A368711 -id:A368711 - cp's OEIS frontend

A368712 The maximal exponent in the prime factorization of the cubefree numbers.

0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1

Offset: 1

Amiram Eldar, Jan 04 2024

The asymptotic density of occurrences of 1 is zeta(3)/zeta(2) = 0.730762... (A253905), and the asymptotic density of occurrences of 2 is 1 - zeta(3)/zeta(2) = 0.269237... .

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

Cf. A004709, A005117, A008966, A013929, A033150, A051903, A067259.
Cf. A002117, A013661, A253905.
Similar sequences: A368710, A368711, A368713.

s[n_] := If[n == 1, 0, Max @@ Last /@ FactorInteger[n]]; s /@ Select[Range[100], Max[FactorInteger[#][[;; , 2]]] < 3 &]
(* or *)
f[n_] := Module[{e = Max @@ FactorInteger[n][[;; , 2]]}, If[e < 3, e, Nothing]]; f[1] = 0; Array[f, 100]

lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = vecmax(factor(k)[,2]); if(e < 3, print1(e, ", ")));}

from sympy import mobius, integer_nthroot, factorint
def A368712(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 max(factorint(m).values(),default=0) # Chai Wah Wu, Aug 12 2024

a(n) = A051903(A004709(n)).
a(n) = 2 - A008966(A004709(n)) for n >= 2.
Except for n = 1, a(n) = 1 or 2.
a(n) = 1 if and only if A004709(n) is squarefree (A005117).
a(n) = 2 if and only if A004709(n) > 1 and is nonsquarefree (A013929), i.e., A004709(n) is in A067259.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 - zeta(3)/zeta(2) = 2 - A253905 = 1.269237030598... .

A368710 The maximal exponent in the prime factorization of the powerful numbers.

Original entry on oeis.org

0, 2, 3, 2, 4, 2, 3, 5, 2, 2, 6, 3, 4, 2, 3, 2, 3, 7, 4, 2, 2, 3, 3, 2, 5, 8, 5, 2, 4, 3, 2, 3, 4, 4, 2, 2, 3, 9, 2, 6, 4, 4, 3, 2, 6, 4, 5, 2, 5, 2, 2, 3, 5, 3, 10, 2, 3, 7, 2, 2, 4, 3, 3, 3, 2, 3, 2, 2, 5, 6, 2, 6, 2, 3, 2, 4, 5, 4, 4, 11, 2, 7, 3, 2, 8, 3, 4

Offset: 1

Amiram Eldar, Jan 04 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Generalizations of the Niven constant and the Feller-Tornier constant, International Mathematical Forum, Vol. 13, No. 9 (2018), pp. 415-425.

Cf. A001694, A033150, A048250, A051903, A059956, A090699.

Similar sequences: A368711, A368712, A368713.

s[n_] := If[n == 1, 0, Max @@ Last /@ FactorInteger[n]]; s /@ Select[Range[3000], # == 1 || Min[FactorInteger[#][[;;, 2]]] > 1 &]
(* or *)
f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[n == 1, 0, If[Min[e] > 1, Max[e], Nothing]]]; Array[f, 3000]

lista(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[,2]; if(k == 1, print1(0, ", "), if(vecmin(e) > 1, print1(vecmax(e), ", "))));}

a(n) = A051903(A001694(n)).
a(n) >= 2 for n >= 2.
Sum_{a(n)<=x} = D_{2,1} * sqrt(x) + O(sqrt(x)), where D_{2,1} = (6/Pi^2) * (2 + Sum_{k>=1} (A051903(k)+2)/(sqrt(k) * A048250(k))) (Jakimczuk, 2018; Theorem 2.1 and Remark 2.3).
Asymptotic mean (consequence of the formula above): Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = D_{2,1} * zeta(3)/zeta(3/2) = D_{2,1} / A090699.
The sum in the formula for D_{2,1} converges slowly: for k up to 10^8, 10^9 and 10^10 the sums are 14.845..., 14.908... and 14.938..., respectively. Thus, a lower bound for the value of this mean, calculated by summing over k=1..10^10, is 4.738... .

A368713 The maximal exponent in the prime factorization of the nonsquarefree numbers.

Original entry on oeis.org

2, 3, 2, 2, 4, 2, 2, 3, 2, 3, 2, 5, 2, 3, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 6, 2, 3, 2, 2, 4, 4, 2, 3, 2, 2, 5, 2, 2, 2, 3, 3, 4, 2, 2, 3, 2, 2, 3, 2, 7, 2, 3, 3, 2, 4, 2, 2, 2, 3, 2, 2, 5, 4, 2, 3, 2, 2, 2, 2, 4, 2, 3, 2, 3, 6, 2, 2, 3, 2, 2, 4, 2, 3, 2, 5, 2, 2, 3, 2, 2, 4

Offset: 1

Amiram Eldar, Jan 04 2024

The terms of A051903 that are larger than 1.

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

Cf. A013661, A013929, A033150, A051903.

Similar sequences: A368710, A368711, A368712.

s[n_] := Max @@ Last /@ FactorInteger[n]; s /@ Select[Range[250], !SquareFreeQ[#] &]
(* or *)
f[n_] := Module[{e = Max @@ FactorInteger[n][[;; , 2]]}, If[e > 1, e, Nothing]]; Array[f, 250]

lista(kmax) = {my(e); for(k = 2, kmax, e = vecmax(factor(k)[,2]); if(e > 1, print1(e, ", ")));}

a(n) = A051903(A013929(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (c * zeta(2) - 1)/(zeta(2) - 1) = 2.798673520766..., where c = 1.705211... is Niven's constant (A033150).

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... .

A372601 The maximal exponent in the prime factorization of the largest exponentially odd divisor of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, May 07 2024

First differs from A331273 at n = 64.

Differs from A363332 at n = 1, 216, 432, 648, 864, 1000, ... .

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

Cf. A051903, A109613, A350390, A368711.
Cf. A331273, A363332.
Similar sequences: A007424, A368781, A372602, A372603, A372604.

f[n_] := n - If[EvenQ[n], 1, 0]; a[n_] := f[Max[FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100]

s(n) = (n+1) \ 2 * 2 - 1;
a(n) = if(n>1, s(vecmax(factor(n)[,2])), 0);

a(n) = A051903(A350390(n)).
a(n) = A109613(A051903(n)-1) for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + 2 * Sum_{i>=1} (1 - (1/zeta(2*i+1))) = 1.42929441950714075659... .

A372466 The maximal exponent in the prime factorization of the numbers whose number of divisors is a power of 2 (A036537).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 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, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3

Offset: 1

Amiram Eldar, May 01 2024

All the terms are of the form 2^k-1 (A000225).

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

Cf. A000225, A036537, A051903, A327839, A372467.

Similar sequences: A368710, A368711, A368712, A368713, A369933, A369935.

pow2Q[n_] := n == 2^IntegerExponent[n, 2]; f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, pow2Q[# + 1] &], Max @@ e, Nothing]]; f[1] = 0; Array[f, 150]

ispow2(n) = n >> valuation(n, 2) == 1;
lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = factor(k)[, 2]; if(ispow2(vecprod(apply(x -> x + 1, e))), print1(vecmax(e), ", "))); }

a(n) = A051903(A036537(n)).
a(n) = 2^A372467(n) - 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (d(1) + Sum_{k>=2} (2^k-1) * (d(k) - d(k-1))) / A327839 = 1.25306367526166810834..., where d(k) = Product_{p prime} (1 - 1/p + Sum_{i=1..k} (1/p^(2^i-1)-1/p^(2^i))).

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.

cp's OEIS Frontend

A368712 The maximal exponent in the prime factorization of the cubefree numbers.

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A368710 The maximal exponent in the prime factorization of the powerful numbers.

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A368713 The maximal exponent in the prime factorization of the nonsquarefree numbers.

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

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A372601 The maximal exponent in the prime factorization of the largest exponentially odd divisor of n.

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A372466 The maximal exponent in the prime factorization of the numbers whose number of divisors is a power of 2 (A036537).

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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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