A295663 -id:A295663 - cp's OEIS frontend

A295664 Exponent of the highest power of 2 dividing number of divisors of n: a(n) = A007814(A000005(n)); 2-adic valuation of tau(n).

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

Offset: 1

Antti Karttunen, Nov 28 2017

In the prime factorization of n = p1^e1 * ... pk^ek, add together the number of trailing 1-bits in each exponent e when they are written in binary.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A007814, A028234, A056169, A067029, A077761, A162642, A295663.

Cf. A000290 (positions of zeros).

Table[IntegerExponent[DivisorSigma[0, n], 2], {n, 120}] (* Michael De Vlieger, Nov 28 2017 *)

a(n) = valuation(numdiv(n), 2); \\ Michel Marcus, Nov 30 2017

from sympy import divisor_count
def A295664(n): return (~(m:=int(divisor_count(n))) & m-1).bit_length() # Chai Wah Wu, Jul 05 2022

Additive with a(p^e) = A007814(1+e).
a(1) = 0; for n > 1, a(n) = A007814(1+A067029(n)) + a(A028234(n)).
a(n) = A007814(A000005(n)).
a(n) >= A162642(n) >= A056169(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} f(1/p) =-0.223720656976344505701..., where f(x) = -x + (1-x) * Sum_{k>=1} x^(2^k-1)/(1-x^(2^k)). - Amiram Eldar, Sep 28 2023

A295883 Number of exponents that are 3 in the prime factorization of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Antti Karttunen, Nov 29 2017

nonn

			For n = 8 = 2^3, there is one exponent that is exactly 3, thus a(8) = 1.
For n = 216 = 2^3 * 3^3 there are two exponents that are exactly 3, thus a(216) = 2.
For n = 432 = 2^4 * 3^3, there is one exponent that is exactly 3, thus a(432) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.

Cf. A085541, A085964, A295659, A295662, A295663, A295884.

Array[Total@ Map[Boole[# == 3] &, FactorInteger[#][[All, -1]]] &, 120] (* Michael De Vlieger, Nov 29 2017 *)
Count[FactorInteger[#][[All,2]],3]&/@Range[120] (* Harvey P. Dale, Apr 13 2019 *)

a(n) = vecsum(apply(x->(x==3), factor(n)[,2])); \\ Michel Marcus, Jul 25 2022

Additive with a(p^3) = 1, a(p^e) = 0 when e <> 3.
a(n) = A295659(n) - A295884(n).
a(n) <= A295662(n) <= A295663(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (1/p^3 - 1/p^4) = A085541 - A085964 = 0.0977694995... . - Amiram Eldar, Jul 25 2022

A295662 Number of odd exponents larger than one in the canonical prime factorization of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Antti Karttunen, Nov 28 2017

			For n = 24 = 2^3 * 3^1 there are two odd exponents, but only the other is larger than 1, thus a(24) = 1.
For n = 216 = 2^3 * 3^3 there are two odd exponents larger than 1, thus a(216) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.

Cf. A056169, A162642, A295659, A295663, A295664.

Cf. A295661 (positions of nonzero terms).

Array[Count[FactorInteger[#][[All, -1]], ?(And[OddQ@ #, # > 1] &)] &, 105] (* _Michael De Vlieger, Nov 28 2017 *)

a(n) = vecsum(apply(x -> x%2 - (x==1), factor(n)[, 2])); \\ Amiram Eldar, Sep 28 2023

Additive with a(p) = 0, a(p^e) = A000035(e) if e > 1.
a(1) = 0; and for n > 1, if A067029(n) = 1, a(n) = a(A028234(n)), otherwise A000035(A067029(n)) + a(A028234(n)).
a(n) = A162642(n) - A056169(n).
a(n) <= A295659(n).
a(n) = 0 iff A295663(n) = 0, and when A295663(n) > 0, a(n) <= A295663(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p^2*(p+1)) = 0.122017493776862257491... . - Amiram Eldar, Sep 28 2023

A295661 Numbers with at least one odd exponent larger than one in their prime factorization.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 432, 440, 456, 459, 472, 480, 486, 488, 500, 504, 512, 513, 520, 536, 540

Offset: 1

Antti Karttunen, Nov 28 2017

nonn

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.1184861602... (= 1 - A065465). - Amiram Eldar, May 18 2022

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positions of nonzero terms in A295662 and A295663.
Subsequence of A046099 (64 = 2^6, although a cube, is not in this sequence).
Differs from A060476 (256 = 2^8 is not a member of this sequence).
Complement of A335275.
Cf. A065465.

Select[Range[540], Count[FactorInteger[#][[All, -1]], ?(And[OddQ@ #, # > 1] &)] > 0 &] (* _Michael De Vlieger, Nov 28 2017 *)

from sympy import factorint
def ok(n):
    return max((e for e in factorint(n).values() if e%2), default=-1) > 1
print(list(filter(ok, range(541)))) # Michael S. Branicky, Aug 24 2021

cp's OEIS Frontend

A295664 Exponent of the highest power of 2 dividing number of divisors of n: a(n) = A007814(A000005(n)); 2-adic valuation of tau(n).

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A295883 Number of exponents that are 3 in the prime factorization of n.

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A295662 Number of odd exponents larger than one in the canonical prime factorization of n.

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PARI

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A295661 Numbers with at least one odd exponent larger than one in their prime factorization.

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Author

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Mathematica

Python