A350388 -id:A350388 - cp's OEIS frontend

A350386 a(n) is the sum of the even exponents in the prime factorization of n.

0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 2, 4, 0, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Dec 28 2021

a(n) is the number of prime divisors of n, counted with multiplicity, with an even exponent in the prime factorization of n.

All the terms are even by definition.

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

Cf. A000290, A001222, A162641, A268335, A350387, A350388.

f[p_, e_] := If[EvenQ[e], e, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = my(f=factor(n)); sum(k=1, #f~, if (!(f[k, 2] % 2), f[k, 2])); \\ Michel Marcus, Dec 29 2021

from sympy import factorint
def a(n): return sum(e for e in factorint(n).values() if e%2 == 0)
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Dec 28 2021

Additive with a(p^e) = e if e is even and 0 otherwise.
a(n) = A001222(A350388(n)).
a(n) = 0 if and only if n is an exponentially odd number (A268335).
a(n) = A001222(n) - A350387(n).
a(n) = A001222(n) if and only if n is a positive square (A000290 \ {0}).
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 2*p/((p-1)*(p+1)^2) = 0.7961706018...

A056623 If n=LLgggf (see A056192) and a(n) = LL, then its complementary divisor n/LL = gggf and gcd(L^2, n/LL) = 1.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 1, 25, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1, 1, 4, 1, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 4, 1, 49, 9

Offset: 1

Labos Elemer, Aug 08 2000

The part of the name "Largest unitary square divisor of n" was removed because it is correct only for numbers whose odd exponents in their prime factorization are all smaller than 5. For the correct largest unitary square divisor of n see A350388. - Amiram Eldar, Jul 26 2024

			a(200) = A008833(200)/A055229(200)^2 = 100/2^2 = 25.
a(250) = A008833(250)/A055229(250)^2 = 25/5^2 = 1.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A008833, A055229, A000188, A046951, A034444, A056192, A056622, A350388.

f[p_, 1] := 1; f[p_, e_] := If[EvenQ[e], p^e, p^(e-3)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)

A055229(n) = { my(c=core(n)); gcd(c, n/c); }; \\ Charles R Greathouse IV, Nov 20 2012
A008833(n) = n/core(n) \\ Michael B. Porter, Oct 17 2009
A056623(n) = (A008833(n)/(A055229(n)^2)); \\ Antti Karttunen, Nov 19 2017

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(e == 1, 1, if(e%2, p^(e-3), p^e)));} \\ Amiram Eldar, Dec 25 2023

(definec (A056623 n) (if (= 1 n) n (let ((e (A067029 n)) (rest (A056623 (A028234 n)))) (cond ((even? e) (* (A028233 n) rest)) ((= 1 e) rest) (else (* (expt (A020639 n) (- e 3)) rest)))))) ;; After Jovovic's multiplicative formula, using memoization-macro definec - Antti Karttunen, Nov 19 2017

a(n) = A008833(n)/A055229(n)^2 = K^2/g^2, which coincides with the largest square divisor iff the g-factor is 1.
Multiplicative with a(p^e)=p^e for even e, a(p)=1, a(p^e)=p^(e-3) for odd e > 1. - Vladeta Jovovic, Apr 30 2002
From Amiram Eldar, Dec 25 2023 (Start)
Dirichlet g.f.: zeta(2*s-2) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-2) + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = Product_{p prime} (1 + 1/p^(3/2) - 1/p^(5/2) + 1/p^(9/2)) = 1.81133051934001073532... . (End)
a(n) = A056622(n)^2. - Amiram Eldar, Jul 26 2024

Name edited by Amiram Eldar, Jul 26 2024

A351570 Arithmetic derivative of the sum of the divisors of the largest unitary divisor of n that is a square.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 22, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 22, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 22, 1, 38

Offset: 1

Antti Karttunen, Feb 23 2022

nonn

Observation: There seems to be no terms in range 2..19 in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A003415, A342925, A350388, A351568, A351571, A351572, A351575 (positions of ones).

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); };
A351570(n) = A003415(sigma(A350388(n)));

a(n) = A003415(A351568(n)) = A342925(A350388(n)) = A003415(A000203(A350388(n))).

A365401 The number of 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, 3, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 5, 3, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 5, 5, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Sep 03 2023

First differs from A212181 at n = 32.
The sum of these divisors is A351568(n).
All the terms are odd.

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

Cf. A000005, A000290, A212181, A268335 A350388, A351568, A365402.

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

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

a(n) = A000005(A350388(n)).
a(n) = A000005(n) / A365402(n).
a(n) <= A000005(n) with equality if and only if n is a square (A000290).
a(n) >= 1 with equality if and only if n is an exponentially odd number (A268335).
Multiplicative with a(p^e) = 1 if e is odd, and e+1 if e is even.
Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 1/p^s + 1/p^(2*s) - 1/p^(3*s)).
From Vaclav Kotesovec, Sep 05 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s)^2 * Product_{p prime} (1 - 2/p^(3*s) + 1/p^(4*s)).
Let f(s) = Product_{p prime} (1 - 2/p^(3*s) + 1/p^(4*s)).
Sum_{k=1..n} a(k) ~ f(1) * Pi^4 * n / 36 + sqrt(n) * zeta(1/2) * f(1/2)/2 * (log(n) + 4*gamma - 2 + zeta'(1/2)/zeta(1/2) + f'(1/2)/f(1/2)), where
f(1) = Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.7446954979060674204391238715944543281179691329049241118630718137015097502...,
f(1/2) = Product_{p prime} (1 - 2/p^(3/2) + 1/p^2) = 0.2312522106782016049013780988087017618011735848676872392115785564006277675...,
f'(1/2) = f(1/2) * Sum_{p prime} 2*(3*sqrt(p) - 2) * log(p) / (1 - 2*sqrt(p) + p^2) = f(1/2) * 6.937179176924511608542644054340717439502789953858512457656... and gamma is the Euler-Mascheroni constant A001620. (End)

A366126 The largest unitary divisor of n that is a cube.

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, 1, 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

Offset: 1

Amiram Eldar, Sep 30 2023

First differs from A056191 at n = 32.

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

Cf. A001221, A001222, A000578, A056191, A350388, A366124, A366125.

f[p_, e_] := If[Divisible[e, 3], p^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] % 3, 1, f[i,1]^f[i,2]));}

Multiplicative with a(p^e) = p^e if e is divisible by 3 and 1 otherwise.
a(n) = n if and only if n is a positive cube (A000578 \ {0}).
A001221(a(n)) = A366124(n).
A001222(a(n)) = A366125(n).
Sum_{k=1..n} a(k) ~ c * n^(4/3), where c = (1/4) * Product_{p prime} (1 + (p^(1/3) + p^(5/3))/(1 + p + p^2 + p^3)) = 0.61488587249270755696... .

A367513 The exponentially evil part of n: the largest unitary divisor of n that is an exponentially evil number (A262675).

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

Offset: 1

Amiram Eldar, Nov 21 2023

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

Cf. A001969, A010059, A034444, A102391, A262675, A270428, A366902, A366904, A367512, A367516.

Similar sequences: A350388, A350389, A366906, A367168, A367514.

f[p_, e_] := p^(e * (1 - ThueMorse[e])); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

from math import prod
from sympy import factorint
def A367513(n): return prod(p**e for p, e in factorint(n).items() if e.bit_count()&1^1) # Chai Wah Wu, Nov 23 2023

Multiplicative with a(p^e) = p^(e*A010059(e)) = p^A102391(e).
a(n) = n/A367514(n).
A001221(a(n)) = A367512(n).
A034444(a(n)) = A367516(n).
a(n) >= 1, with equality if and only if n is an exponentially odious number (A270428).
a(n) <= n, with equality if and only if n is an exponentially evil number (A262675).

A367514 The exponentially odious part of n: the largest unitary divisor of n that is an exponentially odious number (A270428).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70

Offset: 1

Amiram Eldar, Nov 21 2023

First differs from A056192 at n = 32, and from A270418 and A367168 at n = 128.

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

Cf. A000069, A001221, A010060, A034444, A102392, A262675, A270428, A293439, A366901, A366903, A367515.
Cf. A056192, A270418.
Similar sequences: A350388, A350389, A366905, A367168, A367513.

f[p_, e_] := p^(e*ThueMorse[e]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

from math import prod
from sympy import factorint
def A367514(n): return prod(p**e for p, e in factorint(n).items() if e.bit_count()&1) # Chai Wah Wu, Nov 23 2023

Multiplicative with a(p^e) = p^(e*A010060(e)) = p^A102392(e).
a(n) = n/A367513(n).
A001221(a(n)) = A293439(n).
A034444(a(n)) = A367515(n).
a(n) >= 1, with equality if and only if n is an exponentially evil number (A262675).
a(n) <= n, with equality if and only if n is an exponentially odious number (A270428).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 0.88585652437242918295..., and f(x) = (x+2)/(2*(x+1)) + (x/2) * Product_{k>=0} (1 - x^(2^k)).

A375033 The maximum even exponent in the prime factorization of n, or 0 if no such exponent exists.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Jul 28 2024

First differs from A350386 at n = 36.
The asymptotic density of the occurrences of 0's is d(0) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463; the asymptotic density of the exponentially odd numbers, A268335).
The asymptotic density of the occurrences of 2*k, for k = 1, 2, ..., is d(k) = Product_{p prime} (1 - 1/(p^(2*k+1)*(p+1))) - Product_{p prime} (1 - 1/(p^(2*k-1)*(p+1))).
For example, the asymptotic density of the occurrences of 2's is d(1) = Product_{p prime} (1 - 1/(p^3*(p+1))) - Product_{p prime}(1 - 1/(p*(p+1))) = 0.243291... (the asymptotic density of A375031).

Cf. A051903, A065463, A162642, A268335, A350386, A350388, A375031, A375032.

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

a(n) = {my(e = select(x -> !(x % 2), factor(n)[,2])); if(#e == 0, 0, vecmax(e));}

max(a(n), A375032(n)) = A051903(n).
a(n) = 0 if and only if n is an exponentially odd number (A268335).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (2*k) * d(k) = 0.72584606502990528747..., where d(k) is defined in the Comments section above.
a(n) = A051903(A350388(n)). - Amiram Eldar, Aug 17 2024

A376553 Largest unitary square divisor of binomial(n, floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 1, 1, 9, 36, 1, 4, 4, 1, 9, 9, 1, 4, 1, 4, 4, 1, 1, 4, 100, 25, 100, 25, 9, 144, 9, 9, 1, 4, 25, 100, 100, 25, 9, 36, 4, 1, 4, 1, 25, 400, 225, 900, 1764, 441, 196, 49, 49, 784, 4, 1, 1, 16, 1, 16, 16, 1, 441, 441, 49, 196, 49, 196, 36, 9

Offset: 0

Amiram Eldar, Sep 28 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 0..1000

Cf. A001405, A350388, A376554, A376555, A376556.

f[p_, e_] := If[EvenQ[e], p^e, 1]; a[0] = a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[Binomial[n, Floor[n/2]]]; Array[a, 100, 0]

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

a(n) = A350388(A001405(n)).

a(n) = A376554(n)^2.

A377816 Numbers that have a single even exponent in their prime factorization.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 90, 92, 98, 99, 108, 112, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 188, 192, 198, 200, 204, 207, 208, 212, 220, 228

Offset: 1

Amiram Eldar, Nov 09 2024

First differs from A162645 at n = 239: A162645(239) = 900 = 2^2 * 3^2 * 5^2 is not a term of this sequence.
Each term can be represented in a unique way as m * p^(2*k), k >= 1, where m is an exponentially odd number (A268335) and p is a prime that does not divide m.
Numbers k such that A350388(k) is a prime power with an even positive exponent (A056798 \ {1}).
The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p*(p+1))) * Sum_{p prime} 1/(p^2+p-1) = 0.26256423811374124133... .

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

Cf. A056798, A065463, A162645, A229125 (odd analog), A268335, A350388, A377817.

A377818 is a subsequence.

Select[Range[250], Count[FactorInteger[#][[;; , 2]], _?EvenQ] == 1 &]

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

cp's OEIS Frontend

A350386 a(n) is the sum of the even exponents in the prime factorization of n.

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A056623 If n=LLgggf (see A056192) and a(n) = LL, then its complementary divisor n/LL = gggf and gcd(L^2, n/LL) = 1.

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A351570 Arithmetic derivative of the sum of the divisors of the largest unitary divisor of n that is a square.

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A365401 The number of divisors of the largest unitary divisor of n that is a square.

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A366126 The largest unitary divisor of n that is a cube.

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A367513 The exponentially evil part of n: the largest unitary divisor of n that is an exponentially evil number (A262675).

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A367514 The exponentially odious part of n: the largest unitary divisor of n that is an exponentially odious number (A270428).

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