, 137, 264, 391, 518, 645, 772, 899, 1026, 1153, 1280, 2196, 2323, 2450, 2577, 2704, 2831, 2958, 3085, 3212, 3339, 4382, 4509, 4636, 4763, 4890, 5017, 5144, 5271, 5398, 6568, 6695, 6822, 6949, 7076, 7203, 7330, 7457, 8754, 8881, 9008, 9135, 9262, 9389, 9516, 10940

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
356407 is in the sequence as 356407 = 3^7 + 3^7 + 3^7 + 3^7 + 3^7 + 4^7 + 4^7 + 4^7 + 4^7 + 6^7.
926683 is in the sequence as 926683 = 1^7 + 1^7 + 3^7 + 3^7 + 3^7 + 3^7 + 5^7 + 6^7 + 6^7 + 6^7.
1216570 is in the sequence as 1216570 = 1^7 + 1^7 + 2^7 + 3^7 + 4^7 + 5^7 + 6^7 + 6^7 + 6^7 + 6^7. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001015 (seventh powers).

Offset corrected by David A. Corneth, Aug 03 2020

A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 8, 1, 0, 1, 5, 16, 27, 16, 1, 0, 1, 6, 25, 64, 81, 32, 1, 0, 1, 7, 36, 125, 256, 243, 64, 1, 0, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 0, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 0, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 0

Offset: 0

Marc LeBrun

If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - Robert A. Russell, Nov 08 2018

T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - Geoffrey Critzer, Apr 03 2020

			Rows begin:
[1, 0,  0,   0,    0,     0,      0,      0, ...],
[1, 1,  1,   1,    1,     1,      1,      1, ...],
[1, 2,  4,   8,   16,    32,     64,    128, ...],
[1, 3,  9,  27,   81,   243,    729,   2187, ...],
[1, 4, 16,  64,  256,  1024,   4096,  16384, ...],
[1, 5, 25, 125,  625,  3125,  15625,  78125, ...],
[1, 6, 36, 216, 1296,  7776,  46656, 279936, ...],
[1, 7, 49, 343, 2401, 16807, 117649, 823543, ...], ...

T. D. Noe, Rows n = 0..50 of triangle, flattened

Rows 0-49 are A000007, A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A011557, A001020, A001021, A001022, A001023, A001024, A001025, A001026, A001027, A001029, A009964-A009992, A087752.
Columns 0-26 are A000012, A001477, A000290, A000578, A000583, A000584, A001014, A001015, A001016, A001017, A008454, A008455, A008456, A010801-A010813, A089081.
Main diagonal is A000312. Other diagonals include A000169, A007778, A000272, A008788. Antidiagonal sums are in A026898.
Cf. A099555.
Transpose is A004248. See A051128, A095884, A009999 for other versions.
Cf. A277504 (unoriented), A293500 (chiral).

[[(n-k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 08 2018

Table[If[k == 0, 1, (n - k)^k], {n, 0, 11}, {k, 0, n}]//Flatten

T(n,k) = (n-k)^k \\ Charles R Greathouse IV, Feb 07 2017

E.g.f.: Sum T(n,k)*x^n*y^k/k! = 1/(1-x*exp(y)). - Paul D. Hanna, Oct 22 2004

E.g.f.: Sum T(n,k)*x^n/n!*y^k/k! = e^(x*e^y). - Franklin T. Adams-Watters, Jun 23 2006

More terms from David W. Wilson

Edited by Paul D. Hanna, Oct 22 2004

A022523 Nexus numbers (n+1)^7-n^7.

Original entry on oeis.org

1, 127, 2059, 14197, 61741, 201811, 543607, 1273609, 2685817, 5217031, 9487171, 16344637, 26916709, 42664987, 65445871, 97576081, 141903217, 201881359, 281651707, 386128261, 521088541, 693269347, 910467559, 1181645977, 1517044201, 1928294551, 2428543027

Offset: 0

N. J. A. Sloane

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 54.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
H. D. Nguyen, D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013; citseerx. Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

First differences of A001015.
Cf. A008292, A022522, A022524.
Column k=6 of array A047969.

[(n+1)^7-n^7: n in [0..30]]; // Vincenzo Librandi, Nov 22 2011

q=7;lst={};Do[AppendTo[lst,(n+1)^q-n^q],{n,0,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 23 2009 *)
Table[(n+1)^7-n^7,{n,0,20}] (* Vincenzo Librandi, Nov 22 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,127,2059,14197,61741,201811,543607},30] (* Harvey P. Dale, Apr 17 2017 *)
Differences[Range[0,50]^7] (* Harvey P. Dale, Jun 07 2023 *)

a(n)=(n+1)^7-n^7 \\ Charles R Greathouse IV, Sep 28 2015

G.f.: -(x^6+120*x^5+1191*x^4+2416*x^3+1191*x^2+120*x+1) / (x-1)^7. - Colin Barker, Dec 21 2012

G.f.: polylog(-7, x)*(1-x)/x. See the g.f. of the rows of A008292 by Vladeta Jovovic, Sep 02 2002. - Wolfdieter Lang, May 10 2021

cp's OEIS Frontend

A092759 a(n) = prime(n)^7.

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A003370 Numbers that are the sum of 3 positive 7th powers.

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A003374 Numbers that are the sum of 7 positive 7th powers.

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Mathematica

A003378 Numbers that are the sum of 11 positive 7th powers.

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Mathematica

A003372 Numbers that are the sum of 5 positive 7th powers.

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A003375 Numbers that are the sum of 8 positive 7th powers.

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A003376 Numbers that are the sum of 9 positive 7th powers.

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A003377 Numbers that are the sum of 10 positive 7th powers.

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A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

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