Search: id:A137502
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%I A137502
%S A137502 2,2,4,2,6,2,8,4,10,2,18,2,14,6,16,2,12,2,50,10,22,2,54,4,26,8,98,2,30,
%T A137502 2,32,14,34,6,36,2,38,22,250,2,70,2,242,18,46,2,162,4,4,26,338,2,24,10,
%U A137502 686,2,58,2,150,2,150,50,64,14,154,2,578,38,42,2,108,2,74,12,1058,6,506
%N A137502 Reverse sequence of powers in prime decomposition of n.
%C A137502 In the prime decomposition of n we use all the primes up to the highest 
               prime divisor, exponents of zero being allowed except for the largest 
               prime.
%C A137502 If n=(p(1)^e1)*(p(2)^e2)*.......*(p(k)^ek) (ek>0, other ei>=0 and p(n) 
               = n-th prime) then A137502 we reverse the sequence e1,e2,... ek to 
               build a(n): a(n) = (p(1)^ek)*(p(2)^e(k-1))* . . . . *(p(k)^e1)
%C A137502 As p(1)=2 and ek =/=0, a(n) is always even
%C A137502 If n is prime then a(n) =2 and if n is a power of prime, a(n) is the 
               same power of 2
%C A137502 If the sequence e1,e2,. . . . ek is palindromic, a(n)=n
%C A137502 For any given even number Q, we can by reversing the sequence of its 
               powers define not only one but an infinity (by adding as many zeros 
               as we want on the left end) of n such that a(n)=Q. Hence the sequence 
               is a permutation of even integers where each even integer is infinitely 
               repeated.
%C A137502 For example as Q=1224=(2^3)*(3^2)*(5^0)*(7^0)*(11^0)*(13^0)*(17^1),
%C A137502 Q =a((2^1)*(3^0)*(5^0)*(7^0)*(11^0)*(13^2)*(17^3))=a(1660594) but also 
               of an infinity of other ones, the first one being a((2^0)*(3^1)*(5^0)*(7^0)*(11^0)*(13^0)*(17^2)*(19^1))=\
               a(5946753)
%e A137502 n=9=(2^0)*(3^2), hence a(9) = (2^2)*(3^0)=4
%Y A137502 Sequence in context: A067824 A107067 A046801 this_sequence A143112 A167272 
               A090624
%Y A137502 Adjacent sequences: A137499 A137500 A137501 this_sequence A137503 A137504 
               A137505
%K A137502 base,easy,nonn
%O A137502 2,1
%A A137502 Philippe Lallouet (philip.lallouet(AT)orange.fr), Apr 22 2008
%E A137502 Edited by N. J. A. Sloane (njas(AT)research.att.com), Jan 16 2009

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