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A064547 Sum of binary digits (or count of 1-bits) in the exponents of the prime factorization of n. 15
0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 3, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

This sequence is different from A058061 for n containing 6th, 8th, ..., k-th powers in its prime decomposition, where k runs through the integers missing from A064548.

For n > 1, n is a product of a(n) distinct members of A050376. - Matthew Vandermast, Jul 13 2004

For n > 1: a(n) = length of n-th row in A213925. - Reinhard Zumkeller, Mar 20 2013

LINKS

Harry J. Smith (terms 1..2000) & Antti Karttunen, Table of n, a(n) for n = 1..32768

FORMULA

a(m*n) <= a(m)*a(n). - Reinhard Zumkeller, Mar 20 2013

From Antti Karttunen, Feb 09 2016: (Start)

a(1) = 0, and for n > 1, a(n) = A000120(A067029(n)) + a(A028234(n)).

a(1) = 0, and for n > 1, a(n) = A000120(A007814(n)) + a(A064989(n)).

(End)

a(n) = log_2(A037445(n)). - Vladimir Shevelev, May 13 2016

EXAMPLE

a(54)=3 since 54=2^1 * 3^3 with exponents (1) and (11) in binary.

MAPLE

expts:=proc(n) local t1, t2, t3, t4, i; if n=1 then RETURN([0]); fi; if isprime(n) then RETURN([1]); fi; t1:=ifactor(n); if nops(factorset(n))=1 then RETURN([op(2, t1)]); fi; t2:=nops(t1); t3:=[]; for i from 1 to t2 do t4:=op(i, t1); if nops(t4) = 1 then t3:=[op(t3), 1]; else t3:=[op(t3), op(2, t4)]; fi; od; RETURN(t3); end;

A000120 := proc(n) local w, m, i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end:

LamMos:= proc(n) local t1, t2, t3, i; t1:=expts(n); add( A000120(t1[i]), i=1..nops(t1)); end; # N. J. A. Sloane_, Dec 20 2007

A064547:= proc(n) local F;

F:= ifactors(n)[2];

add(convert(convert(f[2], base, 2), `+`), f=F)

end proc:

map(A064547, [$1..100]); # Robert Israel, May 17 2016

MATHEMATICA

Table[Plus@@(DigitCount[Last/@FactorInteger[k], 2, 1]), {k, 105}]

PROG

(PARI) SumD(x)= { local(s); s=0; while (x>9, s+=x-10*(x\10); x\=10); return(s + x) }

baseE(x, b)= { local(d, e, f); e=0; f=1; while (x>0, d=x-b*(x\b); x\=b; e+=d*f; f*=10); return(e) }

{ for (n=1, 2000, f=factor(n)~; a=0; for (i=1, length(f), a+=SumD(baseE(f[2, i], 2))); write("b064547.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 18 2009

(PARI) a(n) = {my(f = factor(n)[, 2]); sum(k=1, #f, hammingweight(f[k])); } \\ Michel Marcus, Feb 10 2016

(Haskell)

a064547 1 = 0

a064547 n = length $ a213925_row n  -- Reinhard Zumkeller, Mar 20 2013

(Scheme, two variants, both using memoizing-macro definec)

(definec (A064547 n) (cond ((= 1 n) 0) (else (+ (A000120 (A067029 n)) (A064547 (A028234 n))))))

(definec (A064547 n) (if (= 1 n) 0 (+ (A000120 (A007814 n)) (A064547 (A064989 n)))))

;; Antti Karttunen, Feb 09 2016

CROSSREFS

Cf. A000028 (positions of odd terms), A000379 (of even terms).

Cf. A050376 (positions of ones), A268388 (terms larger than ones).

Cf. A000120, A007814, A028234, A058061, A064989, A067029, A213925.

Cf. also A176699, A181819, A267116, A268387.

Sequence in context: A160980 A065031 A058061 * A214715 A244145 A086435

Adjacent sequences:  A064544 A064545 A064546 * A064548 A064549 A064550

KEYWORD

nonn,base

AUTHOR

Wouter Meeussen, Oct 09 2001

STATUS

approved

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Last modified March 27 22:02 EDT 2017. Contains 284182 sequences.