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A245449 Fixed points of A245447 and A245448.

%I A245449 #30 Apr 01 2025 18:01:26
%S A245449 1,2,3,4,5,9,13,25,26,30,33,53,93,1023,1034,1203,1330,2657,8584,17159,
%T A245449 779212,970225,1558409,8550146,240902643,244608573,325422273,
%U A245449 414690595,570131490,1020233393,1864797542,2438037206
%N A245449 Fixed points of A245447 and A245448.
%C A245449 First apply A003961(n), where the primes in the prime factorization of natural number n are shifted one step left [i.e. each p_i changes to p_{i+1}]. Then increment the resulting odd number by one to get an even number, which is divided by 2, and the same three operations are done second time to that quotient. This sequence consists of such numbers for which the final result is equal to the original n which we started from.
%C A245449 8550146 is the largest term <= 123456789.
%C A245449 Numbers which are in 1- and 2-cycles of A048673 and A064216.
%e A245449 For n = 30 = 2*3*5 = p_1 * p_2 * p_3, the first shift operation results p_2 * p_3 * p_4 = 3*5*7 = 105, and (105+1)/2 = 53, which is the 16th prime, p_16. Shifting this once left results p_17 = 59, and (59+1)/2 = 30 again. Thus 30 is included in the sequence. For the same reason 53 is also included in the sequence.
%o A245449 (PARI)
%o A245449 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of _Michel Marcus_
%o A245449 A048673(n) = (A003961(n)+1)/2;
%o A245449 isA245449(n) = ((A048673(A048673(n)) == n))
%o A245449 i=0; for(n=1, 123456789, if(isA245449(n), i++; write("b245449.txt", i, " ", n)))
%o A245449 (Scheme)
%o A245449 ;; With _Antti Karttunen_'s IntSeq-library.
%o A245449 (define A245449 (FIXED-POINTS 1 1 A245447))
%Y A245449 A048674 is a subsequence.
%Y A245449 Cf. A048673, A064216, A245447, A245448.
%K A245449 nonn,more
%O A245449 1,2
%A A245449 _Antti Karttunen_, Jul 22 2014
%E A245449 a(25)-a(32) added by _Antti Karttunen_, Sep 13 2014