A010812 -id:A010812 - 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

A137488 Numbers with 25 divisors.

Original entry on oeis.org

1296, 10000, 38416, 50625, 194481, 234256, 456976, 1185921, 1336336, 1500625, 2085136, 2313441, 4477456, 6765201, 9150625, 10556001, 11316496, 14776336, 16777216, 17850625, 22667121, 29986576, 35153041, 45212176, 52200625

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^24 (24th powers of A000040, subset of A010812) or p^4*q^4 (A189991), where p and q are distinct primes. - R. J. Mathar, Mar 01 2010

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000005, A010812, A030513-A030516, A030626, A030627, A030634-A030638, A005179, A003680, A096932, A061286, A061283, A135581, A175755.

a137488 n = a137488_list !! (n-1)
a137488_list = m (map (^ 24) a000040_list) (map (^ 4) a006881_list) where
   m xs'@(x:xs) ys'@(y:ys) | x < y = x : m xs ys'
                           | otherwise = y : m xs' ys
-- Reinhard Zumkeller, Nov 29 2011

lst = {}; Do[If[DivisorSigma[0, n] == 25, Print[n]; AppendTo[lst, n]], {n, 55000000}]; lst (* Vladimir Joseph Stephan Orlovsky, May 03 2011 *)
Select[Range[5221*10^4],DivisorSigma[0,#]==25&] (* Harvey P. Dale, Mar 11 2019 *)

is(n)=numdiv(n)==25 \\ Charles R Greathouse IV, Jun 19 2016

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A137488(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x+(t:=primepi(s:=isqrt(y:=integer_nthroot(x,4)[0])))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)))-primepi(integer_nthroot(x,24)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

A000005(a(n)) = 25.

Sum_{n>=1} 1/a(n) = (P(4)^2 - P(8))/2 + P(24) = 0.000933328..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

A122968 27th powers: a(n) = n^27.

Original entry on oeis.org

0, 1, 134217728, 7625597484987, 18014398509481984, 7450580596923828125, 1023490369077469249536, 65712362363534280139543, 2417851639229258349412352, 58149737003040059690390169

Offset: 0

Franklin T. Adams-Watters, Oct 27 2006

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A010812, A010813, A089081.

lst={}; Do[AppendTo[lst,n^27],{n,0,4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
Range[0,10]^27 (* Harvey P. Dale, Dec 17 2011 *)

Totally multiplicative sequence with a(p) = p^27 for prime p. Multiplicative sequence with a(p^e) = p^(27e). - Jaroslav Krizek, Nov 01 2009
From Amiram Eldar, Oct 09 2020: (Start)
Dirichlet g.f.: zeta(s-27).
Sum_{n>=1} 1/a(n) = zeta(27).
Sum_{n>=1} (-1)^(n+1)/a(n) = 67108863*zeta(27)/67108864. (End)

A036102 Centered cube numbers: (n+1)^24 + n^24.

Original entry on oeis.org

1, 16777217, 282446313697, 281757406247137, 59886119752101281, 4797985983097007521, 196319612718888031297, 4913947714250211628097, 84488809559742155077057, 1079766443076872509863361

Offset: 0

N. J. A. Sloane

Can never be prime, as a(n) = (2n^8 + 8n^7 + 28n^6 + 56n^5 + 70n^4 + 56n^3 + 28n^2 + 8n + 1) * (n^16 + 8n^15 + 92n^14 + 504n^13 + 1750n^12 + 4312n^11 + 7980n^10 + 11432n^9 + 12869n^8 + 11440n^7 + 8008n^6 + 4368n^5 + 1820n^4 + 560n^3 + 120n^2 + 16n + 1). [Jonathan Vos Post, Aug 28 2011]

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. A010812, A036100, A036101.

[(n+1)^24+n^24: n in [0..20]]; // Vincenzo Librandi, Aug 28 2011

Total/@Partition[Range[0,10]^24,2,1] (* Harvey P. Dale, Nov 23 2013 *)

A022540 Nexus numbers (n+1)^24 - n^24.

Original entry on oeis.org

1, 16777215, 282412759265, 281192547174175, 59323169798679969, 4678776693546226271, 186842850042244797505, 4530785251489078799295, 75044076594002864649665, 920233556923127490136639, 8849732675807611094711841, 69647114527583233038729695, 463303923170979668638153825

Offset: 0

N. J. A. Sloane

nonn

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 54.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Column k=23 of A047969.

Cf. A010812 (n^24).

[(n+1)^24 - n^24: n in [0..20]]; // G. C. Greubel, Feb 27 2018

b:=24: a:=n->(n+1)^b-n^b: seq(a(n),n=0..18); # Muniru A Asiru, Feb 28 2018

Last[#]-First[#]&/@Partition[Range[0,10]^24,2,1] (* Harvey P. Dale, Apr 19 2014 *)
Table[(n+1)^24 - n^24, {n,0,20}] (* G. C. Greubel, Feb 27 2018 *)

for(n=0,20, print1((n+1)^24 - n^24, ", ")) \\ G. C. Greubel, Feb 27 2018

a(n) = A010812(n+1) - A010812(n). - Michel Marcus, Feb 27 2018

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A268335 Exponentially odd numbers.

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A262675 Exponentially evil numbers.

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A137488 Numbers with 25 divisors.

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A122968 27th powers: a(n) = n^27.

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A036102 Centered cube numbers: (n+1)^24 + n^24.

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A022540 Nexus numbers (n+1)^24 - n^24.

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