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A153212 A permutation of the natural numbers: in the prime factorization of n, swap each prime's index difference (from the previous distinct prime that divides n) and the prime's power.

%I A153212 #26 Dec 18 2014 02:19:53
%S A153212 1,2,4,3,8,6,16,5,9,18,32,15,64,54,12,7,128,10,256,75,36,162,512,35,
%T A153212 27,486,25,375,1024,30,2048,11,108,1458,24,21,4096,4374,324,245,8192,
%U A153212 150,16384,1875,45,13122,32768,77,81,50,972,9375,65536,14,72,1715,2916,39366,131072,105,262144,118098,225,13
%N A153212 A permutation of the natural numbers: in the prime factorization of n, swap each prime's index difference (from the previous distinct prime that divides n) and the prime's power.
%C A153212 In order for the "index difference" to make sense, we consider the factorization to be sorted with respect to the primes but not the powers to which they are raised; that is, first comes the smallest prime and each subsequent prime is larger than the previous disregarding their powers.
%C A153212 For every n it is true that a(a(n)) = n.
%C A153212 From _Antti Karttunen_, May 29 2014: (Start)
%C A153212 In other words, this is a self-inverse permutation (involution) of natural numbers.
%C A153212 This permutation maps primes (A000040) to the powers of two larger than one (A000079(n>=1)) and vice versa.
%C A153212 The term a(1) = 1 was added on the grounds that as 1 has an empty prime factorization, there is nothing to swap, thus it stays same. It is also needed as a base case for the given recurrence.
%C A153212 Considered as a function on partitions encoded by the indices of primes in the prime factorization of n (as in table A112798), this implements an operation which exchanges the horizontal and vertical line segment of each "step" in Young (or Ferrers) diagram of a partition. Please see the last example in the example section.
%C A153212 (End)
%H A153212 Antti Karttunen, <a href="/A153212/b153212.txt">Table of n, a(n) for n = 1..1024</a>
%H A153212 Wikipedia, <a href="http://en.wikipedia.org/wiki/Young_diagram">Young diagram</a>
%H A153212 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A153212 Denote the i-th prime with p(i): p(1)=2, p(2)=3, p(3)=5, p(4)=7, etc. Let n = p(a1)^b1 * p(a2)^b2 * ... * p(ak)^bk is the factorization of n where p(i)^j is the i-th prime raised to power j. As mentioned above, we assume that the primes are sorted, i.e., a1 < a2 < a3 ... < ak. Then a(n) = p(c1)^d1 * p(c2)^d2 * ... * p(ck)^dk where c1 = b1 and c(i) = b(i) + c(i-1) for i > 1 d1 = a1 and d(i) = a(i) - a(i-1) for i > 1.
%F A153212 From _Antti Karttunen_, May 16 2014: (Start)
%F A153212 a(1) = 1 and for n>1, let r = a(A051119(n)). Then a(n) = r * (A000040(A061395(r)+A071178(n)) ^ A241919(n)).
%F A153212 a(n) = A122111(A242419(n)) = A242419(A122111(n)).
%F A153212 (End)
%e A153212 For n = 10 we have 10 = 2^1 * 5^1 = p(1)^1 * p(3)^1 then a(10) = p(1)^1 * p(2)^2 = 2^1 * 3^2 = 18.
%e A153212 For n = 18 we have 18 = 2^1 * 3^2 = p(1)^1 * p(2)^2 then a(18) = p(1)^1 * p(3)^1 = 2^1 * 5^1 = 10.
%e A153212 For n = 19 we have 19 = 19^1 = p(8)^1 then a(19) = p(1)^8 = 2^8 = 256.
%e A153212 For n = 2200, we see that it encodes the partition (1,1,1,3,3,5) in A112798 as 2200 = p_1 * p_1 * p_1 * p_3 * p_3 * p_5 = 2^3 * 5^2 * 11. This in turn corresponds to the following Young diagram in French notation:
%e A153212    _
%e A153212   | |
%e A153212   | |
%e A153212   | |_ _
%e A153212   |     |
%e A153212   |     |_ _
%e A153212   |_ _ _ _ _|
%e A153212 Exchanging the order of the horizontal and vertical line segment of each "step", results the following Young diagram:
%e A153212    _ _ _
%e A153212   |     |_ _
%e A153212   |         |
%e A153212   |         |_
%e A153212   |           |
%e A153212   |_ _ _ _ _ _|
%e A153212 which represents the partition (3,5,5,6,6), encoded in A112798 by p_3 * p_5^2 * p_6^2 = 5 * 11^2 * 13^2 = 102245, thus a(2200) = 102245.
%o A153212 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A153212 (definec (A153212 n) (if (<= n 1) n (let ((r (A153212 (A051119 n)))) (* r (expt (A000040 (+ (A061395 r) (A071178 n))) (A241919 n))))))
%o A153212 (PARI) a(n) = {my(f = factor(n)); my(g = f); for (i=1, #f~, if (i==1, g[i,1] = prime(f[i,2]), g[i,1] = prime(f[i,2]+ primepi(g[i-1,1]))); if (i==1, g[i,2] = primepi(f[i,1]), g[i,2] = primepi(f[i,1]) - primepi(f[i-1,1]));); factorback(g);} \\ _Michel Marcus_, Dec 16 2014
%Y A153212 Cf. A000040, A051119, A061395, A071178.
%Y A153212 Fixed points: A242421.
%Y A153212 {A000027, A122111, A153212, A242419} form a 4-group.
%Y A153212 Cf. also A112798, A069799, A242415, A241909, A241916.
%K A153212 nonn
%O A153212 1,2
%A A153212 Luchezar Belev (l_belev(AT)yahoo.com), Dec 20 2008
%E A153212 Term a(1)=1 prepended, and also more terms computed by _Antti Karttunen_, May 16 2014