A270428 -id:A270428 - cp's OEIS frontend

A367514 The exponentially odious part of n: the largest unitary divisor of n that is an exponentially odious number (A270428).

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70

Offset: 1

Amiram Eldar, Nov 21 2023

First differs from A056192 at n = 32, and from A270418 and A367168 at n = 128.

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

Cf. A000069, A001221, A010060, A034444, A102392, A262675, A270428, A293439, A366901, A366903, A367515.
Cf. A056192, A270418.
Similar sequences: A350388, A350389, A366905, A367168, A367513.

f[p_, e_] := p^(e*ThueMorse[e]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(hammingweight(f[i, 2])%2, f[i, 1]^f[i, 2], 1));}

from math import prod
from sympy import factorint
def A367514(n): return prod(p**e for p, e in factorint(n).items() if e.bit_count()&1) # Chai Wah Wu, Nov 23 2023

Multiplicative with a(p^e) = p^(e*A010060(e)) = p^A102392(e).
a(n) = n/A367513(n).
A001221(a(n)) = A293439(n).
A034444(a(n)) = A367515(n).
a(n) >= 1, with equality if and only if n is an exponentially evil number (A262675).
a(n) <= n, with equality if and only if n is an exponentially odious number (A270428).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 0.88585652437242918295..., and f(x) = (x+2)/(2*(x+1)) + (x/2) * Product_{k>=0} (1 - x^(2^k)).

A367515 The number of unitary divisors of n that are exponentially odious numbers (A270428).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 1, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 2, 2, 4, 1, 4, 2, 8, 2, 1, 4, 4, 4, 4, 2, 4, 4, 2, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 2, 4, 2, 4, 4, 2, 8, 2, 4, 4, 1, 4, 8, 2, 4, 4, 8, 2, 2, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Nov 21 2023

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

Cf. A001285, A262675, A270428, A293439.

Similar sequences: A034444, A055076, A056624, A366901, A366902, A367516.

f[p_, e_] := If[OddQ[DigitCount[e, 2, 1]], 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> hammingweight(x)%2+1, factor(n)[, 2]));

from sympy import factorint
def A367515(n): return 1<Chai Wah Wu, Nov 23 2023

Multiplicative with a(p^e) = A001285(e).
a(n) = A034444(n)/A367516(n).
a(n) = 2^A293439(n).
a(n) >= 1, with equality if and only if n is an exponentially evil number (A262675).
a(n) <= A034444(n), with equality if and only if n is an exponentially odious number (A270428).

A367801 Numbers that are both exponentially odd (A268335) and exponentially odious (A270428).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106

Offset: 1

Amiram Eldar, Dec 01 2023

First differs from its subsequence A005117 at n = 79: a(79) = 128 is not a squarefree number.
First differs from A077377 at n = 63, and from A348506 at n = 68.
Numbers whose prime factorization contains only exponents that are odd odious numbers (A092246).
The asymptotic density of this sequence is Product_{p prime} f(1/p) = 0.61156148494581943994..., where f(x) = (1-x) * (1 + x/(2*(1-x^2)) + (Product_{k>=0} (1-(-x)^(2^k)) - Product_{k>=0} (1-x^(2^k))))/2.

Amiram Eldar, Table of n, a(n) for n = 1..12232 (terms below 20000)
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175 (2016), pp. 385-395.

Intersection of A268335 and A270428.
Cf. A367802, A367803, A367804.
Subsequences: A005117, A092759.
Cf. A092246.
Cf. A077377, A348506.

odQ[n_] := OddQ[n] && OddQ[DigitCount[n, 2, 1]]; Select[Range[150], AllTrue[FactorInteger[#][[;;, 2]], odQ] &]

is(n) = {my(f = factor(n)); for (i = 1, #f~, if(!(f[i, 2]%2 && hammingweight(f[i, 2])%2), return (0))); 1;}

A367931 a(n) is the smallest number k such that k*n is an exponentially odious number (A270428).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 05 2023

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

Cf. A000069, A270428, A367933.

Similar sequences: A365298, A365685, A367932.

f[p_, e_] := Module[{k = e}, While[! OddQ[DigitCount[k, 2 ,1]], k++]; p^(k-e)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = {my(k = e); while(!(hammingweight(k)%2), k++); k - e; };
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2]));}

Multiplicative with a(p^e) = p^s(e), where s(e) = min{k >= e, k is odious} - e.
a(n) = A367933(n)/n.
a(n) >= 1, with equality if and only if n is an exponentially odious number (A270428).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.30023300..., where f(x) = (1-x) * (1 + Sum_{k>=1} x*(k-s(k))), and s(k) is defined above.

A367696 Numbers k such that k and k+1 are both exponentially odious numbers (A270428).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, 28, 29, 30, 33, 34, 35, 36, 37, 38, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Amiram Eldar, Nov 27 2023

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 8, 78, 762, 7615, 76113, 761127, 7611222, 76111895, 761119135, 7611190807, ... . Apparently, the asymptotic density of this sequence exists and equals 0.761119... .

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

Subsequence of A270428.
Subsequences: A007674, A367697.
Similar sequences: A071318, A121495, A340152, A367695.

expOdQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ[DigitCount[#, 2, 1]] &]; Select[Range[100], And @@ expOdQ /@ {#, # + 1} &]

isexpod(n) = {my(f = factor(n)); for(i=1, #f~, if (!(hammingweight(f[i, 2]) % 2), return (0))); 1;}
is(n) = isexpod(n) && isexpod(n+1)

A367933 a(n) is the smallest multiple of n that is an exponentially odious number (A270428).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 48, 25, 26, 81, 28, 29, 30, 31, 128, 33, 34, 35, 36, 37, 38, 39, 80, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 162, 55, 112, 57, 58, 59, 60, 61, 62, 63, 128, 65, 66, 67

Offset: 1

Amiram Eldar, Dec 05 2023

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

Cf. A270428, A367931.

Similar sequences: A356192, A365684, A367934.

f[p_, e_] := Module[{k = e}, While[! OddQ[DigitCount[k, 2 ,1]], k++]; p^k]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = {my(k = e); while(!(hammingweight(k)%2), k++); k; };
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2])); }

Multiplicative with a(p^e) = p^s(e), s(e) = min{k >= e, k is odious}.
a(n) = n * A367931(n).
a(n) >= n, with equality if and only if n is an exponentially odious number (A209061).
Sum_{k=1..n} a(k) ~  c * n^2 / 2, where c = Product_{p prime} f(1/p) = 1.30023300... , where f(x) = (1-x) * (1 + Sum_{k>=1} x^(2*k-s(k))), and s(k) is defined above.

A367697 Starts of runs of 15 consecutive integers that are exponentially odious numbers (A270428).

Original entry on oeis.org

9, 73, 137, 169, 201, 393, 521, 553, 633, 649, 713, 761, 809, 841, 889, 1001, 1033, 1065, 1129, 1145, 1193, 1225, 1273, 1289, 1353, 1385, 1513, 1545, 1577, 1609, 1657, 1769, 1785, 1865, 1897, 1929, 2025, 2089, 2169, 2217, 2297, 2345, 2377, 2409, 2441, 2505, 2569

Offset: 1

Amiram Eldar, Nov 27 2023

The maximal length of a run of consecutive exponentially odious numbers is 15 since numbers of the form 16*k + 8 are not exponentially odious. Thus all the terms of this sequence are of the form 16*k + 9 with k = 0, 4, 8, 10, 12, 24, 32, 34, 39, 40, ... .

The numbers of terms not exceeding 10^k for k = 1, 2, ... , are 1, 2, 15, 176, 1821, 18120, 181277, 1812917, 18129256, 181290721, ... . Apparently, the asymptotic density of this sequence exists and equals 0.018129... .

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

Subsequence of A270428 and A367696.

Similar sequences: A007675, A194002, A325058, A328016.

expOdQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ[DigitCount[#, 2, 1]] &]; q[n_] := AllTrue[16*n + Range[9, 23], expOdQ]; 16 * Select[Range[0, 160], q] + 9

isexpod(n) = {my(f = factor(n)); for(i=1, #f~, if (!(hammingweight(f[i, 2]) % 2), return (0))); 1;}
is(n) = {my(k = (n-9)/16); if(denominator(k) > 1, return(0)); for(i=9, 23, if(!isexpod(16*k + i), return(0))); 1;}

A367698 The smallest divisor d of n such that n/d is an exponentially odious number (A270428).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Nov 27 2023

First differs from A055229 at n = 64.

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

Cf. A055229, A270428, A366905, A367699.

maxOdious[e_] := Module[{k = e}, While[EvenQ[DigitCount[k, 2, 1]], k--]; k]; f[p_, e_] := p^(e - maxOdious[e]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = {my(k = n); while(!(hammingweight(k)%2), k--); n-k; }
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2])); }

a(n) = n/A366905(n).
Multiplicative with a(p^e) = p^(e-s(e)), where s(e) = max({k=1..e, k odious}).
a(n) >= 1, with equality if and only if n is an exponentially odious number (A270428).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.25857819194624249136..., where f(x) = (1-x)*(1+Sum_{k>=1} x^s(k)), s(k) is defined above for k >= 1, and s(0) = 0.

A209061 Exponentially squarefree numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 1

Reinhard Zumkeller, Mar 13 2012

nonn

Numbers having only squarefree exponents in their canonical prime factorization.
According to the formula of Theorem 3 [Toth], the density of the exponentially squarefree numbers is 0.9559230158619... (A262276). - Peter J. C. Moses and Vladimir Shevelev, Sep 10 2015
From Vladimir Shevelev, Sep 24 2015: (Start)
A generalization. Let S be a finite or infinite increasing integer sequence s=s(n), s(0)=0.
Let us call a positive number N an exponentially S-number, if all exponents in its prime power factorization are in the sequence S.
Let {u(n)} be the characteristic function of S. Then, for the density h=h(S) of the exponentially S-numbers, we have the representations
h(S) = Product_{prime p} Sum_{j in S} (p-1)/p^(j+1) = Product_{p} (1 + Sum_{j>=1} (u(j) - u(j-1))/p^j). In particular, if S = {0,1}, then the exponentially S-numbers are squarefree numbers; if S consists of 0 and {2^k}A138302%20(see%20%5BShevelev%5D,%202007);%20if%20S%20consists%20of%200%20and%20squarefree%20numbers,%20then%20u(n)=%7Cmu(n)%7C,%20where%20mu(n)%20is%20the%20M%C3%B6bius%20function%20(A008683),%20we%20obtain%20the%20density%20h%20of%20the%20exponentially%20squarefree%20numbers%20(cf.%20Toth's%20link,%20Theorem%203);%20the%20calculation%20of%20h%20with%20a%20very%20high%20degree%20of%20accuracy%20belongs%20to%20_Juan%20Arias-de-Reyna">{k>=0}, then the exponentially S-numbers form A138302 (see [Shevelev], 2007); if S consists of 0 and squarefree numbers, then u(n)=|mu(n)|, where mu(n) is the Möbius function (A008683), we obtain the density h of the exponentially squarefree numbers (cf. Toth's link, Theorem 3); the calculation of h with a very high degree of accuracy belongs to _Juan Arias-de-Reyna (A262276). Note that if S contains 1, then h(S) >= 1/zeta(2) = 6/Pi^2; otherwise h(S) = 0. Indeed, in the latter case, the density of the sequence of exponentially S-numbers does not exceed the density of A001694, which equals 0. (End)
The term "exponentially squarefree number" was apparently coined by Subbarao (1972). - Amiram Eldar, May 28 2025

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Xiaodong Cao and Wenguang Zhai, Some arithmetic functions involving exponential divisors, JIS 13 (2010), Article 10.3.7.
Y.-F. S. Petermann, Arithmetical functions involving exponential divisors: note on two papers by L. Toth, Ann. Univ. Sci. Budapest, Sect. Comp. 32 (2010) 143-149.
Vladimir Shevelev, Compact integers and factorials, Acta Arithmetica 126:3 (2007), pp. 195-236.
Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv:1511.03860 [math.NT], 2015.
Vladimir Shevelev, A fast computation of density of exponentially S-numbers, arXiv:1602.04244 [math.NT], 2016.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.
H. M. Stark, On the asymptotic density of the k-free integers, Proc. Amer. Soc. 17 (1966), 1211-1214.
M. V. Subbarao, On some arithmetic convolutions, in: A. A. Gioia and D. L. Goldsmith (eds.), The Theory of Arithmetic Functions: Proceedings of the Conference at Western Michigan University, April 29 - May 1, 1971, Berlin, Heidelberg: Springer Berlin Heidelberg, 2006, pp. 247-271; alternative link.
László Tóth, On certain arithmetic functions involving exponential divisors, II., Annales Univ. Sci. Budapest., Sect. Comp., 27 (2007), 155-166 and arXiv:0708.3557 [math.NT], 2007-2009.

Complement of A130897.
A005117, A004709, and A046100 are subsequences.
Cf. A001221, A001694, A008683, A008966, A036537, A115063, A138302, A197680, A262276, A262675, A268335, A270428.

a209061 n = a209061_list !! (n-1)
a209061_list = filter
   (all (== 1) . map (a008966 . fromIntegral) . a124010_row) [1..]

Select[Range@ 69, Times @@ Boole@ Map[SquareFreeQ, Last /@ FactorInteger@ #] > 0 &] (* Michael De Vlieger, Sep 07 2015 *)

is(n)=my(f=factor(n)[,2]); for(i=1,#f,if(!issquarefree(f[i]), return(0))); 1 \\ Charles R Greathouse IV, Sep 02 2015

A166234(a(n)) <> 0.
Product_{k=1..A001221(n)} A008966(A124010(n,k)) = 1.
One can prove that the principal term of Toth's asymptotics for the density of this sequence (cf. Toth's link, Theorem 3) equals also Product_{prime p}(Sum_{j in S}(p-1)/p^{j+1})*x, where S is the set of 0 and squarefree numbers. The remainder term O(x^(0.2+t)), where t>0 is arbitrarily small, was obtained by L. Toth while assuming the Riemann Hypothesis. - Vladimir Shevelev, Sep 12 2015

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

cp's OEIS Frontend

A367514 The exponentially odious part of n: the largest unitary divisor of n that is an exponentially odious number (A270428).

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A367515 The number of unitary divisors of n that are exponentially odious numbers (A270428).

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A367801 Numbers that are both exponentially odd (A268335) and exponentially odious (A270428).

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A367931 a(n) is the smallest number k such that k*n is an exponentially odious number (A270428).

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A367696 Numbers k such that k and k+1 are both exponentially odious numbers (A270428).

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A367933 a(n) is the smallest multiple of n that is an exponentially odious number (A270428).

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A367697 Starts of runs of 15 consecutive integers that are exponentially odious numbers (A270428).

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A367698 The smallest divisor d of n such that n/d is an exponentially odious number (A270428).

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