A368167 -id:A368167 - cp's OEIS frontend

A368169 The number of divisors of the largest unitary divisor of n that is a cubefull exponentially odd number (A368167).

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 14 2023

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

Cf. A000005, A013661, A098198, A335275, A335988, A368167, A368168.

f[p_, e_] := If[e == 1 || EvenQ[e], 1, e+1]; 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] == 1 || !(f[i, 2]%2), 1, f[i, 2]+1));}

Multiplicative with a(p^e) = e+1 if e is odd that is larger than 1, and 1 otherwise.
a(n) = A000005(A368167(n)).
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= A000005(n), with equality if and only if n is in A335988.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 3/p^3 - 2/p^4 - 1/p^5 + 1/p^6) = 1.47140789970892803631... .

A368168 The number of unitary divisors of n that are cubefull exponentially odd numbers (A335988).

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, 2, 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, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 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 14 2023

First differs from A359411 and A367516 at n = 64.

Also, the number of unitary divisors of the largest unitary divisor of n that is a cubefull exponentially odd number (A368167).

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

Cf. A013661, A034444, A055076, A325837, A335275, A335988, A368167, A368169.

Cf. A359411, A367516.

f[p_, e_] := If[e == 1 || EvenQ[e], 1, 2]; 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] == 1 || !(f[i, 2]%2), 1, 2));}

a(n) = A034444(A368167(n)).
Multiplicative with a(p^e) = 2 if e is odd that is larger than 1, and 1 otherwise.
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= n, with equality if and only if n is in A335988.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 1.12560687309375943599... .

A368170 The largest cubefull exponentially odd divisor of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 27, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 14 2023

First differs from A008834 at n = 32, and from A366906 at n = 64.

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

Cf. A004709, A335988, A350390, A356192, A368167, A368171.

Cf. A008834, A366906.

f[p_, e_] := If[e <= 2, 1, If[EvenQ[e], p^(e-1), p^e]]; 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, 1, if(!(f[i,2]%2), f[i,1]^(f[i, 2]-1), f[i, 1]^f[i, 2])))};

Multiplicative with a(p^e) = 1 if e <= 2, a(p^e) = p^e if e is odd and e > 1, and p^(e-1) otherwise.
a(n) = n/A368171(n).
a(n) >= 1, with equality if and only if n is cubefree (A004709).
a(n) <= n, with equality if and only if n is in A335988.

A368540 The smallest unitary divisor d of n such that n/d is a term of A138302.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 27, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 125, 1, 1

Offset: 1

Amiram Eldar, Dec 29 2023

First differs from A368167 at n = 64 and from A367513 at n = 128.

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

Cf. A048298, A077610, A138302, A367168, A367513, A368167.

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], 1, p^e]; 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] == 1 << valuation(f[i, 2], 2), 1, f[i, 1]^f[i, 2]));}

from math import prod
from sympy import factorint
def A368540(n): return prod(p**e for p, e in factorint(n).items() if not e or (e&-e)^e) # Chai Wah Wu, Dec 30 2023

a(n) = n / A367168(n).
Multiplicative with a(p^e) = p^(e-A048298(e)).
a(n) >= 1, with equality if and only if n is in A138302.

cp's OEIS Frontend

A368169 The number of divisors of the largest unitary divisor of n that is a cubefull exponentially odd number (A368167).

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A368168 The number of unitary divisors of n that are cubefull exponentially odd numbers (A335988).

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A368170 The largest cubefull exponentially odd divisor of n.

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A368540 The smallest unitary divisor d of n such that n/d is a term of A138302.

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