A108548 - OEIS

%I #30 Apr 26 2022 16:09:18

%S 1,2,3,4,5,6,7,8,9,10,13,12,11,14,15,16,17,18,19,20,21,26,29,24,25,22,

%T 27,28,23,30,37,32,39,34,35,36,31,38,33,40,41,42,43,52,45,58,53,48,49,

%U 50,51,44,47,54,65,56,57,46,61,60,59,74,63,64,55,78,73,68,87,70,67,72

%N Fully multiplicative with a(prime(j)) = A108546(j), where A108546 is the lexicographically earliest permutation of primes such that after 2 the forms 4*k+1 and 4*k+3 alternate, and prime(j) is the j-th prime in A000040.

%C Multiplicative with a(2^e) = 2^e, else if p is the m-th prime then a(p^e) = q^e where q is the m/2-th prime of the form 4*k + 3 (A002145) for even m and a(p^e) = r^e where r is the (m-1)/2-th prime of the form 4*k + 1 (A002144) for odd m. - _David A. Corneth_, Apr 25 2022

%C Permutation of the natural numbers with fixed points A108549: a(A108549(n)) = A108549(n).

%H Antti Karttunen, <a href="/A108548/b108548.txt">Table of n, a(n) for n = 1..26927</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%t terms = 72;

%t A111745 = Module[{prs = Prime[Range[2 terms]], m3, m1, min},

%t      m3 = Select[prs, Mod[#, 4] == 3&];

%t      m1 = Select[prs, Mod[#, 4] == 1&];

%t      min = Min[Length[m1], Length[m3]];

%t      Riffle[Take[m3, min], Take[m1, min]]];

%t A108546[n_] := If[n == 1, 2, A111745[[n - 1]]];

%t A049084[n_] := PrimePi[n]*Boole[PrimeQ[n]];

%t a[n_] := If[n == 1, 1, Module[{p, e}, Product[{p, e} = pe; A108546[A049084[p]]^e, {pe, FactorInteger[n]}]]];

%t Array[a, terms] (* _Jean-François Alcover_, Nov 19 2021, using _Harvey P. Dale_'s code for A111745 *)

%o (PARI)

%o up_to = 26927; \\ One of the prime fixed points.

%o A108546list(up_to) = { my(v=vector(up_to), p,q); v[1] = 2; v[2] = 3; v[3] = 5; for(n=4,up_to, p = v[n-2]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[n] = q); (v); };

%o v108546 = A108546list(up_to);

%o A108546(n) = v108546[n];

%o A108548(n) = { my(f=factor(n)); f[,1] = apply(A108546,apply(primepi,f[,1])); factorback(f); }; \\ _Antti Karttunen_, Apr 25 2022

%Y Cf. A002144, A002145, A049084, A108546, A108549 (fixed points), A332808 (inverse permutation).

%Y Cf. also A332815, A332817 (this permutation applied to Doudna tree and its mirror image), also A332818, A332819.

%Y Cf. also A267099, A332212 and A348746 for other similar mappings.

%K nonn,mult,look

%O 1,2

%A _Reinhard Zumkeller_, Jun 10 2005

%E Name edited by _Antti Karttunen_, Apr 25 2022