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A137502 Reverse sequence of powers in prime decomposition of n.

%I A137502 #22 Aug 27 2016 09:56:36
%S A137502 1,2,2,4,2,6,2,8,4,10,2,18,2,14,6,16,2,12,2,50,10,22,2,54,4,26,8,98,2,
%T A137502 30,2,32,14,34,6,36,2,38,22,250,2,70,2,242,18,46,2,162,4,20,26,338,2,
%U A137502 24,10,686,34,58,2,150,2,62,50,64,14,154,2,578,38,42,2,108,2,74,12,722,6,286
%N A137502 Reverse sequence of powers in prime decomposition of n.
%C A137502 The term a(1) = 1 added on the grounds that as 1 has an empty prime factorization, it stays same when reversed. - _Antti Karttunen_, May 20 2014
%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 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 is never zero for n>1, a(n) is always even for n>1.
%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 (That is, a(A000961(n)) = A000079(A025474(n)) for all n.) - _Antti Karttunen_, May 20 2014.
%C A137502 If the sequence e1, e2, ..., ek is palindromic, a(n)=n. (A242418 gives such 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).
%C A137502 Please see A241916 for a variant which results an ordinary permutation of all natural numbers. - _Antti Karttunen_, May 20 2014
%H A137502 Antti Karttunen, <a href="/A137502/b137502.txt">Table of n, a(n) for n = 1..10000</a>
%F A137502 a(1) = 1, and for n>1, a(n) = 2*A241916(n) / A006530(n). - _Antti Karttunen_, May 20 2014
%e A137502 As 9 = (2^0)*(3^2), hence a(9) = (2^2)*(3^0) = 4.
%e A137502 As 50 = (2^1)*(3^0)*(5^2), hence a(50) = (2^2)*(3^0)*(5^1) = 2*2*5 = 20.
%e A137502 As 57 = (2^0)*(3^1)*(5^0)*(7^0)*(11^0)*(13^0)*(17^0)*(19^1), hence a(57) = (2^1)*(3^0)*(5^0)*(7^0)*(11^0)*(13^0)*(17^1)*(19^0) = 2*17 = 34.
%t A137502 f[n_] := If[n == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ n]; g[w_List] := Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, w]; Table[g@ Reverse@ f@ n, {n, 120}] (* _Michael De Vlieger_, Aug 27 2016 *)
%o A137502 (Scheme) (define (A137502 n) (if (< n 2) n (/ (* 2 (A241916 n)) (A006530 n)))) ;; _Antti Karttunen_, May 20 2014
%Y A137502 A242418 gives the fixed points.
%Y A137502 Cf. A241916, A006530, A000961, A025474.
%K A137502 easy,nonn
%O A137502 1,2
%A A137502 Philippe Lallouet (philip.lallouet(AT)orange.fr), Apr 22 2008
%E A137502 Edited by _N. J. A. Sloane_, Jan 16 2009.
%E A137502 Term a(1)=1 prepended, and erroneous terms (first at n=50) corrected, _Antti Karttunen_, May 20 2014