A008456 -id:A008456 - cp's OEIS frontend

A253712 Second partial sums of 12th powers (A008456).

1, 4098, 539636, 17852390, 279305769, 2717541484, 18997064400, 103996064052, 471424600185, 1838853136318, 6344710049172, 19766667410282, 56486709893873, 149900664752760, 373060957502272, 877696226962440, 1964953733652369, 4209042621768474, 8666446428950740, 17219850236133006, 33129081554701913, 61893315504320036

Offset: 1

Luciano Ancora, Jan 12 2015

nonn

The formula for the second partial sums of m-th powers is: b(n,m) = (n+1)*F(m) - F(m+1), where F(m) are the m-th Faulhaber's formulas.

Luciano Ancora, Recurrence relation for the second partial sums of m-th powers
Luciano Ancora, Second partial sums of the m-th powers

[(n+1)^2*n*(n+2)*(30*n^10+300*n^9+925*n^8+200*n^7-3022*n^6-772*n^5+7073*n^4-1228*n^3-7888*n^2+5528*n-691)/5460: n in [1..30]]; // Vincenzo Librandi, Jan 19 2015

RecurrenceTable[{a[n] == 2 a[n - 1] - a[n - 2] + n^12, a[1] == 1, a[2] == 4098}, a, {n, 1, 25}] (* Bruno Berselli, Jan 19 2015 *)
Table[(n + 1)^2 n (n + 2) (30 n^10 + 300 n^9 + 925 n^8 + 200 n^7 - 3022 n^6 - 772 n^5 + 7073 n^4 - 1228 n^3 - 7888 n^2 + 5528 n - 691)/5460, {n, 1, 25}] (* Vincenzo Librandi, Jan 19 2015 *)
Nest[Accumulate[#]&,Range[30]^12,2] (* Harvey P. Dale, Aug 17 2020 *)

a(n) = (n+1)^2*n*(n+2)*(30*n^10+300*n^9+925*n^8+200*n^7-3022*n^6-772*n^5+7073*n^4-1228*n^3-7888*n^2+5528*n-691)/5460.
a(n) = 2*a(n-1)-a(n-2)+n^12.
G.f.: x*(1 + 4083*x + 478271*x^2 + 10187685*x^3 + 66318474*x^4 + 162512286*x^5 + 162512286*x^6 + 66318474*x^7 + 10187685*x^8 + 478271*x^9 + 4083*x^10 + x^11)/(1-x)^15. - Vincenzo Librandi, Jan 19 2015

a(22) corrected by Vincenzo Librandi, Jan 19 2015

A268335 Exponentially odd numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97

Offset: 1

Vladimir Shevelev, Feb 01 2016

nonn

The sequence is formed by 1 and the numbers whose prime power factorization contains only odd exponents.
The density of the sequence is the constant given by A065463.
Except for the first term the same as A002035. - R. J. Mathar, Feb 07 2016
Also numbers k all of whose divisors are bi-unitary divisors (i.e., A286324(k) = A000005(k)). - Amiram Eldar, Dec 19 2018
The term "exponentially odd integers" was apparently coined by Cohen (1960). These numbers were also called "unitarily 2-free", or "2-skew", by Cohen (1961). - Amiram Eldar, Jan 22 2024

Peter J. C. Moses, Table of n, a(n) for n = 1..2000
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74 (1960), pp. 66-80.
Eckford Cohen, Some sets of integers related to the k-free integers, Acta Sci. Math. (Szeged), Vol. 22, No. 3-4 (1961), pp. 223-233.
Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv preprint arXiv:1511.03860 [math.NT], 2015.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.
D. Suryanarayana and R. Sita Rama Chandra Rao, Distribution of unitarily k-free integers, Journal of the Australian Mathematical Society, Vol. 20 , No. 2 (1975), pp. 129-141.

Cf. A002035, A209061, A138302, A197680, A000578, A000584, A001014, A001017, A008456, A010803, A010805, A010806, A010808, A010811, A010812, A001221, A124010.

Select[Range@ 100, AllTrue[Last /@ FactorInteger@ #, OddQ] &] (* Version 10, or *)
Select[Range@ 100, Times @@ Boole[OddQ /@ Last /@ FactorInteger@ #] == 1 &] (* Michael De Vlieger, Feb 02 2016 *)

isok(n)=my(f = factor(n)); for (k=1, #f~, if (!(f[k,2] % 2), return (0))); 1; \\ Michel Marcus, Feb 02 2016

from itertools import count, islice
from sympy import factorint
def A268335_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(e&1 for e in factorint(n).values()),count(max(startvalue,1)))
A268335_list = list(islice(A268335_gen(),20)) # Chai Wah Wu, Jun 22 2023

Sum_{a(n)<=x} 1 = C*x + O(sqrt(x)*log x*e^(c*sqrt(log x)/(log(log x))), where c = 4*sqrt(2.4/log 2) = 7.44308... and C = Product_{prime p} (1 - 1/p*(p + 1)) = 0.7044422009991... (A065463).

Sum_{n>=1} 1/a(n)^s = zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)), s>1. - Amiram Eldar, Sep 26 2023

A030631 Numbers with 13 divisors.

Original entry on oeis.org

4096, 531441, 244140625, 13841287201, 3138428376721, 23298085122481, 582622237229761, 2213314919066161, 21914624432020321, 353814783205469041, 787662783788549761, 6582952005840035281, 22563490300366186081, 39959630797262576401, 116191483108948578241

Offset: 1

Jeff Burch

12th powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. - Omar E. Pol, May 06 2008

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. A000040, A013670, A030629.

Subsequence of A008456.

[p^12: p in PrimesUpTo(100)]; // Vincenzo Librandi, Mar 27 2014

(Prime@Range@30)^12  (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)

is(n)=isprimepower(n)==12 \\ Charles R Greathouse IV, Jun 19 2016

a(n) = A000040(n)^(13-1) = A000040(n)^(12). - Omar E. Pol, May 06 2008
From Amiram Eldar, Jan 24 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(12)/zeta(24) = 218517792968475/(236364091*Pi^12).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(12) = 638512875/(691*Pi^12) = 1/A013670. (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

A123868 a(n) = n^12 - 1.

Original entry on oeis.org

0, 4095, 531440, 16777215, 244140624, 2176782335, 13841287200, 68719476735, 282429536480, 999999999999, 3138428376720, 8916100448255, 23298085122480, 56693912375295, 129746337890624, 281474976710655, 582622237229760, 1156831381426175, 2213314919066160

Offset: 1

Reinhard Zumkeller, Oct 16 2006

a(n) mod 13 = 0 iff n mod 13 > 0; a(A008595(n)) = 12; a(A113763(n)) = 0.

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A024010, A008456, A005563, A123865, A123866, A123867.

List([1..20], n-> n^12 -1); # G. C. Greubel, Aug 08 2019

[n^12 -1:n in [1..20]]; // Vincenzo Librandi, Dec 27 2010

seq(n^(12) -1, n=1..20); # G. C. Greubel, Aug 08 2019

Range[20]^12 -1 (* G. C. Greubel, Aug 08 2019 *)

vector(20, n, n^12 -1) \\ G. C. Greubel, Aug 08 2019

[n^12 -1 for n in (1..20)] # G. C. Greubel, Aug 08 2019

From Chai Wah Wu, Jun 18 2016: (Start)
a(n) = 13*a(n-1) - 78*a(n-2) + 286*a(n-3) - 715*a(n-4) + 1287*a(n-5) - 1716*a(n-6) + 1716*a(n-7) - 1287*a(n-8) + 715*a(n-9) - 286*a(n-10) + 78*a(n-11) - 13*a(n-12) + a(n-13) for n > 12.
G.f.: x*(4095 + 478205*x + 10187905*x^2 + 66317979*x^3 + 162513078*x^4 + 162511362*x^5 + 66319266*x^6 + 10187190*x^7 + 478491*x^8 + 4017*x^9 + 13*x^10 - x^11)/(1 - x)^13. (End)

A010805 17th powers: a(n) = n^17.

Original entry on oeis.org

0, 1, 131072, 129140163, 17179869184, 762939453125, 16926659444736, 232630513987207, 2251799813685248, 16677181699666569, 100000000000000000, 505447028499293771, 2218611106740436992, 8650415919381337933, 30491346729331195904, 98526125335693359375, 295147905179352825856, 827240261886336764177

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 (18,-153,816,-3060,8568,-18564,31824,-43758,48620,-43758,31824,-18564,8568,-3060,816,-153,18,-1).

Cf. A013675 (zeta(17)).

Cf. A000290 (squares), A000578 (cubes), A000583 (4th powers), A001016 (8th powers), A008456 (12th powers).

[n^17: n in [0..15]]; // Vincenzo Librandi, Jun 19 2011

A010805 := n -> n^17; # M. F. Hasler, Jul 03 2025

Range[0,15]^17 (* Harvey P. Dale, Sep 14 2011 *)

for(n=0,15,print1(n^17,", ")) \\ Derek Orr, Feb 27 2017

apply( {A010805(n)=n^17}, [0..20]) \\ Defines the function and (as "proof of concept") applies it to [0..20]. - M. F. Hasler, Jul 03 2025

A010805 = lambda n: n**17 # M. F. Hasler, Jul 03 2025

Totally multiplicative sequence with a(p) = p^17 for prime p. Multiplicative sequence with a(p^e) = p^(17e). - Jaroslav Krizek, Nov 01 2009
From Ilya Gutkovskiy, Feb 27 2017: (Start)
Dirichlet g.f.: zeta(s-17).
Sum_{n>=1} 1/a(n) =  zeta(17) = A013675. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 65535*zeta(17)/65536. - Amiram Eldar, Oct 09 2020

A010812 24th powers: a(n) = n^24.

Original entry on oeis.org

0, 1, 16777216, 282429536481, 281474976710656, 59604644775390625, 4738381338321616896, 191581231380566414401, 4722366482869645213696, 79766443076872509863361, 1000000000000000000000000

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 (25, -300, 2300, -12650, 53130, -177100, 480700, -1081575, 2042975, -3268760, 4457400, -5200300, 5200300, -4457400, 3268760, -2042975, 1081575, -480700, 177100, -53130, 12650, -2300, 300, -25, 1).

Cf. A008456 (n^12).

[n^24: n in [0..15]]; // Vincenzo Librandi, Jun 19 2011

Range[0,10]^24 (* Harvey P. Dale, Sep 04 2017 *)

a(n) = n^24; \\ Michel Marcus, Feb 27 2018

Totally multiplicative sequence with a(p) = p^24 for prime p. Multiplicative sequence with a(p^e) = p^(24e). - Jaroslav Krizek, Nov 01 2009
a(n) = A008456(n)^2. - Michel Marcus, Feb 27 2018
From Amiram Eldar, Oct 09 2020: (Start)
Dirichlet g.f.: zeta(s-24).
Sum_{n>=1} 1/a(n) = zeta(24) = 236364091*Pi^24/201919571963756521875.
Sum_{n>=1} (-1)^(n+1)/a(n) = 8388607*zeta(24)/8388608 = 1982765468311237*Pi^24/1693824136731743669452800000. (End)

A089081 26th powers: a(n) = n^26.

Original entry on oeis.org

0, 1, 67108864, 2541865828329, 4503599627370496, 1490116119384765625, 170581728179578208256, 9387480337647754305649, 302231454903657293676544, 6461081889226673298932241, 100000000000000000000000000

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Dec 04 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to divisibility sequences

Cf. A008454-A008456, A010801-A010813.

[n^26: n in [0..15]]; // Vincenzo Librandi, Jun 19 2011

Range[0, 9]^26 (* Alonso del Arte, Apr 26 2015 *)

a(n)=n^26 \\ Charles R Greathouse IV, Jun 28 2015

a(n) = n^26.
Completely multiplicative sequence with a(p) = p^26 for prime p. Multiplicative sequence with a(p^e) = p^(26e). - Jaroslav Krizek, Nov 01 2009
From Amiram Eldar, Oct 09 2020: (Start)
Dirichlet g.f.: zeta(s-26).
Sum_{n>=1} 1/a(n) = zeta(26) = 1315862*Pi^26/11094481976030578125.
Sum_{n>=1} (-1)^(n+1)/a(n) = 33554431*zeta(26)/33554432 = 22076500342261*Pi^26/186134520519971831808000000. (End)

A022528 Nexus numbers (n+1)^12-n^12.

Original entry on oeis.org

1, 4095, 527345, 16245775, 227363409, 1932641711, 11664504865, 54878189535, 213710059745, 717570463519, 2138428376721, 5777672071535, 14381984674225, 33395827252815, 73052425515329, 151728638820031, 301147260519105, 574209144196415

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
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Column k=11 of array A047969.

Cf. A008292, A022527, A022529.

[(n+1)^12-n^12: n in [0..20]]; // Vincenzo Librandi, Nov 22 2011

lst={};Do[a=n^6;b=(n+1)^6;s=(a+b)*(b-a);AppendTo[lst,s],{n,0,4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 23 2009 *)
Table[(n+1)^12-n^12,{n,0,20}] (* Vincenzo Librandi, Nov 22 2011 *)
LinearRecurrence[{12,-66,220,-495,792,-924,792,-495,220,-66,12,-1},{1,4095,527345,16245775,227363409,1932641711,11664504865,54878189535,213710059745,717570463519,2138428376721,5777672071535},20] (* Harvey P. Dale, Apr 23 2019 *)

vector(30, n, n--; (n+1)^12-n^12) \\ Colin Barker, Nov 30 2014

a(n) = A008456(n+1) - A008456(n). - Colin Barker, Nov 30 2014
G.f.: (x +1)*(x^10 +4082*x^9 +474189*x^8 +9713496*x^7 +56604978*x^6 +105907308*x^5 +56604978*x^4 +9713496*x^3 +474189*x^2 +4082*x +1) / (x -1)^12. - Colin Barker, Nov 30 2014
G.f.: polylog(-12, x)*(1-x)/x. See the g.f. of Colin Barker (with expanded numerator), and the g.f. of the rows of A008292 by Vladeta Jovovic, Sep 02 2002. - Wolfdieter Lang, May 10 2021

cp's OEIS Frontend

A253712 Second partial sums of 12th powers (A008456).

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A268335 Exponentially odd numbers.

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A030631 Numbers with 13 divisors.

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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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A123868 a(n) = n^12 - 1.

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