A046645 - OEIS

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A046645

a(n) = log_2(A046644(n)); also the 2-adic valuation of A046644(n).

0, 1, 1, 3, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 2, 7, 1, 4, 1, 4, 2, 2, 1, 5, 3, 2, 4, 4, 1, 3, 1, 8, 2, 2, 2, 6, 1, 2, 2, 5, 1, 3, 1, 4, 4, 2, 1, 8, 3, 4, 2, 4, 1, 5, 2, 5, 2, 2, 1, 5, 1, 2, 4, 10, 2, 3, 1, 4, 2, 3, 1, 7, 1, 2, 4, 4, 2, 3, 1, 8, 7, 2, 1, 5, 2, 2, 2, 5, 1, 5, 2, 4, 2

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OFFSET

1,4

COMMENTS

A268375 gives numbers n for which a(n) = A289617(n) = A005187(A001222(n)). - Antti Karttunen, Jul 08 2017

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65536

Index entries for sequences computed from exponents in factorization of n.

FORMULA

a(n) = A007814(A046644(n)). - Michel Marcus, Apr 16 2015

Additive with a(p^n) = A005187(n). - Antti Karttunen, Jul 08 2017

a(n) = A293447(A293442(n)). - Antti Karttunen, Nov 10 2017

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) = 1.410258867603361890498..., where f(x) = -x + Sum_{k>=0} (2^(k+1)-1)*x^(2^k)/(1+x^(2^k)). - Amiram Eldar, Sep 29 2023

MATHEMATICA

f[p_, e_] := 2*e - DigitCount[2*e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

PROG

(PARI)

A007814(n) = (valuation(n, 2));

A046643perA046644(n) = { my(c=1); if(1==n, c, fordiv(n, d, if((d>1)&&(d<n), c -= (A046643perA046644(d)*A046643perA046644(n/d)))); (c/2)); } \\ After the Maple-program given in A046643.

A046645(n) = A007814(denominator(A046643perA046644(n))); \\ Antti Karttunen, Jul 08 2017

(PARI)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };

A046645(n) = vecsum(apply(e -> A005187(e), factorint(n)[, 2])); \\ A faster implementation. - Antti Karttunen, Jul 08 2017

CROSSREFS

See A046643, A046644 for more details.

Cf. A001222, A005187, A007814, A077761, A268375, A289617, A289618, A293442, A293447.

Sequence in context: A257980 A203531 A324885 * A284639 A320887 A295923

Adjacent sequences: A046642 A046643 A046644 * A046646 A046647 A046648

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved