A010805 -id:A010805 - cp's OEIS frontend

A155019 Integer part of square root of n^17 = A010805(n).

0, 1, 362, 11363, 131072, 873464, 4114202, 15252229, 47453132, 129140163, 316227766, 710947978, 1489500287, 2941158941, 5521897022, 9926032708, 17179869184, 28761784747, 46753733110, 74029634996, 114486680447

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jan 19 2009

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Integer part of square root of n^k: A000196 (k=1), A000093 (k=3), A155013 (k=5), A155014 (k=7), A155015 (k=11), A155016 (k=13), A155018 (k=15), this sequence (k=17).

[Floor(Sqrt(n^17)): n in [0..30]]; // G. C. Greubel, Dec 30 2017

a={};Do[AppendTo[a,IntegerPart[(n^17)^(1/2)]],{n,0,5!}];a
Table[Floor[Sqrt[n^17]], {n,0,30}] (* G. C. Greubel, Dec 30 2017 *)

for(n=0,30, print1(floor(sqrt(n^17)), ", ")) \\ G. C. Greubel, Dec 30 2017

Offset corrected by Alois P. Heinz, Sep 27 2014

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

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

A022533 Nexus numbers (n+1)^17 - n^17.

Original entry on oeis.org

1, 131071, 129009091, 17050729021, 745759583941, 16163719991611, 215703854542471, 2019169299698041, 14425381885981321, 83322818300333431, 405447028499293771, 1713164078241143221, 6431804812640900941, 21840930809949857971, 68034778606362163471, 196621779843659466481

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=16 of A047969.

Cf. A010805 (n^17).

[(n+1)^17-n^17: n in [0..20]]; // Vincenzo Librandi, Nov 22 2011

b:=17: a:=n->(n+1)^b-n^b: seq(a(n),n=0..18); # Muniru A Asiru, Feb 28 2018

Table[(n+1)^17-n^17,{n,0,20}] (* Vincenzo Librandi, Nov 22 2011 *)
Differences[Range[0,20]^17] (* Harvey P. Dale, May 21 2019 *)

for(n=0,20, print1((n+1)^17 - n^17, ", ")) \\ G. C. Greubel, Feb 27 2018

a(n) = A010805(n+1) - A010805(n). - Michel Marcus, Feb 27 2018

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

A170790 a(n) = n^9*(n^8 + 1)/2.

Original entry on oeis.org

0, 1, 65792, 64579923, 8590065664, 381470703125, 8463334761216, 116315277170407, 1125899973951488, 8338591043543529, 50000000500000000, 252723515428620731, 1109305555950108672, 4325207964992918653

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 17 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)=65792, there are 2^17=131072 oriented arrangements of two colors. Of these, 2^9=512 are achiral. That leaves (131072-512)/2=65280 chiral pairs. Adding achiral and chiral, we get 65792. - 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 (18,-153,816,-3060,8568,-18564, 31824,-43758,48620,-43758,31824,-18564,8568,-3060,816,-153,18,-1).

Row 17 of A277504.

Cf. A010805 (oriented), A001017 (achiral).

List([0..30], n -> n^9*(n^8+1)/2); # G. C. Greubel, Nov 15 2018

[n^9*(n^8+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 26 2011

Table[(n^9 (n^8+1))/2,{n,0,20}] (* Harvey P. Dale, Oct 03 2016 *)

for(n=0,30, print1(n^9*(n^8+1)/2, ", ")) \\ G. C. Greubel, Dec 06 2017

for n in range(0,20): print(int(n**9*(n**8 + 1)/2), end=', ') # Stefano Spezia, Nov 15 2018

[n^9*(n^8+1)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

G.f.: (x + 65774*x^2 + 63395820*x^3 + 7437692410*x^4 + 236676566180*x^5 + 2858646249342*x^6 + 15527826341908*x^7 + 41568611082650*x^8 + 57445191259830*x^9 + 41568611082650*x^10 + 15527826341908*x^11 + 2858646249342*x^12 + 236676566180*x^13 + 7437692410*x^14 + 63395820*x^15 + 65774*x^16 + x^17)/(1-x)^18. - G. C. Greubel, Dec 06 2017
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010805(n) + A001017(n)) / 2 = (n^17 + n^9) / 2.
G.f.: (Sum_{j=1..17} S2(17,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..16} A145882(17,k) * x^k / (1-x)^18.
E.g.f.: (Sum_{k=1..17} S2(17,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>17, a(n) = Sum_{j=1..18} -binomial(j-19,j) * a(n-j). (End)

A165254 a(n) = 9 + n^17.

Original entry on oeis.org

9, 10, 131081, 129140172, 17179869193, 762939453134, 16926659444745, 232630513987216, 2251799813685257, 16677181699666578, 100000000000000009, 505447028499293780, 2218611106740437001, 8650415919381337942

Offset: 0

Philippe Deléham, Sep 10 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A010805.

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

a(n) = 9 + n^17; \\ Michel Marcus, Oct 04 2016

a(n) = A010805(n) + 9.

Corrected and edited by Vincenzo Librandi, Jun 19 2011

A276962 Numbers n such that n^17 - 1 is semiprime.

Original entry on oeis.org

20, 62, 84, 368, 410, 614, 720, 740, 762, 1230, 1280, 1988, 1998, 2064, 2100, 2268, 2312, 2468, 2678, 2940, 3002, 3324, 3392, 3462, 3768, 3848, 3968, 4178, 4244, 4680, 4968, 5022, 5024, 5198, 5304, 5382, 5624, 5822, 5850, 6048, 6248, 6338, 6354, 6398, 6428

Offset: 1

Gary E. Davis, Sep 22 2016

nonn

Least number such that n^17-1 and n^17+1 are both semiprime is 93888. - Altug Alkan, Sep 30 2016

			a(1) = 20 because 20^17-1 = 13107199999999999999999 = 19*689852631578947368421 is the first occurrence of n^17 - 1 as a product of two distinct primes.

Cf. A001358, A010805, A104494.

Select[Range[3000],PrimeOmega[#^17-1] == 2 &]

isok(n) = bigomega(n^17-1)==2; \\ Michel Marcus, Sep 23 2016

lista(nn) = forprime(p=2, nn, if(ispseudoprime(((p+1)^17-1)/p), print1(p+1, ", "))); \\ Altug Alkan, Sep 30 2016

More terms from Altug Alkan, Sep 30 2016

cp's OEIS Frontend

A155019 Integer part of square root of n^17 = A010805(n).

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A268335 Exponentially odd numbers.

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A262675 Exponentially evil numbers.

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A022533 Nexus numbers (n+1)^17 - n^17.

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A170790 a(n) = n^9*(n^8 + 1)/2.

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A165254 a(n) = 9 + n^17.

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A276962 Numbers n such that n^17 - 1 is semiprime.