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Let b(n,p(k)) be the exponent raising the k-th prime in the prime-factorization of n. (b(n,p(k)) may equal 0.) Write each b down in order and in binary (and exponent of zero is written as '0') with b(n,2) on the right and b(n,P) on the left, where P is the largest prime dividing n. Concatenate to form one binary number. a(n) is the decimal equivalent.
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%I #10 Mar 04 2019 00:19:19

%S 0,1,2,2,4,3,8,3,4,5,16,6,32,9,6,4,64,5,128,10,10,17,256,7,8,33,6,18,

%T 512,7,1024,5,18,65,12,10,2048,129,34,11,4096,11,8192,34,12,257,16384,

%U 12,16,9,66,66,32768,7,20,19,130,513,65536,14,131072,1025,20,6,36,19

%N Let b(n,p(k)) be the exponent raising the k-th prime in the prime-factorization of n. (b(n,p(k)) may equal 0.) Write each b down in order and in binary (and exponent of zero is written as '0') with b(n,2) on the right and b(n,P) on the left, where P is the largest prime dividing n. Concatenate to form one binary number. a(n) is the decimal equivalent.

%C If p(m) is the m-th prime, then a(p(m)) = 2^(m-1).

%e 1125 has the prime-factorization, with the power of 2 on the right and power of the largest prime on the left, of 5^3 * 3^4 * 2^0. Writing down the exponents in base 2, we have 11, 100, 0. Concatenating, we have 111000, which in decimal is 56. So a(1125) = 56.

%p Lton := proc(L) add(op(i,L)*2^(i-1),i=1..nops(L)) ; end: A162474 := proc(n) local nred,L,p,e ; nred := n ; L := [] ; p := 2 ; while nred > 1 do e := 0 ; while nred mod p = 0 do e := e+1 ; nred := nred/p ; od: if e = 0 then L := [op(L), 0 ] ; else L := [op(L), op(convert(e,base,2)) ] ; fi; p := nextprime(p) ; od: Lton(L) ; end: seq(A162474(n),n=1..100) ; # _R. J. Mathar_, Jul 30 2009

%K base,nonn

%O 1,3

%A _Leroy Quet_, Jul 04 2009

%E More terms from _R. J. Mathar_, Jul 30 2009