A350387 -id:A350387 - cp's OEIS frontend

A350389 a(n) is the largest unitary divisor of n that is an exponentially odd number (A268335).

1, 2, 3, 1, 5, 6, 7, 8, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 24, 1, 26, 27, 7, 29, 30, 31, 32, 33, 34, 35, 1, 37, 38, 39, 40, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 54, 55, 56, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71

Offset: 1

Amiram Eldar, Dec 28 2021

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

Cf. A000290, A001222, A007913, A268335, A350387, A350388, A350390.

f[p_, e_] := If[OddQ[e], 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]%2, f[i,1]^f[i,2], 1));} \\ Amiram Eldar, Sep 18 2023

from math import prod
from sympy import factorint
def A350389(n): return prod(p**e if e % 2 else 1 for p, e in factorint(n).items()) # Chai Wah Wu, Feb 24 2022

Multiplicative with a(p^e) = p^e if e is odd and 1 otherwise.
a(n) = n/A350388(n).
A001222(a(n)) = A350387(n).
a(n) = 1 if and only if n is a positive square (A000290 \ {0}).
a(n) = n if and only if n is an exponentially odd number (A268335).
Sum_{k=1..n} a(k) ~ (1/2)*c*n^2, where c = Product_{p prime} (1 - p/(1+p+p^2+p^3)) = 0.7406196365...
Dirichlet g.f.: zeta(2*s-2) * zeta(2*s) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(3*s-1)). - Amiram Eldar, Sep 18 2023

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

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, 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...

A367169 a(n) is the sum of the exponents in the prime factorization of n that are powers of 2.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Nov 07 2023

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

Cf. A001222, A048298, A077761, A138302, A367168, A367170, A367171.

Similar sequences: A350386, A350387.

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], e, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); sum(i = 1, #f~, if(f[i, 2] == 1 << valuation(f[i, 2], 2), f[i, 2], 0));}

from sympy import factorint
def A367169(n): return sum(e for e in factorint(n).values() if not(e&-e)^e) # Chai Wah Wu, Nov 10 2023

a(n) = A001222(A367168(n)).
Additive with a(p^e) = A048298(e).
a(n) <= A001222(n), with equality if and only if n is in A138302.
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = -P(2) + Sum_{k>=1} 2^k * (P(2^k) - P(2^k+1)) = 0.28425245481079272416..., where P(s) is the prime zeta function.

A332423 If n = Product (p_j^k_j) then a(n) = Sum ((-1)^(k_j + 1) * k_j).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Feb 12 2020

Sum of odd exponents in prime factorization of n minus the sum of even exponents in prime factorization of n.

			a(2700) = a(2^2 * 3^3 * 5^2) = -2 + 3 - 2 = -1.

Cf. A001222, A037916, A067255, A071321, A077761, A195017, A316523, A319273, A332422, A350386, A350387.

a[n_] := Plus @@ ((-1)^(#[[2]] + 1) #[[2]] & /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 1, 80}]

a(n) = vecsum(apply(x -> (-1)^(x+1) * x, factor(n)[, 2])); \\ Amiram Eldar, Oct 09 2023

From Amiram Eldar, Oct 09 2023: (Start)
Additive with a(p^e) = (-1)^(e+1) * e.
a(n) = A350387(n) - A350386(n).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B - C), where B is Mertens's constant (A077761) and C = Sum_{p prime} (3*p+1)/(p*(p+1)^2) = 0.81918453457738985491 ... . (End)

A356413 Numbers with an equal sum of the even and odd exponents in their prime factorizations.

Original entry on oeis.org

1, 60, 84, 90, 126, 132, 140, 150, 156, 198, 204, 220, 228, 234, 260, 276, 294, 306, 308, 315, 340, 342, 348, 350, 364, 372, 380, 414, 444, 460, 476, 490, 492, 495, 516, 522, 525, 532, 550, 558, 564, 572, 580, 585, 620, 636, 644, 650, 666, 693, 708, 726, 732, 735

Offset: 1

Amiram Eldar, Aug 06 2022

nonn

Numbers k such that A350386(k) = A350387(k).

A085987 is a subsequence. Terms that are not in A085987 are 1, 2160, 3024, ...

			60 is a term since A350386(60) = A350387(60) = 2.

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

Cf. A350386, A350387.
Subsequence of A028260.
Subsequences: A085987, A179698, A190109, A190110.
Similar sequences: A048109, A187039, A348097.

f[p_, e_] := (-1)^e*e; q[1] = True; q[n_] := Plus @@ f @@@ FactorInteger[n] == 0; Select[Range[1000], q]

isok(n) = {my(f = factor(n)); sum(i = 1, #f~, (-1)^f[i,2]*f[i,2]) == 0};

A370079 The product of the odd exponents of the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 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, 1, 1, 1, 1, 1, 1, 3, 1, 3, 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, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Feb 09 2024

First differs from A363329 at n = 32.

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

Cf. A005361, A013661, A268335, A335275, A350387, A350389, A363329, A368472, A370080.

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

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

a(n) = A005361(A350389(n)).
Multiplicative with a(p^e) = e if e is odd, and 1 if e is even.
a(n) = A005361(n)/A370080(n).
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= A005361(n), 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^(2*s) + 1/p^(3*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 1/p^4) = 1.32800597172596287374... .
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 2/((p^s - 1)*(p^s + 1)^2)). - Vaclav Kotesovec, Feb 11 2024

A371600 Numbers of least prime signature (A025487) whose prime factorization has equal sum of even and odd exponents.

Original entry on oeis.org

1, 60, 2160, 12600, 18480, 77760, 180180, 216000, 453600, 665280, 2646000, 2799360, 3880800, 7776000, 10810800, 16329600, 16336320, 23950080, 32016600, 45360000, 66528000, 95256000, 100776960, 139708800, 214414200, 232792560, 279936000, 389188800, 555660000, 587865600

Offset: 1

Amiram Eldar, Mar 29 2024

nonn

			The prime signatures of the first 12 terms are:
   n     a(n)     signature  A350386(a(n)) = A350387(a(n))
  --  -------  ------------  -------------   -------------
   1        1            {}             0                0
   2       60       {1,1,2}             2            1+1=2
   3     2160       {1,3,4}             4            1+3=4
   4    12600     {1,2,2,3}         2+2=4            1+3=4
   5    18480   {1,1,1,1,4}             4        1+1+1+1=4
   6    77760       {1,5,6}             6            1+5=6
   7   180180 {1,1,1,1,2,2}         2+2=4        1+1+1+1=4
   8   216000       {3,3,6}             6            3+3=6
   9   453600     {1,2,4,5}         2+4=6            1+5=6
  10   665280   {1,1,1,3,6}             6        1+1+1+3=6
  11  2646000     {2,3,3,4}         2+4=6            3+3=6
  12  2799360       {1,7,8}             8            1+7=8

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

Intersection of A025487 and A356413.

Cf. A350386, A350387, A371599.

fun[p_, e_] := (-1)^e * e; q[n_] := Module[{f = FactorInteger[n]}, n == 1 || (f[[-1, 1]] == Prime[Length[f]] && Plus @@ fun @@@ f == 0 && Max@ Differences[f[[;; , 2]]] < 1)]; Select[Range[4*10^6], q]

is(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); n == 1 || (sum(i = 1, #e, (-1)^e[i] * e[i]) == 0 && e == vecsort(e, , 4) && primepi(p[#p]) == #p);}

A356414 Number k such that k and k+1 both have an equal sum of even and odd exponents in their prime factorization (A356413).

Original entry on oeis.org

819, 1035, 1196, 1274, 1275, 1449, 1665, 1924, 1925, 1988, 2324, 2331, 2540, 3068, 3195, 3324, 3339, 3549, 3555, 3626, 3717, 4164, 4220, 4235, 4556, 4598, 4635, 4675, 4796, 5084, 5525, 5634, 5660, 6003, 6027, 6068, 6164, 6363, 6740, 6867, 6908, 7028, 7227, 7275

Offset: 1

Amiram Eldar, Aug 06 2022

nonn

Numbers k such that A350386(k) = A350387(k) and A350386(k+1) = A350387(k+1).

			819 is a term since A350386(819) = A350387(819) = 2 and A350386(820) = A350387(820) = 2.

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

Cf. A350386, A350387, A356413.

f[p_, e_] := (-1)^e*e; q[1] = True; q[n_] := Plus @@ f @@@ FactorInteger[n] == 0; Select[Range[10^4], q[#] && q[# + 1] &]

is(n) = {my(f = factor(n)); sum(i = 1, #f~, (-1)^f[i,2]*f[i,2]) == 0};
is1 = is(1); for(k = 2, 10^4, is2 = is(k); if(is1 && is2, print1(k-1,", ")); is1 = is2);

cp's OEIS Frontend

A350389 a(n) is the largest unitary divisor of n that is an exponentially odd number (A268335).

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

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A367169 a(n) is the sum of the exponents in the prime factorization of n that are powers of 2.

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A332423 If n = Product (p_j^k_j) then a(n) = Sum ((-1)^(k_j + 1) * k_j).

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A356413 Numbers with an equal sum of the even and odd exponents in their prime factorizations.

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A370079 The product of the odd exponents of the prime factorization of n.

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A371600 Numbers of least prime signature (A025487) whose prime factorization has equal sum of even and odd exponents.

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A356414 Number k such that k and k+1 both have an equal sum of even and odd exponents in their prime factorization (A356413).