A010806 -id:A010806 - cp's OEIS frontend

A268335 Exponentially odd numbers.

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

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

A036096 Centered cube numbers: (n+1)^18 + n^18.

Original entry on oeis.org

1, 262145, 387682633, 69106897225, 3883416742361, 105374653934041, 1729973554578865, 19642812107392433, 168109033806481105, 1150094635296999121, 6559917313492231481, 32183250594377475385

Offset: 0

N. J. A. Sloane

Never prime nor semiprime, as a(n) = (2n^2 + 2n +1) * (n^4 + 2n^3 + 5n^2 + 4n +1) * (n^12 + 6n^11 + 51n^10 + 200n^9 + 480n^8 + 786n^7 + 923n^6 + 792n^5 + 495n^4 + 220n^3 + 66n^2 + 12n + 1). Triprime for n in {9, 347, 1069, 1072, ...}. - Jonathan Vos Post, Aug 27 2011

			9^18 + (9+1)^18 = 1150094635296999121 = 181 * 8461 * 750988536481, the minimum nontrivial number of prime factors.

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

Cf. A010806, A036094, A036095.

[(n+1)^18+n^18: n in [0..20]]; // Vincenzo Librandi, Aug 28 2011

Total/@Partition[Range[0,20]^18,2,1] (* Harvey P. Dale, May 10 2022 *)

a(n)=(n+1)^18+n^18 \\ Charles R Greathouse IV, Aug 30 2017

A022534 Nexus numbers (n+1)^18 - n^18.

Original entry on oeis.org

1, 262143, 387158345, 68332056247, 3745977788889, 97745259402791, 1526853641242033, 16385984911571535, 132080236787517137, 849905364703000879, 4559917313492231481, 21063415967393012423, 85832073671072149225, 314423447258679349527, 1051013025824763647969

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

Column k=17 of A047969.

Cf. A010806 (n^18).

[(n+1)^18-n^18: n in [0..20]]; // Vincenzo Librandi, Nov 22 2011

b:=18: a:=n->(n+1)^b-n^b: seq(a(n),n=0..18); # Muniru A Asiru, Feb 28 2018

Table[(n+1)^18-n^18,{n,0,20}] (* Vincenzo Librandi, Nov 22 2011 *)

for(n=0,20, print1((n+1)^18 - n^18, ", ")) \\ G. C. Greubel, Feb 27 2018

a(n) = A010806(n+1) - A010806(n). - Michel Marcus, Feb 27 2018

More terms added by G. C. Greubel, Feb 27 2018

A170791 a(n) = n^9*(n^9 + 1)/2.

Original entry on oeis.org

0, 1, 131328, 193720086, 34359869440, 1907349609375, 50779983373056, 814206819132028, 9007199321849856, 75047317842209805, 500000000500000000, 2779958657925089586, 13311666643022512128, 56227703481280946251

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 18 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=131328, there are 2^18=262144 oriented arrangements of two colors. Of these, 2^9=512 are achiral. That leaves (262144-512)/2=130816 chiral pairs. Adding achiral and chiral, we get 131328. - Robert A. Russell, Nov 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (19,-171,969,-3876,11628,-27132, 50388,-75582,92378,-92378,75582,-50388,27132,-11628,3876,-969,171, -19,1).

Row 18 of A277504.

Cf. A010806 (oriented), A001017 (achiral).

List([0..30], n -> n^9*(n^9 + 1)/2); # G. C. Greubel, Nov 15 2018

[n^9*(n^9+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 26 2011

f[n_]:=Module[{n9=n^9},(n9(n9+1))/2]; Array[f,20,0] (* Harvey P. Dale, Nov 24 2012 *)
Table[n^9*(n^9+1)/2, {n,0,30}] (* G. C. Greubel, Dec 06 2017 *)

for(n=0,30, print1(n^9*(n^9+1)/2, ", ")) \\ G. C. Greubel, Dec 06 2017

for n in range(0,20): print(int(n**9*(n**9 + 1)/2), end=', ') # Stefano Spezia, Nov 15 2018

[n^9*(1 + n^9)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

G.f.: (x + 131309*x^2 + 191225025*x^3 + 30701643925*x^4 + 1287510971765*x^5 + 20228672721537*x^6 + 142998536758213*x^7 + 503354983579865*x^8 + 932692830330915*x^9 + 932692827449735*x^10 + 503354984335363*x^11 + 142998537549087*x^12 + 20228672026535*x^13 + 1287511125835*x^14 + 30701669175*x^15 + 191214899*x^16 + 130816*x^17) /(1-x)^19. - G. C. Greubel, Dec 06 2017
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010806(n) + A001017(n)) / 2 = (n^18 + n^9) / 2.
G.f.: (Sum_{j=1..18} S2(18,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..9} S2(9,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..17} A145882(18,k) * x^k / (1-x)^19.
E.g.f.: (Sum_{k=1..18} S2(18,k)*x^k + Sum_{k=1..9} S2(9,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>18, a(n) = Sum_{j=1..19} -binomial(j-20,j) * a(n-j). (End)

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A268335 Exponentially odd numbers.

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A262675 Exponentially evil numbers.

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A036096 Centered cube numbers: (n+1)^18 + n^18.

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A022534 Nexus numbers (n+1)^18 - n^18.

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A170791 a(n) = n^9*(n^9 + 1)/2.

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