A367170 -id:A367170 - cp's OEIS frontend

A367168 The largest unitary divisor of n that is a term of A138302.

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 07 2023

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

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

Cf. A048298, A065465, A138302, A367169, A367170, A367171.
Cf. A056192, A270418.
Similar sequences: A350388, A350389, A360539.

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], 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] == 1 << valuation(f[i, 2], 2), f[i, 1]^f[i, 2], 1));}

from math import prod
from sympy import factorint
def A367168(n): return prod(p**e for p,e in factorint(n).items() if not(e&-e)^e) # Chai Wah Wu, Nov 10 2023

Multiplicative with a(p^e) = p^A048298(e).
a(n) <= n, with equality if and only if n is in A138302.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 0.881513... (A065465).

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.

A367171 The sum of divisors of the largest unitary divisor of n that is a term of A138302.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 1, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 4, 31, 42, 1, 56, 30, 72, 32, 1, 48, 54, 48, 91, 38, 60, 56, 6, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 3, 72, 8, 80, 90, 60, 168, 62, 96, 104, 1, 84, 144, 68, 126

Offset: 1

Amiram Eldar, Nov 07 2023

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

Cf. A000203, A048298, A138302, A306633, A367168, A367169, A367170.

Similar sequences: A351568, A351569.

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], (p^(e+1)-1)/(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] == 1 << valuation(f[i, 2], 2), (f[i, 1]^(f[i, 2]+1)-1)/(f[i, 1]-1), 1));}

Multiplicative with a(p^e) = (p^(A048298(e)+1)-1)/(p-1).
a(n) = A000203(A367168(n)).
a(n) <= A000203(n), with equality if and only if n is in A138302.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2)/zeta(3) = 1.368432... (A306633).

cp's OEIS Frontend

A367168 The largest unitary divisor of n that is a term of A138302.

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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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A367171 The sum of divisors of the largest unitary divisor of n that is a term of A138302.

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