cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A108546 Lexicographically earliest permutation of primes such that for n>1 forms 4*k+1 and 4*k+3 alternate.

Original entry on oeis.org

2, 3, 5, 7, 13, 11, 17, 19, 29, 23, 37, 31, 41, 43, 53, 47, 61, 59, 73, 67, 89, 71, 97, 79, 101, 83, 109, 103, 113, 107, 137, 127, 149, 131, 157, 139, 173, 151, 181, 163, 193, 167, 197, 179, 229, 191, 233, 199, 241, 211, 257, 223, 269, 227, 277, 239, 281, 251, 293
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 10 2005

Keywords

Links

Crossrefs

Cf. A000040, A002144, A002145, A102261, A108547 (fixed points), A108548, A111745, A332806 (inverse), A332807.
Cf. also A267101, A332211.

Programs

  • Haskell
    import Data.List (transpose)
a108546 n = a108546_list !! (n-1)
a108546_list =  2 : concat
   (transpose [a002145_list, a002144_list])
-- Reinhard Zumkeller, Nov 13 2014, Feb 22 2011
  • Mathematica
    terms = 60; A111745 = Module[{prs = Prime[Range[2terms]], 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]]]; a[1] = 2; a[n_] := A111745[[n-1]]; Table[a[n], {n, 1, terms}] (* Jean-François Alcover, May 18 2017, using Harvey P. Dale's code for A111745 *)
  • PARI
    up_to = 10000;
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); };
v108546 = A108546list(up_to);
A108546(n) = v108546[n]; \\ Antti Karttunen, Feb 27 2020

Formula

a(n) mod 4 = 3 - 2 * (n mod 2) for n>1.
For n > 1: a(n) = A111745(n-1).
a(2*n+1) - a(2*n) = A102261(n).
From Antti Karttunen, Feb 27 2020: (Start)
a(1) = 2, a(2n) = A002145(n), a(2n+1) = A002144(n).
a(n) = A000040(A332807(n)).
(End)

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 j-th prime in A000040.

Original entry on oeis.org

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
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 10 2005

Keywords

Comments

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

Links

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.

Programs

  • 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] (* Jean-François Alcover, Nov 19 2021, using Harvey P. Dale's code for A111745 *)
  • 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[n-2]; 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

Extensions

Name edited by Antti Karttunen, Apr 25 2022

A339900 Lexicographically earliest permutation of odd primes such that A007814(a(n)-1) = 1+A007814(n), where A007814 gives the 2-adic valuation of n.

Original entry on oeis.org

3, 5, 7, 41, 11, 13, 19, 17, 23, 29, 31, 73, 43, 37, 47, 97, 59, 53, 67, 89, 71, 61, 79, 113, 83, 101, 103, 137, 107, 109, 127, 193, 131, 149, 139, 233, 151, 157, 163, 241, 167, 173, 179, 281, 191, 181, 199, 353, 211, 197, 223, 313, 227, 229, 239, 337, 251, 269, 263, 409, 271, 277, 283, 641, 307, 293, 311, 457, 331
Offset: 1

Views

Author

Antti Karttunen, Dec 25 2020

Keywords

Links

Crossrefs

Cf. A001511, A007814, A065091.
Cf. A002145 (odd bisection), A007521 (quadrisection starting from 5), A105126, A105127, A105128, A105129, A105130, A105131, A105132.
Cf. also A108546, A111745.

Programs

  • PARI
    A339900(n) = { my(lev=1+valuation(n,2), k=(1+(n>>(lev-1)))/2); forprime(p=3,,if(valuation(p-1,2)==lev, k--; if(!k, return(p)))); };
Showing 1-3 of 3 results.

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