A350389 -id:A350389 - cp's OEIS frontend

A350387 a(n) is the sum of the odd exponents in the prime factorization of n.

0, 1, 1, 0, 1, 2, 1, 3, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 4, 0, 2, 3, 1, 1, 3, 1, 5, 2, 2, 2, 0, 1, 2, 2, 4, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 4, 2, 4, 2, 2, 1, 2, 1, 2, 1, 0, 2, 3, 1, 1, 2, 3, 1, 3, 1, 2, 1, 1, 2, 3, 1, 1, 0, 2, 1, 2, 2, 2, 2, 4, 1, 2, 2, 1, 2, 2, 2, 6, 1, 1, 1, 0, 1, 3, 1, 4, 3

Offset: 1

Amiram Eldar, Dec 28 2021

First differs from A125073 at n = 32.

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

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

Cf. A000290, A001222, A001620, A083342, A125073, A162641, A268335, A350386, A350389.

f[p_, e_] := If[OddQ[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 28 2021

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

Additive with a(p^e) = e if e is odd and 0 otherwise.
a(n) = A001222(A350389(n)).
a(n) = 0 if and only if n is a positive square (A000290 \ {0}).
a(n) = A001222(n) - A350386(n).
a(n) = A001222(n) if and only if n is an exponentially odd number (A268335).
Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = A083342 - Sum_{p prime} 2*p/((p-1)*(p+1)^2) = gamma + Sum_{p prime} (log(1-1/p) + (p^2+1)/((p-1)*(p+1)^2)) = 0.2384832800... and gamma is Euler's constant (A001620).

A351571 Arithmetic derivative of the sum of the divisors of the largest unitary divisor of n that is an exponentially odd number.

Original entry on oeis.org

0, 1, 4, 0, 5, 16, 12, 8, 0, 21, 16, 4, 9, 44, 44, 0, 21, 1, 24, 5, 80, 60, 44, 92, 0, 41, 68, 12, 31, 156, 80, 51, 112, 81, 112, 0, 21, 92, 92, 123, 41, 272, 48, 16, 5, 156, 112, 4, 0, 1, 156, 9, 81, 244, 156, 244, 176, 123, 92, 44, 33, 272, 12, 0, 124, 384, 72, 21, 272, 384, 156, 8, 39, 101, 4, 24, 272, 332, 176, 5

Offset: 1

Antti Karttunen, Feb 23 2022

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A003415, A268335 (exponentially odd numbers), A342925, A350389, A351569, A351570, A351573.

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); };
A351571(n) = A003415(sigma(A350389(n)));

a(n) = A003415(A351569(n)) = A342925(A350389(n)) = A003415(A000203(A350389(n))).

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)).

A368167 The largest unitary divisor of n that is a cubefull exponentially odd number (A335988).

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, 1, 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, Dec 14 2023

First differs from A056191 and A366126 at n = 32, and from A367513 at n = 64.
Also, the largest exponentially odd unitary divisor of the powerful part on n.
Also, the powerful part of the largest exponentially odd unitary divisor of n.

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

Cf. A057521, A077610, A335275, A335988, A350389, A368168, A368169.

Cf. A056191, A366126, A367513.

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

Multiplicative with a(p^e) = p^e if e is odd that is larger than 1, and 1 otherwise.
a(n) = A350389(A057521(n)).
a(n) = A057521(A350389(n)).
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.

A365402 The number of divisors of the largest unitary divisor of n that is an exponentially odd number.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 4, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 8, 1, 4, 4, 2, 2, 8, 2, 6, 4, 4, 4, 1, 2, 4, 4, 8, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 8, 4, 8, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 4, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4

Offset: 1

Amiram Eldar, Sep 03 2023

The sum of these divisors is A351569(n).

All the terms are either 1 or even (A004277).

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

Cf. A000005, A000290, A004277, A268335, A350389, A351569, A365401.

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

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

from math import prod
from sympy import factorint
def A365402(n): return prod(e+1 for e in factorint(n).values() if e&1) # Chai Wah Wu, Nov 17 2023

def a(n): return prod((valuation(n,p)+1) for p in prime_divisors(n) if valuation(n,p)%2==1) # Orges Leka, Nov 16 2023

a(n) = A000005(A350389(n)).
a(n) = A000005(n) / A365401(n).
a(n) <= A000005(n) with equality if and only if n is an exponentially odd number (A268335).
a(n) >= 1 with equality if and only if n is a square (A000290).
Multiplicative with a(p^e) = 1 if e is even, and e+1 if e is odd.
Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 2/p^s - 1/p^(2*s)).
From Vaclav Kotesovec, Sep 05 2023: (Start)
Dirichlet g.f.: zeta(s)^2 * zeta(2*s)^2 * Product_{p prime} (1 - 4/p^(2*s) + 4/p^(3*s) - 1/p^(4*s)).
Let f(s) = Product_{p prime} (1 - 4/p^(2*s) + 4/p^(3*s) - 1/p^(4*s)).
Sum_{k=1..n} a(k) ~ f(1) * Pi^4 * n / 36 * (log(n) + 2*gamma - 1 + 24*Zeta'(2)/Pi^2 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = 0.2177787166195363783230075141194468131307977550013559376482764035236264911...
f'(1) = f(1) * Sum_{p prime} 4*(2*p - 1) * log(p) / (1 - 3*p + p^2 + p^3) = f(1) * 3.3720882314412399056794495057358594564001229865925330149186567502684770675...
and gamma is the Euler-Mascheroni constant A001620. (End)
a(n) = Sum_{d|n} (-1)^(Sum_{p|gcd(d,n/d)} v_p(d)*v_p(n/d)), where v_p(x) denotes the valuation of x at the prime p. - Orges Leka, Nov 16 2023

A375032 The maximum odd exponent in the prime factorization of n, or 0 if no such exponent exists.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 28 2024

The asymptotic density of the occurrences of 0's is 0 (the asymptotic density of squares).
The asymptotic density of the occurrences of 1's is d(0) = Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465, asymptotic density of A335275).
The asymptotic density of the occurrences of 2*k+1, for k = 1, 2, ..., is d(k) = Product_{p prime} (1 - 1/(p^(2*k+2)*(p+1))) - Product_{p prime} (1 - 1/(p^(2*k)*(p+1))).

Cf. A000290, A051903, A065465, A162642, A335275, A350389, A375033.

a[n_] := Max[0, Max[Select[FactorInteger[n][[;; , 2]], OddQ]]]; 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), A375033(n)) = A051903(n).
a(n) = 0 if and only if n is a square (A000290).
a(n) = 1 if and only if n is in A335275 \ A000290.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} (2*k+1) * d(k) = 1.30000522546018852138..., where d(k) is defined in the Comments section above.
a(n) = A051903(A350389(n)). - Amiram Eldar, Aug 17 2024

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)).

A377821 Powerful numbers that have no more than one odd exponent in their prime factorization.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 900, 961, 968, 972, 1024, 1089, 1125, 1152

Offset: 1

Amiram Eldar, Nov 09 2024

Powerful numbers k such that A350389(k) is either 1 or a prime power with an odd exponent (A246551).

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

Disjoint union of A000290 \ {0} and A377820.

Cf. A013661, A085541, A246551, A350389.

With[{max = 1200}, Select[Union@ Flatten@Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], # == 1 || Count[FactorInteger[#][[;; , 2]], _?OddQ] <= 1 &]]

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

Sum_{n>=1} 1/a(n) = zeta(2) * (1 + P(3)) = A013661 * (1 + A085541) = 1.93240708584418977513... .

A368470 a(n) is the number of exponentially odd divisors of the largest unitary divisor of n that is an exponentially odd number (A268335).

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 3, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 6, 1, 4, 3, 2, 2, 8, 2, 4, 4, 4, 4, 1, 2, 4, 4, 6, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 6, 4, 6, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 3, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4

Offset: 1

Amiram Eldar, Dec 26 2023

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

Cf. A000005, A000290, A005117, A033634, A268335, A350389, A368471.

f[p_, e_] := If[OddQ[e], (e + 3)/2, 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,2]+3)/2, 1));}

a(n) = A033634(A350389(n)).
Multiplicative with a(p^e) = (e+3)/2 if e is odd and 1 otherwise.
a(n) >= 1, with equality if and only if n is a square (A000290).
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 2/p^s - 1/p^(2*s) - 1/p^(3*s)).
From Vaclav Kotesovec, Dec 26 2023: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + p^s/(1 + p^s)^2).
Let f(s) = Product_{p prime} (1 - 1/p^s + p^s/(1 + p^s)^2).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - (2*p+1) / (p*(p+1)^2)) = 0.528940778823659679133966695786017426052491935740673837882972347697...,
f'(1) = f(1) * Sum_{p prime} (4*p^2 + 3*p + 1) * log(p) / (p^4 + 3*p^3 + p^2 - 2*p - 1) = f(1) * 1.36109933267802415215189866467122940932493907539386280428818...
and gamma is the Euler-Mascheroni constant A001620. (End)

cp's OEIS Frontend

A350387 a(n) is the sum of the odd exponents in the prime factorization of n.

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A351571 Arithmetic derivative of the sum of the divisors of the largest unitary divisor of n that is an exponentially odd number.

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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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A368167 The largest unitary divisor of n that is a cubefull exponentially odd number (A335988).

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A365402 The number of divisors of the largest unitary divisor of n that is an exponentially odd number.

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A375032 The maximum odd exponent in the prime factorization of n, or 0 if no such exponent exists.

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