A286572 -id:A286572 - cp's OEIS frontend

A053574 Exponent of 2 in phi(n) where phi(n) = A000010(n).

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

Offset: 1

Labos Elemer, Jan 18 2000

			For n = 513 = 27*19, phi(513) = 4*81 so exponent of 2 is 2, thus a(513) = 2.

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

Cf. A000010, A007814, A053575, A286572.

IntegerExponent[Array[EulerPhi, 120], 2] (* Michael De Vlieger, Aug 16 2017 *)

vector(66,n,valuation(eulerphi(n),2)) \\ Joerg Arndt, Apr 22 2011

a(n) = A007814(A000010(n)).
A000010(n) = A053575(n) * 2^a(n). - Antti Karttunen, May 26 2017
Additive with a(2^e) = e-1, and a(p^e) = A007814(p-1) for an odd prime p. - Amiram Eldar, Sep 05 2023

Data section extended to 120 terms by Antti Karttunen, May 26 2017

A286568 Compound filter (phi(n) & 2-adic valuation of sigma(n)): a(n) = P(A000010(n), A286357(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 1, 8, 3, 14, 8, 42, 10, 21, 14, 76, 19, 90, 42, 63, 36, 152, 21, 208, 44, 148, 76, 322, 53, 210, 90, 228, 117, 434, 63, 625, 136, 296, 152, 402, 78, 702, 208, 375, 152, 860, 148, 988, 251, 324, 322, 1271, 169, 903, 210, 627, 324, 1430, 228, 943, 375, 816, 434, 1828, 187, 1890, 625, 777, 528, 1273, 296, 2344, 560, 1220, 402, 2698, 300, 2700, 702, 901

Offset: 1

Antti Karttunen, May 26 2017

nonn

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

Cf. A000010, A000027, A286154, A286160, A286357, A286451, A286572.

A000010(n) = eulerphi(n);
A001511(n) = (1+valuation(n,2));
A286357(n) = A001511(sigma(n));
A286568(n) = (1/2)*(2 + ((A000010(n)+A286357(n))^2) - A000010(n) - 3*A286357(n));

from sympy import divisor_sigma as D, totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a001511(n): return bin(n)[2:][::-1].index("1") + 1
def a286357(n): return a001511(D(n))
def a(n): return T(totient(n), a286357(n)) # Indranil Ghosh, May 26 2017

(define (A286568 n) (* (/ 1 2) (+ (expt (+ (A000010 n) (A286357 n)) 2) (- (A000010 n)) (- (* 3 (A286357 n))) 2)))

a(n) = (1/2)*(2 + ((A000010(n)+A286357(n))^2) - A000010(n) - 3*A286357(n)).

cp's OEIS Frontend

A053574 Exponent of 2 in phi(n) where phi(n) = A000010(n).

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