A268335 -id:A268335 - cp's OEIS frontend

A363825 The number of infinitary divisors of n that are exponentially odd numbers (A268335).

1, 2, 2, 1, 2, 4, 2, 3, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 6, 1, 4, 3, 2, 2, 8, 2, 3, 4, 4, 4, 1, 2, 4, 4, 6, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 6, 4, 6, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 3, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4

Offset: 1

Amiram Eldar, Oct 19 2023

First differs from A295878 at n = 32.

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

Cf. A000005, A000120, A005117, A037445, A077609, A268335, A295878, A358260.

f[p_, e_] := 1 + If[OddQ[e], 2^DigitCount[e-1, 2, 1], 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

Multiplicative with a(p^e) = 1 if e is even, and a(p^e) = 2^A000120(e-1) + 1 if e is odd.

a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).

A374456 The Euler phi function values of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 4, 4, 10, 12, 6, 8, 16, 18, 12, 10, 22, 8, 12, 18, 28, 8, 30, 16, 20, 16, 24, 36, 18, 24, 16, 40, 12, 42, 22, 46, 32, 52, 18, 40, 24, 36, 28, 58, 60, 30, 48, 20, 66, 44, 24, 70, 72, 36, 60, 24, 78, 40, 82, 64, 42, 56, 40, 88, 72, 60, 46, 72, 32, 96

Offset: 1

Amiram Eldar, Jul 09 2024

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

Cf. A000010, A065463, A268335, A307868.
Similar sequences related to phi: A002618, A049200, A323333, A358039.
Similar sequences related to exponentially odd numbers: A366438, A366439, A366534, A366535, A367417, A368711, A374457.

f[p_, e_] := If[OddQ[e], (p-1) * p^(e-1), 0]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 100], # > 0 &]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, (f[i, 1]-1) * f[i, 1]^(f[i, 2] - 1), 0));}
lista(kmax) = {my(s1); for(k = 1, kmax, s1 = s(k); if(s1 > 0, print1(s1, ", ")));}

a(n) = A000010(A268335(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A307868 / A065463^2 = 0.95051132596733153581... .

A385135 The sum of divisors d of n such that n/d is an exponentially odd number (A268335).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 13, 12, 18, 12, 24, 14, 24, 24, 26, 18, 36, 20, 36, 32, 36, 24, 52, 30, 42, 37, 48, 30, 72, 32, 53, 48, 54, 48, 72, 38, 60, 56, 78, 42, 96, 44, 72, 72, 72, 48, 104, 56, 90, 72, 84, 54, 111, 72, 104, 80, 90, 60, 144, 62, 96, 96, 106, 84, 144

Offset: 1

Amiram Eldar, Jun 19 2025

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

Cf. A013662, A033634, A268335.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), this sequence (exponentially odd), A385136 (cubefull), A385137 (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

f[p_, e_] := p^e + (p^(e+1) - If[EvenQ[e], p, 1])/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; p^e + (p^(e + 1) - if(e%2, 1, p))/(p^2 - 1));}

Multiplicative with a(p^e) = p^e + (p^(e+1) - 1)/(p^2-1) if e is odd, and p^e + (p^(e+1) - p)/(p^2-1) if e is even.
Dirichlet g.f.: zeta(s-1) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^4) = 1.542116283140158741... .

A365173 The number of divisors d of n such that gcd(d, n/d) is an exponentially odd number (A268335).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4

Offset: 1

Amiram Eldar, Aug 25 2023

First differs from A252505 at n = 64.

The sum of these divisors is A365174(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A004524, A046101, A005117, A252505, A268335, A350390, A325837, A365171, A365174.

f[p_, e_] := Floor[(e + 5)/4] + Floor[(e + 6)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> (x+5)\4 + (x+6)\4, factor(n)[, 2]));

for(n=1, 100, print1(direuler(p=2, n, (1 + X^2 - X^4)/((1 - X)^2*(1 + X^2)))[n], ", ")) \\ Vaclav Kotesovec, Jan 20 2024

Multiplicative with a(p^e) = floor((e+5)/4) + floor((e+6)/4) = A004524(e+5).
a(n) <= A000005(n), with equality if and only if n is not a biquadrateful number (A046101).
a(n) >= A034444(n), with equality if and only if n is squarefree (A005117).
a(n) == 1 (mod 2) if and only if n is a square of an exponentially odd number (i.e., a number whose prime factorization include only exponents e such that e == 2 (mod 4)).
From Vaclav Kotesovec, Jan 20 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/(p^(2*s)*(1 + p^(2*s)))).
Let f(s) = Product_{p prime} (1 - 1/(p^(2*s)*(1 + p^(2*s)))).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/(p^2*(1 + p^2))) = 0.937494282731300250789438325050116436995101826036120273493270589183132928...,
f'(1) = f(1) * Sum_{p prime} (4*p^2 + 2) * log(p) / (p^6 + 2*p^4 - 1) = f(1) * 0.192452062257404507109731932640803706644036700262364333369815000973104583...
and gamma is the Euler-Mascheroni constant A001620. (End)

A325058 Starts of runs of 7 consecutive exponentially odd numbers (A268335).

Original entry on oeis.org

29, 37, 53, 101, 133, 181, 213, 373, 453, 509, 541, 613, 677, 757, 893, 901, 917, 997, 1109, 1117, 1157, 1189, 1237, 1253, 1333, 1405, 1429, 1477, 1509, 1541, 1589, 1621, 1701, 1749, 1757, 1765, 1829, 1885, 1941, 2077, 2117, 2133, 2181, 2213, 2261, 2333, 2341

Offset: 1

Amiram Eldar, Sep 04 2019

nonn

The maximal run of consecutive exponentially odd numbers is of length 7 since numbers of the form 8k + 4 are not exponentially odd. Thus all the terms of this sequence are of the form 8k + 5 with k = 3, 4, 6, 12, 16, 22, 26, 46, 56, 63, 67, 76, 84, 94, ...

The number of terms below 10^k for k = 2, 3, ... is 3, 18, 201, 1878, 18902, 189515, 1895392, 18954089, ... Apparently this sequence has an asymptotic density of 0.01895...

			29 is in the sequence since 29, 30 = 2 * 3 * 5, 31, 32 = 2^5, 33 = 3 * 11, 34 = 2 * 17 and 35 = 5 * 7 are 7 consecutive exponentially odd numbers, all having prime factorization with only odd exponents.

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

Cf. A268335.

expOddQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; aQ[n_] := AllTrue[8n + Range[5, 11], expOddQ]; 8 * Select[Range[300], aQ] + 5

A367406 The exponentially odd numbers (A268335) multiplied by their squarefree kernels (A007947).

Original entry on oeis.org

1, 4, 9, 25, 36, 49, 16, 100, 121, 169, 196, 225, 289, 361, 441, 484, 529, 144, 676, 81, 841, 900, 961, 64, 1089, 1156, 1225, 1369, 1444, 1521, 400, 1681, 1764, 1849, 2116, 2209, 2601, 2809, 324, 3025, 784, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489, 4761

Offset: 1

Amiram Eldar, Nov 17 2023

Analogous to A355038, with the exponentially odd numbers instead of the square numbers (A000290).

This sequence is a permutation of the square numbers.

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

Cf. A000290, A007947, A053143, A064549, A065463, A268335, A355038, A367407, A367417, A367418.

s[n_] := n * Times @@ FactorInteger[n][[;;, 1]]; s /@ Select[Range[100], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]

b(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, f[i,1]^(f[i,2]+1), 0));}
lista(kmax) = {my(b1); for(k = 1, kmax, b1 = b(k); if(b1 > 0, print1(b1, ", ")));}

a(n) = A064549(A268335(n)).
a(n) = A268335(n)*A367417(n).
a(n) = A367407(n)^2.
a(n) = A268335(n)^2/A367418(n).
Sum_{k=1..n} a(k) = c * n^3 / 3, where c = (Pi^2/(15*d^3)) * Product_{p prime} (1 - 1/(p^3*(p+1))) = 1.78385074227198915372..., and d = A065463 is the asymptotic density of the exponentially odd numbers.
a(n) = A053143(A268335(n)). - Amiram Eldar, Nov 30 2023

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;}

A367804 Numbers that are both exponentially odd (A268335) and exponentially evil (A262675).

Original entry on oeis.org

1, 8, 27, 32, 125, 216, 243, 343, 512, 864, 1000, 1331, 1944, 2197, 2744, 3125, 3375, 4000, 4913, 6859, 7776, 9261, 10648, 10976, 12167, 13824, 16807, 17576, 19683, 24389, 25000, 27000, 29791, 30375, 32768, 35937, 39304, 42592, 42875, 50653, 54872, 59319, 64000

Offset: 1

Amiram Eldar, Dec 01 2023

Numbers whose prime factorization contains only exponents that are odd evil numbers (A129771).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175 (2016), pp. 385-395.

Intersection of A262675 and A268335.
Cf. A367801, A367802, A367803.
Cf. A129771.

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

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

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^A129771(k)) = Product_{p prime} f(1/p) = 1.22183814098622400889..., where f(x) = 1 + (2*x/(1-x^2) + Product_{k>=0} (1 - x^(2^k)) - Product_{k>=0} (1 - (-x)^(2^k)))/4.

A365174 The sum of divisors d of n such that gcd(d, n/d) is an exponentially odd number (A268335).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 27, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 51, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 108, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 107, 84, 144

Offset: 1

Amiram Eldar, Aug 25 2023

The number of these divisors is A365173(n).

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

Cf. A000203, A033634, A034448, A268335, A365172, A365173.

Cf. A002117, A013661, A013664.

f[p_, e_] := 1 + p^e + If[EvenQ[e], (p^(e + 1) - p)/(p^2 - 1), (1 + p^(2*Floor[e/4] + 1))*(p^(2*Floor[(e + 1)/4] + 1) - p)/(p^2 - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; 1 + p^e + if(e%2, (1 + p^(2*(e\4) + 1))*(p^(2*((e+1)\4) + 1) - p)/(p^2 - 1), (p^(e+1)-p)/(p^2-1)));}

Multiplicative with 1 + p^e + (p^(e + 1) - p)/(p^2 - 1) if e is even, and 1 + p^e + (1 + p^(2*floor(e/4)+1))*(p^(2*floor((e+1)/4)+1) - p)/(p^2 - 1) if e is odd.
a(n) <= A000203(n), with equality if and only if n is not a biquadrateful number (A046101).
a(n) >= A034448(n), with equality if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)*zeta(6)/(2*zeta(3))) * Product_{p prime} (1 + 1/p^3 - 1/p^6) = 0.809912096042... .

A367418 The exponentially odd numbers (A268335) divided by their squarefree kernels (A007947).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 9, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1

Offset: 1

Amiram Eldar, Nov 17 2023

Analogous to A102631, with the exponentially odd numbers instead of the square numbers (A000290).

All the terms are square numbers.

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

Cf. A000290, A008833, A102631, A268335, A367406, A367417, A367419.

s[n_] := n / Times @@ FactorInteger[n][[;; , 1]]; s /@ Select[Range[200], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]

b(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, f[i, 1]^(f[i, 2]-1), 0)); }
lista(kmax) = {my(b1); for(k = 1, kmax, b1 = b(k); if(b1 > 0, print1(b1, ", "))); }

a(n) = A003557(A268335(n)).
a(n) = A268335(n)/A367417(n).
a(n) = A367419(n)^2.
a(n) = A268335(n)^2/A367406(n).
a(n) = A008833(A268335(n)). - Amiram Eldar, Nov 30 2023

cp's OEIS Frontend

A363825 The number of infinitary divisors of n that are exponentially odd numbers (A268335).

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A374456 The Euler phi function values of the exponentially odd numbers (A268335).

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A385135 The sum of divisors d of n such that n/d is an exponentially odd number (A268335).

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A365173 The number of divisors d of n such that gcd(d, n/d) is an exponentially odd number (A268335).

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A325058 Starts of runs of 7 consecutive exponentially odd numbers (A268335).

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A367406 The exponentially odd numbers (A268335) multiplied by their squarefree kernels (A007947).

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

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A367804 Numbers that are both exponentially odd (A268335) and exponentially evil (A262675).

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