A366439 -id:A366439 - cp's OEIS frontend

A366438 The number of divisors of the exponentially odd numbers (A268335).

1, 2, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 8, 4, 4, 2, 8, 2, 6, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 2, 4, 2, 8, 4, 8, 4, 4, 2, 2, 4, 4, 8, 2, 4, 8, 2, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 8, 2, 4, 4, 4, 4, 12, 2, 2, 8, 2, 8, 8, 4, 2, 2, 8, 4, 2, 8, 4, 4, 4, 16, 4

Offset: 1

Amiram Eldar, Oct 10 2023

1 is the only odd term in this sequence.

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

Cf. A000005, A268335, A366439.

Similar sequences: A048691, A072048, A076400, A358040, A363194, A363195.

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, OddQ], Times @@ (e + 1), Nothing]]; f[1] = 1; Array[f, 150]

lista(max) = for(k = 1, max, my(e = factor(k)[, 2], isexpodd = 1); for(i = 1, #e, if(!(e[i] % 2), isexpodd = 0; break)); if(isexpodd, print1(vecprod(apply(x -> x+1, e)), ", ")));

from math import prod
from itertools import count, islice
from sympy import factorint
def A366438_gen(): # generator of terms
    for n in count(1):
        f = factorint(n).values()
        if all(e&1 for e in f):
            yield prod(e+1 for e in f)
A366438_list = list(islice(A366438_gen(),30)) # Chai Wah Wu, Oct 10 2023

a(n) = A000005(A268335(n)).

A366535 The sum of unitary divisors of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 3, 4, 6, 12, 8, 9, 18, 12, 14, 24, 24, 18, 20, 32, 36, 24, 36, 42, 28, 30, 72, 32, 33, 48, 54, 48, 38, 60, 56, 54, 42, 96, 44, 72, 48, 72, 54, 84, 72, 72, 80, 90, 60, 62, 96, 84, 144, 68, 96, 144, 72, 74, 114, 96, 168, 80, 126, 84, 108, 132, 120, 108, 90, 112

Offset: 1

Amiram Eldar, Oct 12 2023

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

Cf. A034448, A065463, A077610, A268335, A366439, A366534.

Similar sequences: A034676, A366537, A366539.

s[n_] := Module[{f = FactorInteger[n], e}, e = f[[;;, 2]]; If[AllTrue[e, OddQ], Times @@ (1 + Power @@@ f), Nothing]]; s[1] = 1; Array[s, 100]

lista(max) = for(k = 1, max, my(f = factor(k), e = f[, 2], isexpodd = 1); for(i = 1, #e, if(!(e[i] % 2), isexpodd = 0; break)); if(isexpodd, print1(prod(i = 1, #e, 1 + f[i, 1]^e[i]), ", ")));

a(n) = A034448(A268335(n)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (zeta(4)/d^2) * Product_{p prime} (1 - 2/p^4 + 1/p^5) = 1.92835521961603199612..., d = A065463 is the asymptotic density of the exponentially odd numbers.
The asymptotic mean of the unitary abundancy index of the exponentially odd numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A268335(k) = c * d = 1.35841479521454692063... .

A366440 The sum of divisors of the cubefree numbers (A004709).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 13, 18, 12, 28, 14, 24, 24, 18, 39, 20, 42, 32, 36, 24, 31, 42, 56, 30, 72, 32, 48, 54, 48, 91, 38, 60, 56, 42, 96, 44, 84, 78, 72, 48, 57, 93, 72, 98, 54, 72, 80, 90, 60, 168, 62, 96, 104, 84, 144, 68, 126, 96, 144, 72, 74, 114, 124, 140

Offset: 1

Amiram Eldar, Oct 10 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634; p. 619, eq. (2).

Cf. A000203, A004709, A358040.
Cf. A002117, A082020.
Similar sequences: A062822, A065764, A180114, A362986, A366439.

f[p_, e_] := (p^(e+1)-1)/(p-1); s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;;, 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; s[1] = 1; Array[s, 100]

lista(max) = for(k = 1, max, my(f = factor(k), iscubefree = 1); for(i = 1, #f~, if(f[i, 2] > 2, iscubefree = 0; break)); if(iscubefree, print1(sigma(f), ", ")));

from sympy import mobius, integer_nthroot, divisor_sigma
def A366440(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return divisor_sigma(m) # Chai Wah Wu, Aug 06 2024

a(n) = A000203(A004709(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 15*zeta(3)/(2*Pi^2) = A082020 * A002117 / 2 = 0.913453711751... .
The asymptotic mean of the abundancy index of the cubefree numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A004709(k) = 15/Pi^2 = 1.519817... (A082020).

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... .

A368472 Product of exponents of prime factorization of the exponentially odd numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Amiram Eldar, Dec 26 2023

The odd terms of A005361.

The first position of 2*k-1, for k = 1, 2, ..., is 1, 7, 24, 91, 154, 1444, 5777, 610, 92349, ..., which is the position of A085629(2*k-1) in A268335.

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

Cf. A005361, A013661, A065463, A085629, A098198, A268335.
Cf. A366438, A366439.
Similar sequences: A322327, A368473, A368474.

f[n_] := Module[{p = Times @@ FactorInteger[n][[;; , 2]]}, If[OddQ[p], p, Nothing]]; Array[f, 150]

lista(kmax) = {my(p); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); if(p%2, print1(p, ", ")));}

a(n) = A005361(A268335(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)^2/d) * Product_{p prime} (1 - 3/p^2 + 3/p^3 - 1/p^5) = 1.38446562720473484463..., where d = A065463 is the asymptotic density of the exponentially odd numbers.

A366442 The sum of divisors of the 5-rough numbers (A007310).

Original entry on oeis.org

1, 6, 8, 12, 14, 18, 20, 24, 31, 30, 32, 48, 38, 42, 44, 48, 57, 54, 72, 60, 62, 84, 68, 72, 74, 96, 80, 84, 108, 90, 112, 120, 98, 102, 104, 108, 110, 114, 144, 144, 133, 156, 128, 132, 160, 138, 140, 168, 180, 150, 152, 192, 158, 192, 164, 168, 183, 174, 248

Offset: 1

Amiram Eldar, Oct 10 2023

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

Cf. A000203, A013661, A007310, A100044, A366441.

Similar sequences: A008438, A062822, A065764, A180114, A362986, A366439, A366440.

a[n_] := DivisorSigma[1, 2*Floor[3*n/2] - 1]; Array[a, 100]

PARI
```
a(n) = sigma((3*n)\2 << 1 - 1)
```

from sympy import divisor_sigma
def A366442(n): return divisor_sigma((n+(n>>1)<<1)-1) # Chai Wah Wu, Oct 10 2023

a(n) = A000203(A007310(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2) = 1.644934... (A013661).
The asymptotic mean of the abundancy index of the 5-rough numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A007310(k) = Pi^2/9 = 1.0966227... (A100044).
In general, the asymptotic mean of the abundancy index of the prime(k)-rough numbers is zeta(2) * Product_{i=1..k-1} (1 - 1/prime(i)^2).

A374457 The Dedekind psi function values of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 3, 4, 6, 12, 8, 12, 18, 12, 14, 24, 24, 18, 20, 32, 36, 24, 48, 42, 36, 30, 72, 32, 48, 48, 54, 48, 38, 60, 56, 72, 42, 96, 44, 72, 48, 72, 54, 108, 72, 96, 80, 90, 60, 62, 96, 84, 144, 68, 96, 144, 72, 74, 114, 96, 168, 80, 126, 84, 108, 132, 120, 144, 90

Offset: 1

Amiram Eldar, Jul 09 2024

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

Cf. A001615, A065463, A268335.
Similar sequences related to psi: A000082, A033196, A323332, A371413, A371415.
Similar sequences related to exponentially odd numbers: A366438, A366439, A366534, A366535, A367417, A368711, A374456.

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) = A001615(A268335(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 / A065463^2 = 2.01515877170903249510... .

A382660 The unitary totient function applied to the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 7, 4, 10, 12, 6, 8, 16, 18, 12, 10, 22, 14, 12, 26, 28, 8, 30, 31, 20, 16, 24, 36, 18, 24, 28, 40, 12, 42, 22, 46, 32, 52, 26, 40, 42, 36, 28, 58, 60, 30, 48, 20, 66, 44, 24, 70, 72, 36, 60, 24, 78, 40, 82, 64, 42, 56, 70, 88, 72, 60, 46, 72

Offset: 1

Amiram Eldar, Apr 02 2025

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

Cf. A013662, A047994, A065463, A268335.

Similar sequences: A358346, A363825, A366438, A366439, A366534, A366535, A367417, A368711, A368979, A374456, A382661.

f[p_, e_] := p^e-1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; expOddQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; uphi /@ Select[Range[100], expOddQ]

uphi(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2]-1);}
isexpodd(n) = {my(f = factor(n)); for(i=1, #f~, if(!(f[i, 2] % 2), return (0))); 1;}
list(lim) = apply(uphi, select(isexpodd, vector(lim, i, i)));

a(n) = A047994(A268335(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(4)/(2*d^2)) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 2/p^4 + 1/p^5) = 0.504949539649594981601..., and d = A065463 is the asymptotic density of the exponentially odd numbers.

A382661 The unitary Jordan totient function applied to the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 3, 8, 24, 24, 48, 63, 72, 120, 168, 144, 192, 288, 360, 384, 360, 528, 504, 504, 728, 840, 576, 960, 1023, 960, 864, 1152, 1368, 1080, 1344, 1512, 1680, 1152, 1848, 1584, 2208, 2304, 2808, 2184, 2880, 3024, 2880, 2520, 3480, 3720, 2880, 4032, 2880, 4488, 4224

Offset: 1

Amiram Eldar, Apr 02 2025

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

Cf. A013664, A065463, A191414, A268335.

Similar sequences: A358346, A363825, A366438, A366439, A366534, A366535, A367417, A368711, A368979, A374456, A382660.

f[p_, e_] := p^(2*e)-1; uj2[1] = 1; uj2[n_] := Times @@ f @@@ FactorInteger[n]; expOddQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; uj2 /@ Select[Range[100], expOddQ]

uj2(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(2*f[i, 2])-1);}
isexpodd(n) = {my(f = factor(n)); for(i=1, #f~, if(!(f[i, 2] % 2), return (0))); 1;}
list(lim) = apply(uj2, select(isexpodd, vector(lim, i, i)));

a(n) = A191414(A268335(n)).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(6)/(3*d^3)) * Product_{p prime} (1 - 1/p^2 + 1/p^5 - 2/p^6 + 1/p^7) = 0.59726984314764530141..., and d = A065463 is the asymptotic density of the exponentially odd numbers.

A380326 a(n) is the sum of squarefree divisors of the n-th exponentially odd number.

Original entry on oeis.org

1, 3, 4, 6, 12, 8, 3, 18, 12, 14, 24, 24, 18, 20, 32, 36, 24, 12, 42, 4, 30, 72, 32, 3, 48, 54, 48, 38, 60, 56, 18, 42, 96, 44, 72, 48, 72, 54, 12, 72, 24, 80, 90, 60, 62, 96, 84, 144, 68, 96, 144, 72, 74, 114, 96, 168, 80, 126, 84, 108, 132, 120, 36, 90, 112

Offset: 1

Amiram Eldar, Jan 20 2025

The number of squarefree divisors of the n-th exponentially odd number is A366534(n).

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

Cf. A005117, A048250, A065463, A268335, A366439, A366534, A366535.

f[p_, e_] := p + 1; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], OddQ], Times @@ f @@@ fct, Nothing]]; s[1] = 1; Array[s, 100]

A048250(f) = vecprod(apply(x -> x+1, f[, 1]));
list(lim) = for(k = 1, lim, my(f = factor(k), isexpodd = 1); for(i = 1, #f~, if(!(f[i, 2] % 2), isexpodd = 0; break)); if(isexpodd, print1(A048250(f), ", ")));

a(n) = A048250(A268335(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (1/d^2) * Product_{p prime} (1 - 1/(p*(p^2+1))) = 1.73161118511498727822..., and d = A065463 is the asymptotic density of the exponentially odd numbers.

cp's OEIS Frontend

A366438 The number of divisors of the exponentially odd numbers (A268335).

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A366535 The sum of unitary divisors of the exponentially odd numbers (A268335).

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A366440 The sum of divisors of the cubefree numbers (A004709).

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

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A368472 Product of exponents of prime factorization of the exponentially odd numbers.

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A366442 The sum of divisors of the 5-rough numbers (A007310).

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A374457 The Dedekind psi function values of the exponentially odd numbers (A268335).

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A382660 The unitary totient function applied to the exponentially odd numbers (A268335).

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