A001017 -id:A001017 - cp's OEIS frontend

A003398 Numbers that are the sum of 9 positive 9th powers.

9, 520, 1031, 1542, 2053, 2564, 3075, 3586, 4097, 4608, 19691, 20202, 20713, 21224, 21735, 22246, 22757, 23268, 23779, 39373, 39884, 40395, 40906, 41417, 41928, 42439, 42950, 59055, 59566, 60077, 60588, 61099, 61610, 62121, 78737, 79248, 79759, 80270, 80781, 81292, 98419

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 02 2020: (Start)
26015282 is in the sequence as 26015282 = 1^9 + 1^9 + 1^9 + 2^9 + 5^9 + 5^9 + 5^9 + 6^9 + 6^9.
56248367 is in the sequence as 56248367 = 2^9 + 5^9 + 5^9 + 5^9 + 6^9 + 6^9 + 6^9 + 6^9 + 6^9.
84613980 is in the sequence as 84613980 = 1^9 + 1^9 + 1^9 + 1^9 + 2^9 + 5^9 + 5^9 + 7^9 + 7^9. (End)

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

Cf. A001017 (9th powers).

Column k=9 of A336725.

M = 10^5; m = M^(1/9) // 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++,
For[h = g, h <= m, h++, For[i = h, i <= m, i++,
s = a^9 + b^9 + c^9 + d^9 + e^9 + f^9 + g^9 + h^9 + i^9;
If[s <= M, Sow[s]]]]]]]]]]]][[2, 1]] // Union (* Jean-François Alcover, Dec 01 2020 *)

A003392 Numbers that are the sum of 3 positive 9th powers.

Original entry on oeis.org

3, 514, 1025, 1536, 19685, 20196, 20707, 39367, 39878, 59049, 262146, 262657, 263168, 281828, 282339, 301510, 524289, 524800, 543971, 786432, 1953127, 1953638, 1954149, 1972809, 1973320, 1992491, 2215270, 2215781, 2234952, 2477413, 3906251, 3906762, 3925933, 4168394, 5859375

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 04 2020: (Start)
391407194198 is in the sequence as 391407194198 = 3^9 + 16^9 + 19^9.
2313193015070 is in the sequence as 2313193015070 = 7^9 + 20^9 + 23^9.
11440294750613 is in the sequence as 11440294750613 = 1^9 + 25^9 + 27^9. (End)

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

Cf. A001017 (ninth powers).

N:= 10^10: # For all terms <= N
B:= floor(N^(1/9)):
S:=select(t -> t <= N, {seq(seq(seq(i^9+j^9+k^9,k=j..B),j=i..B),i=1..B)}):
sort(convert(S,list)); # Robert Israel, Mar 04 2020

A003394 Numbers that are the sum of 5 positive 9th powers.

Original entry on oeis.org

5, 516, 1027, 1538, 2049, 2560, 19687, 20198, 20709, 21220, 21731, 39369, 39880, 40391, 40902, 59051, 59562, 60073, 78733, 79244, 98415, 262148, 262659, 263170, 263681, 264192, 281830, 282341, 282852, 283363, 301512, 302023, 302534, 321194, 321705, 340876, 524291, 524802

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 04 2020: (Start)
1429727733 is in the sequence as 1429727733 = 2^9 + 5^9 + 7^9 + 9^9 + 10^9.
2533043314 is in the sequence as 2533043314 = 4^9 + 4^9 + 7^9 + 8^9 + 11^9.
6322041820 is in the sequence as 6322041820 = 1^9 + 9^9 + 9^9 + 9^9 + 12^9. (End)

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

Cf. A001017 (ninth powers).

A003395 Numbers that are the sum of 6 positive 9th powers.

Original entry on oeis.org

6, 517, 1028, 1539, 2050, 2561, 3072, 19688, 20199, 20710, 21221, 21732, 22243, 39370, 39881, 40392, 40903, 41414, 59052, 59563, 60074, 60585, 78734, 79245, 79756, 98416, 98927, 118098, 262149, 262660, 263171, 263682, 264193, 264704, 281831, 282342, 282853, 283364, 283875

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
205066567 is in the sequence as 205066567 = 4^9 + 6^9 + 6^9 + 6^9 + 7^9 + 8^9.
565898075 is in the sequence as 565898075 = 1^9 + 5^9 + 5^9 + 7^9 + 8^9 + 9^9.
1508481311 is in the sequence as 1508481311 = 1^9 + 7^9 + 7^9 + 7^9 + 9^9 + 10^9. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Cf. A001017 (ninth powers).

With[{nn=4},Select[Total/@Tuples[Range[nn]^9,6]//Union,#<(nn+1)^9+5&]] (* Harvey P. Dale, Apr 28 2018 *)

A179665 a(n) = prime(n)^9.

Original entry on oeis.org

512, 19683, 1953125, 40353607, 2357947691, 10604499373, 118587876497, 322687697779, 1801152661463, 14507145975869, 26439622160671, 129961739795077, 327381934393961, 502592611936843, 1119130473102767

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 23 2010

Product_{n >= 2, m_n = (a(n) mod 4) - 2} ((a(n) + 1) / (a(n) - 1))^m_n = 209865342976 / 209844223875. - Dimitris Valianatos, May 13 2020

			a(1) = 512 since the ninth power of the first prime is 2^9 = 512. - _Wesley Ivan Hurt_, Mar 27 2014

T. D. Noe, Table of n, a(n) for n = 1..1000
Xavier Gourdon and Pascal Sebah, Some Constants from Number theory.
Will Nicholes, Prime Signatures.

Cf. A001248, A030078, A030514, A050997, A030516, A092759, A179645.

Cf. A013667, A013676, A085969.

[p^9: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

A179665:=n->ithprime(n)^9; seq(A179665(n), n=1..30); # Wesley Ivan Hurt, Mar 27 2014

Array[Prime[ # ]^9&, 30]
Prime[Range[30]]^9 (* Harvey P. Dale, Jul 20 2024 *)

a(n)=prime(n)^9 \\ Charles R Greathouse IV, Jul 20 2011

a(n) = A000040(n)^9 = A001017(A000040(n)). - Wesley Ivan Hurt, Mar 27 2014
Sum_{n>=1} 1/a(n) = P(9) = 0.0020044675... (A085969). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 24 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(9)/zeta(18) = A013667/A013676.
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(9) = 1/A013667. (End)

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

A017505 a(n) = (11*n + 9)^9.

Original entry on oeis.org

387420489, 512000000000, 26439622160671, 406671383849472, 3299763591802133, 18014398509481984, 75084686279296875, 257327417311663616, 760231058654565217, 1999004627104432128, 4785448563124474679

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 (10,-45,120,-210,252,-210,120,-45,10,-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), A017503 (m=7), A017504 (m=8), this sequence (m=9), A017506 (m=10), A017607 (m=11), A017508 (m=12).

Subsequence of A001017.

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

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

seq((11*n+9)^9, n=0..20); # G. C. Greubel, Oct 28 2019

(11*Range[20] -2)^9 (* G. C. Greubel, Oct 28 2019 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{387420489,512000000000,26439622160671,406671383849472,3299763591802133,18014398509481984,75084686279296875,257327417311663616,760231058654565217,1999004627104432128},20] (* Harvey P. Dale, Nov 18 2022 *)

makelist((11*n+9)^9, n, 0, 30); /* Martin Ettl, Oct 21 2012 */

vector(21, n, (11*n-2)^9) \\ G. C. Greubel, Oct 28 2019

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

From G. C. Greubel, Oct 28 2019: (Start)
G.f.: (387420489 + 508125795110*x + 21337056082676*x^2 + 165268671784082*x^3 + 361474108840298*x^4 + 251642575443146*x^5 + 52874765679980*x^6 + 2535762569534*x^7 + 10604494253*x^8 + 512*x^9)/(1-x)^10.
E.g.f.: (387420489 + 511612579511*x + 12708004790580*x^2 + 54814688324495* x^3 + 76236174032865*x^4 + 44337148166157*x^5 + 12159505753164*x^6 + 1632362365986*x^7 + 102249186237*x^8 + 2357947691*x^9)*exp(x). (End)

A009714 a(n) = Product_{i=0..8} floor((n+i)/9).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 19683, 26244, 34992, 46656, 62208, 82944, 110592, 147456, 196608, 262144, 327680, 409600, 512000, 640000, 800000, 1000000, 1250000, 1562500

Offset: 0

N. J. A. Sloane

For n >= 9, a(n) is the maximal product of 9 positive integers with sum n. - Wesley Ivan Hurt, Jul 08 2022

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 0, 0, 8, -16, 8, 0, 0, 0, 0, 0, 0, -28, 56, -28, 0, 0, 0, 0, 0, 0, 56, -112, 56, 0, 0, 0, 0, 0, 0, -70, 140, -70, 0, 0, 0, 0, 0, 0, 56, -112, 56, 0, 0, 0, 0, 0, 0, -28, 56, -28, 0, 0, 0, 0, 0, 0, 8, -16, 8, 0, 0, 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), A009641 (k=7), A009694 (k=8), this sequence (k=9), A354600 (k=10).

Cf. A001017 (n^9, a subsequence), A013667.

A009714[n_] := Product[Floor[(n + i)/9], {i, 0, 8}];
Array[A009714, 50, 0] (* Paolo Xausa, Aug 21 2024 *)

a(n) = prod(k=0, 8, floor((n+k)/9)); \\ Georg Fischer, Nov 07 2019

a(9*n) = n^9. - Bernard Schott, Nov 20 2022
a(9*n+j) = n^(9-j)*(n+1)^j for 0 <= j <= 8. - Robert Israel, Nov 21 2022
Sum_{n>=9} 1/a(n) = 1 + zeta(9). - Amiram Eldar, Jan 10 2023

A343619 Decimal expansion of P_{3,2}(9) = Sum 1/p^9 over primes == 2 (mod 3).

Original entry on oeis.org

0, 0, 1, 9, 5, 3, 6, 3, 7, 4, 3, 3, 1, 5, 8, 7, 1, 3, 7, 2, 0, 8, 0, 4, 6, 0, 1, 5, 1, 2, 3, 9, 2, 9, 1, 7, 6, 0, 6, 9, 3, 3, 5, 0, 0, 3, 9, 1, 2, 2, 2, 0, 6, 4, 6, 2, 9, 1, 6, 2, 6, 1, 3, 4, 0, 4, 2, 4, 6, 8, 4, 9, 4, 2, 8, 9, 4, 9, 0, 3, 2, 9, 3, 0, 3, 4, 2, 1, 7, 9, 3, 7, 8, 2, 6, 9, 0, 7, 9, 1, 2, 4, 6, 3, 8

Offset: 0

M. F. Hasler, Apr 25 2021

The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.

			0.0019536374331587137208046015123929176069335003912220646291626134042468494...

Jean-François Alcover, Table of n, a(n) for n = 0..1005
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=3, n=2, s=9), p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A001017 (n^9), A085969 (PrimeZeta(9)).
Cf. A343612 - A343618 (P_{3,2}(s): analog for 1/p^s, s = 2 .. 8).
Cf. A343629 (for primes 3k+1), A086039 (for primes 4k+1), A085998 (for primes 4k+3).

digits = 1004; nmax0 = 50; dnmax = 10;
Clear[PrimeZeta31];
PrimeZeta31[s_, nmax_] := PrimeZeta31[s, nmax] = Sum[Module[{t}, t = s + 2 n*s; MoebiusMu[2 n + 1] ((1/(4 n + 2)) (-Log[1 + 2^t] - Log[1 + 3^t] + Log[Zeta[t]] - Log[Zeta[2 t]] + Log[Zeta[t, 1/6] - Zeta[t, 5/6]]))], {n, 0, nmax}] // N[#, digits + 5] &;
PrimeZeta31[9, nmax = nmax0];
PrimeZeta31[9, nmax += dnmax];
While[Abs[PrimeZeta31[9, nmax] - PrimeZeta31[9, nmax - dnmax]] > 10^-(digits + 5), Print["nmax = ", nmax]; nmax += dnmax];
PrimeZeta32[9] = PrimeZetaP[9] - 1/3^9 - PrimeZeta31[9, nmax];
Join[{0, 0}, RealDigits[PrimeZeta32[9], 10, digits][[1]] ] (* Jean-François Alcover, May 07 2021, after M. F. Hasler's PARI code *)

A343619_upto(N=100)={localprec(N+5); digits((PrimeZeta32(9)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32

P_{3,2}(9) = Sum_{p in A003627} 1/p^9 = P(9) - 1/3^9 - P_{3,1}(9).

cp's OEIS Frontend

A003398 Numbers that are the sum of 9 positive 9th powers.

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A003392 Numbers that are the sum of 3 positive 9th powers.

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A003394 Numbers that are the sum of 5 positive 9th powers.

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A003395 Numbers that are the sum of 6 positive 9th powers.

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A179665 a(n) = prime(n)^9.

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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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A262675 Exponentially evil numbers.

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A017505 a(n) = (11*n + 9)^9.

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