A072010 In prime factorization of n replace all primes of form k*4+1 with k*4+3 and primes of form k*4+3 with k*4+1.

1, 2, 1, 4, 7, 2, 5, 8, 1, 14, 9, 4, 15, 10, 7, 16, 19, 2, 17, 28, 5, 18, 21, 8, 49, 30, 1, 20, 31, 14, 29, 32, 9, 38, 35, 4, 39, 34, 15, 56, 43, 10, 41, 36, 7, 42, 45, 16, 25, 98, 19, 60, 55, 2, 63, 40, 17, 62, 57, 28, 63, 58, 5, 64, 105, 18, 65, 76, 21, 70, 69

Offset: 1

Author

Reinhard Zumkeller, Jun 05 2002

Keywords

• nonn
• mult
• nice

Comments

a(3^n) = 1; a(2^n) = 2^n;

a(n)>2 is prime iff n=m*3^i (i>=0), a(n)=a(m) and (m,a(m)) or (a(m),m) is a twin prime pair of form ((4*k+1),(4*k+3)), a(m)*m=A071697(j)=A071695(j)*A071696(j) for some j.

Examples

			a(26928) = a(2^4*3^2*11*17) = a(2)^4 * a(3)^2 * a(11) * a(17)
= 2^4 * 1^2 * 9 * 19 = 2736.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..1000

Crossrefs

Cf. A002144, A002145, A072012(n) = a(a(n)).

For a(n) = n see A072011.

Cf. A072014, A072015.

Cf. A027746.

Programs

Haskell

a072010 1 = 1
a072010 n = product $ map f $ a027746_row n where
   f 2 = 2
   f p = p + 2 * (2 - p `mod` 4)
-- Reinhard Zumkeller, Apr 09 2012

Mathematica

a[1] = 1; a[p_?PrimeQ] = p + 2*(2 - Mod[p, 4]); a[n_] := Times @@ (a[#[[1]]]^#[[2]] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, May 04 2012 *)

Formula

Multiplicative with a(p) = p + 2*(2 - p mod 4), p prime.