A350388 -id:A350388 - cp's OEIS frontend

A377818 Powerful numbers that have a single even exponent in their prime factorization.

4, 9, 16, 25, 49, 64, 72, 81, 108, 121, 169, 200, 256, 288, 289, 361, 392, 432, 500, 529, 625, 648, 675, 729, 800, 841, 961, 968, 972, 1024, 1125, 1152, 1323, 1352, 1369, 1372, 1568, 1681, 1728, 1849, 2000, 2209, 2312, 2401, 2592, 2809, 2888, 3087, 3200, 3267, 3481

Offset: 1

Amiram Eldar, Nov 09 2024

Each term can be represented in a unique way as m * p^(2*k), k >= 1, where m is a cubefull exponentially odd number (A335988) and p is a prime that does not divide m.

Powerful numbers k such that A350388(k) is a prime power with an even positive exponent (A056798 \ {1}).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Intersection of A001694 and A377816.
Subsequence of A377819.
Cf. A056798, A065487, A335988, A350388.

With[{max = 3500}, Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], Count[FactorInteger[#][[;; , 2]], _?EvenQ] == 1 &]]

is(k) = if(k == 1, 0, my(e = factor(k)[, 2]); vecmin(e) > 1 && #select(x -> !(x%2), e) == 1);

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p*(p^2-1))) * Sum_{p prime} (p/(p^3-p+1)) = 0.61399274770712398109... .

A351572 Arithmetic derivative of the largest unitary divisor of n that is a square.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 0, 6, 0, 0, 4, 0, 0, 0, 32, 0, 6, 0, 4, 0, 0, 0, 0, 10, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 60, 0, 0, 0, 0, 0, 0, 0, 4, 6, 0, 0, 32, 14, 10, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 6, 192, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 10, 4, 0, 0, 0, 32, 108, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 4, 0, 0, 0, 0, 0, 14, 6, 140

Offset: 1

Antti Karttunen, Feb 23 2022

nonn

All terms are even: see comments in A235992 and observe that the terms of A350388 are all either odd or multiples of 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A235992, A350388, A351570, A351573.

f1[p_, e_] := If[EvenQ[e], p^e, 1]; f2[p_, e_] := If[EvenQ[e], e/p, 0]; a[1] = 0; a[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]))*(Plus @@ f2 @@@ f); Array[a, 100] (* Amiram Eldar, Feb 23 2022 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A350388(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(0==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351572(n) = A003415(A350388(n));

a(n) = A003415(A350388(n)).

A351573 Arithmetic derivative of the largest unitary divisor of n that is an exponentially odd number.

Original entry on oeis.org

0, 1, 1, 0, 1, 5, 1, 12, 0, 7, 1, 1, 1, 9, 8, 0, 1, 1, 1, 1, 10, 13, 1, 44, 0, 15, 27, 1, 1, 31, 1, 80, 14, 19, 12, 0, 1, 21, 16, 68, 1, 41, 1, 1, 1, 25, 1, 1, 0, 1, 20, 1, 1, 81, 16, 92, 22, 31, 1, 8, 1, 33, 1, 0, 18, 61, 1, 1, 26, 59, 1, 12, 1, 39, 1, 1, 18, 71, 1, 1, 0, 43, 1, 10, 22, 45, 32, 140, 1, 7, 20, 1, 34

Offset: 1

Antti Karttunen, Feb 23 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003415, A350388, A268335 (exponentially odd numbers), A351571, A351572.

f1[p_, e_] := If[OddQ[e], p^e, 1]; f2[p_, e_] := If[OddQ[e], e/p, 0]; a[1] = 0; a[n_] := (Times @@ f1 @@@ (f = FactorInteger[n])) * (Plus @@ f2 @@@ f); Array[a, 100] (* Amiram Eldar, Feb 23 2022 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A350389(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(1==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351573(n) = A003415(A350389(n));

a(n) = A003415(A350389(n)).

A367987 The number of square divisors of the largest unitary divisor of n that is a square.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 07 2023

Also, the number of divisors of the square root of the largest unitary divisor of n that is a square.

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

Cf. A000005, A046951, A071974, A268335, A335762, A350388, A365401, A367988.

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

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

Multiplicative with a(p^e) = e/2 + 1 if e is even and 1 otherwise.
a(n) = A046951(A350388(n)).
a(n) = A000005(A071974(n)).
a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + p/((p-1)*(p+1)^2)) = 1.450032... (A335762).

A370080 The product of the even exponents of the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Feb 09 2024

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

Cf. A005361, A013661, A268335, A335275, A350386, A350388, A370079.

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

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

a(n) = A005361(A350388(n)).
Multiplicative with a(p^e) = e if e is even, and 1 if e is odd.
a(n) = A005361(n)/A370079(n).
a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
a(n) <= A005361(n), with equality if and only if n is in A335275.
Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 1/p^s - 1/p^(3*s) + 1/p^(4*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)^2 * Product_{p prime} (1 - 1/p^2 - 1/p^3 +  2/p^4 - 1/p^5) = 1.53318063378623623841... .
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + (p^(2*s) + 1)/(p^s*(p^s - 1)*(p^s + 1)^2)). - Vaclav Kotesovec, Feb 11 2024

A374988 Largest unitary square divisor of n!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 144, 144, 9, 81, 518400, 518400, 25600, 25600, 1225, 35721, 35721, 35721, 21069103104, 21069103104, 52672757760000, 163840000, 75625, 75625, 18730002677760000, 468250066944000000, 18867078140625, 319515625, 767157015625, 767157015625, 15759472921106221891584

Offset: 0

Amiram Eldar, Jul 26 2024

nonn

Unitary analog of A055071.

a(n) is even if and only if n > 1 and is in A006364.

Amiram Eldar, Table of n, a(n) for n = 0..673

Cf. A006364, A055071, A350388, A374989.

f[p_, e_] := If[EvenQ[e], p^e, 1]; a[0] = a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 30, 0]

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

from math import prod
from itertools import count, islice
from collections import Counter
from sympy import factorint
def A374988_gen(): # generator of terms
    c = Counter()
    for i in count(0):
        c += Counter(factorint(i))
        yield prod(p**e for p, e in c.items() if e&1^1)
A374988_list = list(islice(A374988_gen(),30)) # Chai Wah Wu, Jul 27 2024

a(n) = A350388(n!).

a(n) = A374989(n)^2.

A377817 Numbers that have more than one even exponent in their prime factorization.

Original entry on oeis.org

36, 100, 144, 180, 196, 225, 252, 300, 324, 396, 400, 441, 450, 468, 484, 576, 588, 612, 676, 684, 700, 720, 784, 828, 882, 900, 980, 1008, 1044, 1089, 1100, 1116, 1156, 1200, 1225, 1260, 1296, 1300, 1332, 1444, 1452, 1476, 1521, 1548, 1575, 1584, 1600, 1620, 1692, 1700, 1764, 1800

Offset: 1

Amiram Eldar, Nov 09 2024

Subsequence of A072413 and differs from it by not having the terms 216, 1000, 1080, 1512, ... .
Each term can be represented in a unique way as m * k^2, where m is an exponentially odd number (A268335) and k is a composite number that is coprime to m.
Numbers k such that A350388(k) is a square of a composite number (A062312 \ {1}).
The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p*(p+1))) * (1 + Sum_{p prime} 1/(p^2+p-1)) = 0.032993560887093165933... .

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

Complement of the union of A268335 and A377816.
Subsequence of A072413.
Cf. A062312, A162645, A350388.

Select[Range[1800], Count[FactorInteger[#][[;; , 2]], _?EvenQ] > 1 &]

is(k) = if(k == 1, 0, my(e = factor(k)[, 2]); #select(x -> !(x%2), e) > 1);

A351575 Positions of primes in A351568.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 44, 45, 48, 50, 52, 60, 63, 64, 68, 72, 75, 76, 80, 84, 90, 92, 99, 108, 112, 116, 117, 124, 126, 132, 140, 148, 150, 153, 156, 164, 171, 172, 175, 176, 188, 192, 198, 200, 204, 207, 208, 212, 220, 228, 234, 236, 240, 244, 260, 261, 268, 272, 275, 276, 279, 284, 288, 289, 292, 304, 306

Offset: 1

Antti Karttunen, Feb 24 2022

nonn

Numbers k such that A350388(k) is one of the terms of A023194.

Positions of primes in A351568, positions of ones in A351570.

Cf. A000203, A023194, A350388.

f[p_, e_] := If[EvenQ[e], (p^(e + 1) - 1)/(p - 1), 1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[300], PrimeQ[s[#]] &] (* Amiram Eldar, Feb 25 2022 *)

A350388(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(0==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351568(n) = sigma(A350388(n));
isA351575(n) = isprime(A351568(n));

A367990 Sum of the squarefree divisors of the largest unitary divisor of n that is a square.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 1, 6, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 3, 8, 6, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 6, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 07 2023

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

Cf. A000203, A048250, A268335, A336643, A350388, A367991.

Cf. A307868.

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

Multiplicative with a(p^e) = p + 1 if e is even and 1 otherwise.
a(n) = A048250(A350388(n)).
a(n) = A000203(A336643(n)).
a(n) = A048250(n)/A367991(n).
a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-1)).
From Vaclav Kotesovec, Apr 20 2025: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * Product_{p prime} ((p^(2*s) - p) * (p^(2*s) + p^s + p) / ((p^s+1) * p^(3*s))).
Let f(s) = Product_{p prime} ((p^(2*s)-p) * (p^(2*s)+p^s+p) / ((p^s+1) * p^(3*s))).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
f(1) = A307868 = Product_{p prime} (1 - 2/(p*(p+1))) = 0.4716806136129978680752356330804820874259263820069868836357372554177321167...
f'(1) = f(1) * Sum_{p prime} (7*p + 5) * log(p) / ((p-1)*(p+1)*(p+2)) = f(1) * 3.0570993566532132522378281945383016697995408795919384628849894110222383828...
and gamma is the Euler-Mascheroni constant A001620. (End)

A375031 Numbers whose prime factorization has at least one exponent that equals 2 and no higher even exponent.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 225, 228, 234, 236, 242, 244, 245

Offset: 1

Amiram Eldar, Jul 28 2024

Subsequence of A304365 and differs from it by not having the terms 1, 144, 216, 324, 400, ... .
Subsequence of A038109 and differs from it by not having the terms 144, 324, 400, 576, 720, ... .
Numbers whose largest unitary divisor that is a square (A350388) is a square of squarefree number (A062503) that is larger than 1.
Each term is a product of two coprime numbers: an exponentially odd number (A268335) and a square of a squarefree number (A062503) that is larger than 1.
The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^3*(p+1))) - Product_{p prime}(1 - 1/(p*(p+1))) = A065466 - A065463 = 0.2432910611445097832029... .

			4 = 2^2 is a term because it has the exponent 2 in its prime factorization, and no higher even exponent.
144 = 2^4 * 3^2 is not a term because it has the exponent 4 in its prime factorization which is even and larger than 2.

Subsequence of A013929, A038109 and A304365.
A062503 \ {1} is a subsequence.
Cf. A065463, A065466, A268335, A335275, A350388, A375033.

q[n_] := Max[Select[FactorInteger[n][[;; , 2]], EvenQ]] == 2; Select[Range[250], q]

is(k) = {my(e = select(x -> !(x % 2), factor(k)[,2])); #e > 0 && vecmax(e) == 2;}

A375033(a(n)) = 2.

cp's OEIS Frontend

A377818 Powerful numbers that have a single even exponent in their prime factorization.

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A351572 Arithmetic derivative of the largest unitary divisor of n that is a square.

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A351573 Arithmetic derivative of the largest unitary divisor of n that is an exponentially odd number.

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A367987 The number of square divisors of the largest unitary divisor of n that is a square.

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A370080 The product of the even exponents of the prime factorization of n.

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A374988 Largest unitary square divisor of n!.

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A377817 Numbers that have more than one even exponent in their prime factorization.

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A351575 Positions of primes in A351568.

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