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A048675 If n = p_i^e_i * ... * p_k^e_k, p_i < ... < p_k primes (with p_i = prime(i)), then a(n) = (1/2) * (e_i * 2^i + ... + e_k * 2^k). 160
0, 1, 2, 2, 4, 3, 8, 3, 4, 5, 16, 4, 32, 9, 6, 4, 64, 5, 128, 6, 10, 17, 256, 5, 8, 33, 6, 10, 512, 7, 1024, 5, 18, 65, 12, 6, 2048, 129, 34, 7, 4096, 11, 8192, 18, 8, 257, 16384, 6, 16, 9, 66, 34, 32768, 7, 20, 11, 130, 513, 65536, 8, 131072, 1025, 12, 6, 36, 19 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The original motivation for this sequence was to encode the prime factorization of n in the binary representation of a(n), each such representation being unique as long as this map is restricted to A005117 (squarefree numbers, resulting a permutation of nonnegative integers A048672) or any of its subsequence, resulting an injective function like A048623 and A048639.

However, also the restriction to A260443 (not all terms of which are squarefree) results a permutation of nonnegative integers, namely A001477, the identity permutation.

When a polynomial with nonnegative integer coefficients is encoded with the prime factorization of n (e.g., as in A206296, A260443), then a(n) gives the evaluation of that polynomial at x=2.

The primitive completely additive integer sequence that satisfies a(n) = a(A225546(n)), n >= 1. By primitive, we mean that if b is another such sequence, then there is an integer k such that b(n) = k * a(n) for all n >= 1. - Peter Munn, Feb 03 2020

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..1024

Hans Havermann, Factorization of the first 10000 terms, in format [[primes], [exponents]]

FORMULA

a(1) = 0, a(n) = 1/2 * (e1*2^i1 + e2*2^i2 + ... + ez*2^iz) if n = p_{i1}^e1*p_{i2}^e2*...*p_{iz}^ez, where p_i is the i-th prime. (e.g. p_1 = 2, p_2 = 3).

Totally additive with a(p^e) = e * 2^(PrimePi(p)-1), where PrimePi(n) = A000720(n). [Missing factor e added to the comment by Antti Karttunen, Jul 29 2015]

From Antti Karttunen, Jul 29 2015: (Start)

a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A032742(n)). [Where A055396(n) gives the index of the smallest prime dividing n and A032742(n) gives the largest proper divisor of n.]

a(1) = 0; for n > 1, a(n) = (A067029(n) * (2^(A055396(n)-1))) + a(A028234(n)).

Other identities. For all n >= 0:

a(A019565(n)) = n.

a(A260443(n)) = n.

a(A206296(n)) = A000129(n).

a(A005940(n+1)) = A087808(n).

a(A007913(n)) = A248663(n).

a(A007947(n)) = A087207(n).

a(A283477(n)) = A005187(n).

a(A284003(n)) = A006068(n).

a(A285101(n)) = A028362(1+n).

a(A285102(n)) = A068052(n).

Also, it seems that a(A163511(n)) = A135529(n) for n >= 1. (End)

a(1) = 0, a(2n) = 1+a(n), a(2n+1) = 2*a(A064989(2n+1)). - Antti Karttunen, Oct 11 2016

From Peter Munn, Jan 31 2020: (Start)

a(n^2) = a(A003961(n)) = 2 * a(n).

a(A297845(n,k)) = a(n) * a(k).

a(n) = a(A225546((n)).

a(A329332(n,k)) = n * k.

a(A329050(n,k)) = 2^(n+k).

(End)

From Antti Karttunen, Feb 02-25 2020: (Start)

a(n) = Sum_{d|n} A297108(d).

a(n) = a(A097248(n)).

For n >= 2:

A001221(a(n)) = A322812(n), A001222(a(n)) = A277892(n).

A000203(a(n)) = A324573(n), A033879(a(n)) = A324575(n).

For n >= 1, A331750(n) = a(A000203(n)).

For n >= 1, the following chains hold:

A293447(n) >= a(n) >= A331740(n) >= A331591(n).

a(n) >= A087207(n) >= A248663(n).

(End)

MAPLE

nthprime := proc(n) local i; if(isprime(n)) then for i from 1 to 1000000 do if(ithprime(i) = n) then RETURN(i); fi; od; else RETURN(0); fi; end; # nthprime(2) = 1, nthprime(3) = 2, nthprime(5) = 3, etc. - this is also A049084.

A048675 := proc(n) local s, d; s := 0; for d in ifactors(n)[ 2 ] do s := s + d[ 2 ]*(2^(nthprime(d[ 1 ])-1)); od; RETURN(s); end;

# simpler alternative

f:= n -> add(2^(numtheory:-pi(t[1])-1)*t[2], t=ifactors(n)[2]):

map(f, [$1..100]); # Robert Israel, Oct 10 2016

MATHEMATICA

a[1] = 0; a[n_] := Total[ #[[2]]*2^(PrimePi[#[[1]]]-1)& /@ FactorInteger[n] ]; Array[a, 100] (* Jean-François Alcover, Mar 15 2016 *)

PROG

(Scheme, with memoization-macro definec, two alternatives)

(definec (A048675 n) (cond ((= 1 n) (- n 1)) (else (+ (A000079 (- (A055396 n) 1)) (A048675 (A032742 n))))))

(definec (A048675 n) (cond ((= 1 n) (- n 1)) (else (+ (* (A067029 n) (A000079 (- (A055396 n) 1))) (A048675 (A028234 n))))))

;; Antti Karttunen, Jul 29 2015

(definec (A048675 n) (cond ((= 1 n) 0) ((even? n) (+ 1 (A048675 (/ n 2)))) (else (* 2 (A048675 (A064989 n)))))) ;; Third one, using the new recurrence. - Antti Karttunen, Oct 11 2016

(PARI) a(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ Michel Marcus, Oct 10 2016

(PARI)

\\ The following program reconstructs terms (e.g. for checking purposes) from the factorization file prepared by Hans Havermann:

v048675sigs = readvec("a048675.txt");

A048675(n) = if(n<=2, n-1, my(prsig=v048675sigs[n], ps=prsig[1], es=prsig[2]); prod(i=1, #ps, ps[i]^es[i])); \\ Antti Karttunen, Feb 02 2020

(Python)

from sympy import factorint, primepi

def a(n):

    if n==1: return 0

    f=factorint(n)

    return sum([f[i]*2**(primepi(i) - 1) for i in f])

print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jun 19 2017

CROSSREFS

Row 2 of A104244.

Similar logarithmic functions: A001414, A056239, A090880, A289506, A293447.

Left inverse of the following sequences: A000079, A019565, A038754, A068911, A134683, A260443, A332824.

A003961, A028234, A032742, A055396, A064989, A067029, A225546, A297845 are used to express relationship between terms of this sequence.

Cf. also A048623, A048676, A099884, A277896 and tables A277905, A285325.

Cf. A297108 (Möbius transform), A332813 and A332823 [= a(n) mod 3].

Pairs of sequences (f,g) that satisfy a(f(n)) = g(n), possibly with offset change: (A000203,A331750), (A005940,A087808), (A007913,A248663), (A007947,A087207), (A097248,A048675), (A206296,A000129), (A283477,A005187), (A284003,A006068), (A285101,A028362), (A285102,A068052), (A293214,A001065), (A318834,A051953), (A319991,A293897), (A319992,A293898), (A320017,A318674), (A329352,A069359), (A332461,A156552), (A332462,A156552), (A332825,A000010) and apparently (A163511,A135529).

See comments/formulas in A277333, A331591, A331740 giving their relationship to this sequence.

The formula section details how the sequence maps the terms of A329050, A329332.

A277892, A322812, A322869, A324573, A324575 give properties of the n-th term of this sequence.

Sequence in context: A324754 A174220 A334871 * A162474 A334878 A285330

Adjacent sequences:  A048672 A048673 A048674 * A048676 A048677 A048678

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jul 14 1999

EXTENSIONS

Entry revised by Antti Karttunen, Jul 29 2015

More linking formulas added by Antti Karttunen, Apr 18 2017

STATUS

approved

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Last modified July 9 01:30 EDT 2020. Contains 335537 sequences. (Running on oeis4.)