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%I #37 Nov 26 2021 17:02:32
%S 1,2,3,25,26,33,93,1034,970225,8550146,325422273,414690595,1864797542,
%T 2438037206
%N Fixed points of A048673 and A064216: Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) = (2*n)-1, where p_k indicates the k-th prime, A000040(k).
%C Equally: after 1, numbers n such that, if the prime factorization of 2n-1 = Product_{k >= 1} (p_k)^(c_k) then Product_{k >= 1} (p_{k-1})^(c_k) = n.
%C Factorization of the initial terms: 1, 2, 3, 5^2, 2*13, 3*11, 3*31, 2*11*47, 5^2*197^2, 2*11*47*8269, 3*11*797*12373, 5*11^2*433*1583, 2*23*59*101*6803, 2*11*53*1201*1741.
%C The only 3-cycle of permutation A048673 in range 1 .. 402653184 is (2821 3460 5639).
%C For 2-cycles, take setwise difference of A245449 and this sequence.
%C Numbers k for which A336853(k) = k-1. - _Antti Karttunen_, Nov 26 2021
%e 25 is present, as 2*25 - 1 = 49 = p_4^2, and p_3^2 = 5*5 = 25.
%e 26 is present, as 2*26 - 1 = 51 = 3*17 = p_2 * p_8, and p_1 * p_7 = 2*13 = 26.
%e Alternatively, as 26 = 2*13 = p_1 * p_7, and ((p_2 * p_8)+1)/2 = ((3*17)+1)/2 = 26 also, thus 26 is present.
%p A048673 := n -> (A003961(n)+1)/2;
%p A048674list := proc(upto_n) local b,i; b := [ ]; for i from 1 to upto_n do if(A048673(i) = i) then b := [ op(b), i ]; fi; od: RETURN(b); end;
%t Join[{1}, Reap[For[n = 1, n < 10^7, n++, ff = FactorInteger[n]; If[Times @@ Power @@@ (NextPrime[ff[[All, 1]]]^ff[[All, 2]]) == 2 n - 1, Print[n]; Sow[n]]]][[2, 1]]] (* _Jean-François Alcover_, Mar 04 2016 *)
%o (PARI)
%o A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o isA048674(n) = ((n+n)==(1+A003961(n))); \\ _Antti Karttunen_, Nov 26 2021
%Y Fixed points of permutation pair A048673/A064216.
%Y Positions of zeros in A349573.
%Y Subsequence of the following sequences: A245449, A269860, A319630, A349622.
%Y This sequence is also obtained as a setwise difference of the following pairs of sequences: A246281 \ A246351, A246352 \ A246282, A246361 \ A246371, A246372 \ A246362.
%Y Cf. A000040, A003961 (A045965), A064989, A285701, A336853.
%Y Cf. also A348514 (fixed points of map A108228, similar to A048673).
%K nonn,hard,more
%O 1,2
%A _Antti Karttunen_, Jul 14 1999
%E Entry revised and the names in Maple-code cleaned by _Antti Karttunen_, Aug 25 2014
%E Terms a(11) - a(14) added by _Antti Karttunen_, Sep 11-13 2014
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