A000584 -id:A000584 - cp's OEIS frontend

A003357 Numbers that are the sum of 12 positive 5th powers.

12, 43, 74, 105, 136, 167, 198, 229, 254, 260, 285, 291, 316, 322, 347, 353, 378, 384, 409, 440, 471, 496, 502, 527, 533, 558, 564, 589, 595, 620, 651, 682, 713, 738, 744, 769, 775, 800, 806, 831, 862, 893, 924, 955, 980, 986, 1011, 1017, 1035, 1042, 1066, 1073, 1097

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
14585 is in the sequence as 14585 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 4^5 + 4^5 + 5^5 + 5^5 + 5^5 + 5^5.
22088 is in the sequence as 22088 = 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5 + 3^5 + 3^5 + 4^5 + 5^5 + 7^5.
24800 is in the sequence as 24800 = 2^5 + 4^5 + 4^5 + 4^5 + 4^5 + 4^5 + 4^5 + 4^5 + 4^5 + 4^5 + 6^5 + 6^5. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A000584 (fifth powers).

Cf. A003347 - A003356 (numbers that are the sum of 2, ..., 11 positive 5th powers); A003335, A003346, A003368, A003379, A003390, A004801, A004812, A004823 (numbers that are the sum of 12 positive 3rd, ..., 11th powers).

(A003357_upto(N, k=12, m=5)=[n|n<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, n)])(1100) \\ 2nd & 3rd optional arg allow to get other sequences of this group. See A003333 for alternate code. - M. F. Hasler, Aug 03 2020

A003354 Numbers that are the sum of 9 positive 5th powers.

Original entry on oeis.org

9, 40, 71, 102, 133, 164, 195, 226, 251, 257, 282, 288, 313, 344, 375, 406, 437, 468, 493, 499, 524, 555, 586, 617, 648, 679, 710, 735, 766, 797, 828, 859, 890, 921, 977, 1008, 1032, 1039, 1063, 1070, 1094, 1101, 1125, 1132, 1156, 1187, 1218, 1219, 1249, 1250, 1274

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
23946 is in the sequence as 23946 = 2^5 + 2^5 + 2^5 + 4^5 + 4^5 + 5^5 + 5^5 + 6^5 + 6^5.
28955 is in the sequence as 28955 = 2^5 + 3^5 + 3^5 + 3^5 + 3^5 + 3^5 + 5^5 + 6^5 + 7^5.
44054 is in the sequence as 44054 = 1^5 + 4^5 + 4^5 + 5^5 + 6^5 + 6^5 + 6^5 + 6^5 + 6^5. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A000584, A003343, A003353, A003355, A345619, A346336.

A008455 11th powers: a(n) = n^11.

Original entry on oeis.org

0, 1, 2048, 177147, 4194304, 48828125, 362797056, 1977326743, 8589934592, 31381059609, 100000000000, 285311670611, 743008370688, 1792160394037, 4049565169664, 8649755859375, 17592186044416, 34271896307633, 64268410079232, 116490258898219, 204800000000000

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A000583, A000584, A001014, A001017, A008454, A013669.

Cf. A004813 - A004823 (sums of 2, ..., 12 positive eleventh powers).

[n^11: n in [0..40]]; // Vincenzo Librandi, Jul 05 2014

Table[n^11, {n, 0, 30}] (* Vincenzo Librandi, Jul 05 2014 *)

A008455(n):=n^11$ makelist(A008455(n),n,0,20); /* Martin Ettl, Dec 17 2012 */

A008455(n)=n^11 \\ M. F. Hasler, Jul 03 2025

A008455 = lambda n: n**11 # M. F. Hasler, Jul 03 2025

a(n) = A000584(n)*A001014(n).
Multiplicative with a(p^e) = p^(11*e). - David W. Wilson, Aug 01 2001
Totally multiplicative with a(p) = p^11 for primes p. - Jaroslav Krizek, Nov 01 2009
From Amiram Eldar, Oct 08 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(11) (A013669).
Sum_{n>=1} (-1)^(n+1)/a(n) = 1023*zeta(11)/1024. (End)

A008454 Tenth powers: a(n) = n^10.

Original entry on oeis.org

0, 1, 1024, 59049, 1048576, 9765625, 60466176, 282475249, 1073741824, 3486784401, 10000000000, 25937424601, 61917364224, 137858491849, 289254654976, 576650390625, 1099511627776, 2015993900449, 3570467226624, 6131066257801, 10240000000000, 16679880978201, 26559922791424

Offset: 0

N. J. A. Sloane

Fifth powers of the squares and the squares of fifth powers. - Wesley Ivan Hurt, Apr 01 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

a(n) = A123867(n) + 1.
Cf. A000290 (n^2), A000584 (n^5), A013668.
Cf. A004802 - A004812 (sums of 2, ..., 12 nonzero tenth powers).

[n^10: n in [0..20]]; // Vincenzo Librandi, Jun 20 2011

A008454:=n->n^10; seq(A008454(n), n=0..20); # Wesley Ivan Hurt, Jan 22 2014

Table[n^10,{n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2010 *)

A008454(n)=n^10 \\ M. F. Hasler, Jul 03 2025

A008454 = lambda n: n**10 # M. F. Hasler, Jul 03 2025

Multiplicative with a(p^e) = p^(10e). - David W. Wilson, Aug 01 2001
Totally multiplicative sequence with a(p) = p^10 for primes p. - Jaroslav Krizek, Nov 01 2009
From Robert Israel, Mar 31 2016: (Start)
G.f.: x*(x + 1)*(x^8 + 1012*x^7 + 46828*x^6 + 408364*x^5 + 901990*x^4 + 408364*x^3 + 46828*x^2 + 1012*x + 1)/(1 - x)^11.
E.g.f.: x*exp(x)*(x^9 + 45*x^8 + 750*x^7 + 5880*x^6 + 22827*x^5 + 42525*x^4 + 34105*x^3 + 9330*x^2 + 511*x + 1). (End)
From Amiram Eldar, Oct 08 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(10) = Pi^10/93555 (A013668).
Sum_{n>=1} (-1)^(n+1)/a(n) = 511*zeta(10)/512 = 73*Pi^10/6842880. (End)

A003355 Numbers that are the sum of 10 positive 5th powers.

Original entry on oeis.org

10, 41, 72, 103, 134, 165, 196, 227, 252, 258, 283, 289, 314, 320, 345, 376, 407, 438, 469, 494, 500, 525, 531, 556, 587, 618, 649, 680, 711, 736, 742, 767, 798, 829, 860, 891, 922, 953, 978, 1009, 1033, 1040, 1064, 1071, 1095, 1102, 1126, 1133, 1157, 1164, 1188, 1219

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
23227 is in the sequence as 23227 = 2^5 + 3^5 + 3^5 + 3^5 + 3^5 + 3^5 + 4^5 + 4^5 + 5^5 + 7^5.
24600 is in the sequence as 24600 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5 + 4^5 + 6^5 + 6^5 + 6^5.
32894 is in the sequence as 32894 = 1^5 + 3^5 + 4^5 + 4^5 + 4^5 + 5^5 + 5^5 + 6^5 + 6^5 + 6^5. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A000584, A003344, A003354, A345634, A346346.

A003348 Numbers that are the sum of 3 positive 5th powers.

Original entry on oeis.org

3, 34, 65, 96, 245, 276, 307, 487, 518, 729, 1026, 1057, 1088, 1268, 1299, 1510, 2049, 2080, 2291, 3072, 3127, 3158, 3189, 3369, 3400, 3611, 4150, 4181, 4392, 5173, 6251, 6282, 6493, 7274, 7778, 7809, 7840, 8020, 8051, 8262, 8801, 8832, 9043, 9375, 9824, 10902, 10933

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
2656719 is in the sequence as 2656719 = 6^5 + 15^5 + 18^5.
8167555 is in the sequence as 8167555 = 14^5 + 19^5 + 22^5.
15646315 is in the sequence as 15646315 = 12^5 + 16^5 + 27^5. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A000584, A003337, A003347, A003349, A344641, A345010.

A003356 Numbers that are the sum of 11 positive 5th powers.

Original entry on oeis.org

11, 42, 73, 104, 135, 166, 197, 228, 253, 259, 284, 290, 315, 321, 346, 352, 377, 408, 439, 470, 495, 501, 526, 532, 557, 563, 588, 619, 650, 681, 712, 737, 743, 768, 774, 799, 830, 861, 892, 923, 954, 979, 985, 1010, 1034, 1041, 1065, 1072, 1096, 1103, 1127, 1134

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
16989 is in the sequence as 16989 = 1^5 + 2^5 + 2^5 + 2^5 + 3^5 + 4^5 + 5^5 + 5^5 + 5^5 + 5^5 + 5^5.
22564 is in the sequence as 22564 = 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5 + 4^5 + 4^5 + 5^5 + 7^5.
30191 is in the sequence as 30191 = 1^5 + 3^5 + 3^5 + 3^5 + 3^5 + 3^5 + 3^5 + 4^5 + 5^5 + 6^5 + 7^5. (End)

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

Cf. A000584 (fifth powers).
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

Incorrect program removed by David A. Corneth, Aug 01 2020

A022521 a(n) = (n+1)^5 - n^5.

Original entry on oeis.org

1, 31, 211, 781, 2101, 4651, 9031, 15961, 26281, 40951, 61051, 87781, 122461, 166531, 221551, 289201, 371281, 469711, 586531, 723901, 884101, 1069531, 1282711, 1526281, 1803001, 2115751, 2467531

Offset: 0

N. J. A. Sloane

Last digit of a(n) is always 1. Last two digits of a(n) (i.e., a(n) mod 100) are repeated periodically with palindromic part of period 20 {1,31,11,81,1,51,31,61,81,51,51,81,61,31,51,1,81,11,31,1}. Last three digits of a(n) (i.e., a(n) mod 1000) are repeated periodically with palindromic part of period 200. - Alexander Adamchuk, Aug 11 2006
In Conway and Guy, these numbers are called nexus numbers of order 5. - M. F. Hasler, Jan 27 2013
Numbers that can be arranged in a triangular-antitegmatic icosachoron (the 4D version of "rhombic dodecahedal numbers" (A005917)). - Steven Lu, Mar 28 2023

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

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Polytope Wiki, Triangular-antitegmatic Icosachoron
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

First differences of A000584.
Column k=4 of array A047969.
Cf. A003215, A005917, A006322, A008292, A019916, A019952, A022522.

[(n+1)^5-n^5: n in [0..30]]; // Vincenzo Librandi, Nov 23 2011

Table[(n+1)^5-n^5, {n,0,30}] (* Vincenzo Librandi, Nov 23 2011 *)

a(n)=(n+1)^5-n^5 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = A003215(n) + 24 * A006322(n). - Xavier Acloque, Oct 11 2003
G.f.: (-1-x^4-26*x^3-66*x^2-26*x)/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
G.f.: polylog(-5, x)*(1-x)/x. See the g.f. of the rows of A008292 by Vladeta Jovovic, Sep 02 2002. - Wolfdieter Lang, May 10 2021
Sum_{n>=0} 1/a(n) = c1*tanh(c2/2) - c2*tanh(c1/2), where c1 = tan(3*Pi/10)*Pi and c2 = tan(Pi/10)*Pi. - Amiram Eldar, Jan 27 2022

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

A262675 Exponentially evil numbers.

Original entry on oeis.org

1, 8, 27, 32, 64, 125, 216, 243, 343, 512, 729, 864, 1000, 1024, 1331, 1728, 1944, 2197, 2744, 3125, 3375, 4000, 4096, 4913, 5832, 6859, 7776, 8000, 9261, 10648, 10976, 12167, 13824, 15552, 15625, 16807, 17576, 19683, 21952, 23328, 24389, 25000, 27000, 27648, 29791

Offset: 1

Vladimir Shevelev, Sep 27 2015

Or the numbers whose prime power factorization contains primes only in evil exponents (A001969): 0, 3, 5, 6, 9, 10, 12, ...
If n is in the sequence, then n^2 is also in the sequence.
A268385 maps each term of this sequence to a unique nonzero square (A000290), and vice versa. - Antti Karttunen, May 26 2016

			864 = 2^5*3^3; since 5 and 3 are evil numbers, 864 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Subsequence of A036966.
Apart from 1, a subsequence of A270421.
Indices of ones in A270418.
Sequence A270437 sorted into ascending order.
Cf. A001969, A209061, A138302, A197680, A000578, A000584, A001014, A001017, A008456, A010803, A010805, A010806, A010808, A010811, A010812, A001221, A010059, A124010, A268385, A270428.

a262675 n = a262675_list !! (n-1)
a262675_list = filter
   (all (== 1) . map (a010059 . fromIntegral) . a124010_row) [1..]
-- Reinhard Zumkeller, Oct 25 2015

{1}~Join~Select[Range@ 30000, AllTrue[Last /@ FactorInteger[#], EvenQ@ First@ DigitCount[#, 2] &] &] (* Michael De Vlieger, Sep 27 2015, Version 10 *)
expEvilQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], EvenQ[DigitCount[#, 2, 1]] &]; With[{max = 30000}, Select[Union[Flatten[Table[i^2*j^3, {j, Surd[max, 3]}, {i, Sqrt[max/j^3]}]]], expEvilQ]] (* Amiram Eldar, Dec 01 2023 *)

isok(n) = {my(f = factor(n)); for (i=1, #f~, if (hammingweight(f[i,2]) % 2, return (0));); return (1);} \\ Michel Marcus, Sep 27 2015

use ntheory ":all"; sub isok { my @f = factor_exp($[0]); return scalar(grep { !(hammingweight($->[1]) % 2) } @f) == @f; } # Dana Jacobsen, Oct 26 2015

Product_{k=1..A001221(n)} A010059(A124010(n,k)) = 1. - Reinhard Zumkeller, Oct 25 2015

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=2} 1/p^A001969(k)) = Product_{p prime} f(1/p) = 1.2413599378..., where f(x) = (1/(1-x) + Product_{k>=0} (1 - x^(2^k)))/2. - Amiram Eldar, May 18 2023, Dec 01 2023

More terms from Michel Marcus, Sep 27 2015

cp's OEIS Frontend

A003357 Numbers that are the sum of 12 positive 5th powers.

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A003354 Numbers that are the sum of 9 positive 5th powers.

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A008455 11th powers: a(n) = n^11.

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A008454 Tenth powers: a(n) = n^10.

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A003355 Numbers that are the sum of 10 positive 5th powers.

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A003348 Numbers that are the sum of 3 positive 5th powers.

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A003356 Numbers that are the sum of 11 positive 5th powers.

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A022521 a(n) = (n+1)^5 - n^5.

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