A010803 -id:A010803 - cp's OEIS frontend

A155018 Integer part of square root of n^15 = A010803(n).

0, 1, 181, 3787, 32768, 174692, 685700, 2178889, 5931641, 14348907, 31622776, 64631634, 124125023, 226242995, 394421215, 661735513, 1073741824, 1691869691, 2597429617, 3896296578, 5724334022, 8253624572, 11699575548, 16328969210

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), this sequence (k=15), A155019 (k=17).

[Floor(Sqrt(n^15)): n in [1..30]]; // G. C. Greubel, Dec 30 2017

a={};Do[AppendTo[a,IntegerPart[(n^15)^(1/2)]],{n,0,5!}];a
Table[Floor[Sqrt[n^15]], {n,1,30}] (* G. C. Greubel, Dec 30 2017 *)

for(n=1,30, print1(floor(sqrt(n^15)), ", ")) \\ 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

A030634 Numbers with 16 divisors.

Original entry on oeis.org

120, 168, 210, 216, 264, 270, 280, 312, 330, 378, 384, 390, 408, 440, 456, 462, 510, 520, 546, 552, 570, 594, 616, 640, 680, 690, 696, 702, 714, 728, 744, 750, 760, 770, 798, 858, 870, 888, 896, 910, 918, 920, 930, 945, 952, 966, 984, 1000

Offset: 1

Jeff Burch

nonn

Numbers of the form p^15 (subset of A010803), p*q^7, p*q*r^3 or p^3*q^3, or p*q*r*s, where p, q, r and s are distinct primes. - R. J. Mathar, Mar 01 2010

R. J. Mathar, Table of n, a(n) for n = 1..1000
Jérôme Germoni, Nombres à huit diviseurs, Images des Mathématiques, CNRS, 2017 (in French).

Cf. A030626, A030631, A030632, A030633.

[n: n in [1..1000] | DivisorSigma(0, n) eq 16]; // Vincenzo Librandi, Oct 05 2017

Select[Range[3000],DivisorSigma[0,#]==16&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

is(n)=numdiv(n)==16 \\ Charles R Greathouse IV, Jun 19 2016

A176509 Composite numbers m for which A064380(m) = A000010(m).

Original entry on oeis.org

8, 27, 125, 128, 343, 1331, 2187, 2197, 4913, 6859, 12167, 24389, 29791, 32768, 50653, 68921, 78125, 79507, 103823, 148877, 205379, 226981, 300763, 357911, 389017, 493039, 571787, 704969, 823543, 912673, 1030301, 1092727, 1225043, 1295029, 1442897, 2048383, 2248091

Offset: 1

Vladimir Shevelev, Apr 19 2010

nonn

Theorem. A064380(m) = A000010(m) iff m has the form m=p^(2^k-1), k>=1, p a prime. Eliminating the primes (k=1), the terms of the sequence have this form for k>1. All terms of A030078 (k=2) and A092759 (k=3) and prime powers of A010803 (k=4) are in the sequence, for example.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A010803, A030078, A050376, A056824, A064380, A092759, A176472.

seq[max_] := Module[{ps = Select[Range[Floor[Surd[max, 3]]], PrimeQ], e, k, s = {}}, Do[e = Floor[Log[ps[[i]], max]]; k = Floor[Log2[e + 1]]; s = Join[s, ps[[i]]^(2^Range[2, k] - 1)], {i, 1, Length[ps]}]; Sort[s]]; seq[3*10^6] (* Amiram Eldar, Mar 26 2023 *)

is(n)=my(e=isprimepower(n));e>2 && 2^valuation(e+1,2)==e+1 \\ Charles R Greathouse IV, Feb 19 2013

a(n) ~ n^3 log^3 n. - Charles R Greathouse IV, Feb 19 2013

Sum_{n>=1} 1/a(n) = Sum_{k>=2} 1/P(2^k-1) = 0.183077059924063305405..., where P(s) is the prime zeta function. - Amiram Eldar, Jul 11 2024

128 inserted, 1024 deleted, 2187 inserted, 32768 inserted, etc. - R. J. Mathar, Nov 21 2010

More terms from Amiram Eldar, Mar 26 2023

A022531 Nexus numbers (n+1)^15 - n^15.

Original entry on oeis.org

1, 32767, 14316139, 1059392917, 29443836301, 439667406451, 4277376525367, 30436810578889, 170706760005817, 794108867905351, 3177248169415651, 11229773405170717, 35778871439504389, 104382202543721467, 282325794823047151, 715027614225987601, 1709501546902968817

Offset: 0

N. J. A. Sloane

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
Eric Weisstein's World of Mathematics, Nexus Number.

Column k = 14 of A047969.

Cf. A010803 (n^15).

[(n+1)^15-n^15: n in [0..20]]; // Vincenzo Librandi, Nov 22 2011

b:=15: a:=n->(n+1)^b-n^b: seq(a(n),n=0..18); # Muniru A Asiru, Feb 28 2018

Table[(n + 1)^15 - n^15, {n, 0, 20}] (* Vincenzo Librandi, Nov 22 2011 *)
#[[2]]-#[[1]]&/@Partition[Range[0,20]^15,2,1] (* Harvey P. Dale, Aug 07 2022 *)

for(n=0,20, print1((n+1)^15 - n^15, ", ")) \\ G. C. Greubel, Feb 27 2018

G.f.: ( -1 - 32752*x - 13824739*x^2 - 848090912*x^3 - 15041229521*x^4 - 102776998928*x^5 - 311387598411*x^6 - 447538817472*x^7 - 311387598411*x^8 - 102776998928*x^9 - 15041229521*x^10 - 848090912*x^11 - 13824739*x^12 - 32752*x^13 - x^14 ) / (x - 1)^15. - R. J. Mathar, Sep 02 2016

a(n) = A010803(n+1) - A010803(n). - Michel Marcus, Feb 28 2018

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

A170779 a(n) = n^8*(n^7 + 1)/2.

Original entry on oeis.org

0, 1, 16512, 7177734, 536903680, 15258984375, 235093332096, 2373783637372, 17592194433024, 102945587570685, 500000050000000, 2088624191887266, 7703511002284032, 25592946914910739, 77784048516800640

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 15 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)=16512, there are 2^15=32768 oriented arrangements of two colors. Of these, 2^8=256 are achiral. That leaves (32768-256)/2=16256 chiral pairs. Adding achiral and chiral, we get 16512. - 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 (16, -120, 560, -1820, 4368, -8008, 11440, -12870, 11440, -8008, 4368, -1820, 560, -120, 16, -1).

Row 15 of A277504.

Cf. A010803 (oriented), A001016 (achiral).

[n^8*(n^7+1)/2: n in [0..30]]; // Vincenzo Librandi, Aug 26 2011

Table[n^8*(n^7+1)/2, {n,0,30}] (* G. C. Greubel, Dec 05 2017 *)

for(n=0, 30, print1(n^8*(n^7+1)/2, ", ")) \\ G. C. Greubel, Dec 05 2017

[n^8*(n^7+1)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

G.f.: (x + 16496*x^2 + 6913662*x^3 + 424040816*x^4 + 7520608675*x^5 + 51388540128*x^6 + 155693747508*x^7 + 223769408736*x^8 + 155693850903*x^9 + 51388458800*x^10 + 7520620846*x^11 + 424050096*x^12 + 6911077*x^13 + 16256*x^14)/(1-x)^16. - G. C. Greubel, Dec 05 2017
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010803(n) + A001016(n)) / 2 = (n^15 + n^8) / 2.
G.f.: (Sum_{j=1..15} S2(15,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..8} S2(8,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..14} A145882(15,k) * x^k / (1-x)^16.
E.g.f.: (Sum_{k=1..15} S2(15,k)*x^k + Sum_{k=1..8} S2(8,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>15, a(n) = Sum_{j=1..16} -binomial(j-17,j) * a(n-j). (End)
E.g.f.: x*(2 +16510*x +2376067*x^2 +42357651*x^3 +210767970*x^4 + 420693539*x^5 +408741361*x^6 +216627841*x^7 +67128490*x^8 + 12662650*x^9 +1479478*x^10 +106470*x^11 +4550*x^12 +105*x^13 +x^14)*exp(x)/2. - G. C. Greubel, Nov 15 2018

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A155018 Integer part of square root of n^15 = A010803(n).

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A268335 Exponentially odd numbers.

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A262675 Exponentially evil numbers.

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A030634 Numbers with 16 divisors.

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A176509 Composite numbers m for which A064380(m) = A000010(m).

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A022531 Nexus numbers (n+1)^15 - n^15.

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A170779 a(n) = n^8*(n^7 + 1)/2.

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