

A108548


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 jth prime in A000040.


14



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, 27, 28, 23, 30, 37, 32, 39, 34, 35, 36, 31, 38, 33, 40, 41, 42, 43, 52, 45, 58, 53, 48, 49, 50, 51, 44, 47, 54, 65, 56, 57, 46, 61, 60, 59, 74, 63, 64, 55, 78, 73, 68, 87, 70, 67, 72
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OFFSET

1,2


COMMENTS

Multiplicative with a(2^e) = 2^e, else if p is the mth prime then a(p^e) = q^e where q is the m/2th prime of the form 4*k + 3 (A002145) for even m and a(p^e) = r^e where r is the (m1)/2th prime of the form 4*k + 1 (A002144) for odd m.  David A. Corneth, Apr 25 2022
Permutation of the natural numbers with fixed points A108549: a(A108549(n)) = A108549(n).


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..26927
Index entries for sequences that are permutations of the natural numbers


MATHEMATICA

terms = 72;
A111745 = Module[{prs = Prime[Range[2 terms]], m3, m1, min},
m3 = Select[prs, Mod[#, 4] == 3&];
m1 = Select[prs, Mod[#, 4] == 1&];
min = Min[Length[m1], Length[m3]];
Riffle[Take[m3, min], Take[m1, min]]];
A108546[n_] := If[n == 1, 2, A111745[[n  1]]];
A049084[n_] := PrimePi[n]*Boole[PrimeQ[n]];
a[n_] := If[n == 1, 1, Module[{p, e}, Product[{p, e} = pe; A108546[A049084[p]]^e, {pe, FactorInteger[n]}]]];
Array[a, terms] (* JeanFrançois Alcover, Nov 19 2021, using Harvey P. Dale's code for A111745 *)


PROG

(PARI)
up_to = 26927; \\ One of the prime fixed points.
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[n2]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[n] = q); (v); };
v108546 = A108546list(up_to);
A108546(n) = v108546[n];
A108548(n) = { my(f=factor(n)); f[, 1] = apply(A108546, apply(primepi, f[, 1])); factorback(f); }; \\ Antti Karttunen, Apr 25 2022


CROSSREFS

Cf. A002144, A002145, A049084, A108546, A108549 (fixed points), A332808 (inverse permutation).
Cf. also A332815, A332817 (this permutation applied to Doudna tree and its mirror image), also A332818, A332819.
Cf. also A267099, A332212 and A348746 for other similar mappings.
Sequence in context: A249157 A033621 A332808 * A333692 A333693 A322464
Adjacent sequences: A108545 A108546 A108547 * A108549 A108550 A108551


KEYWORD

nonn,mult,look


AUTHOR

Reinhard Zumkeller, Jun 10 2005


EXTENSIONS

Name edited by Antti Karttunen, Apr 25 2022


STATUS

approved



