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

1,2

Comments

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.

Links

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

Formula

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

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (p^2-p)/(p^2-a(p)) = 0.7902497571052642... . - Amiram Eldar, Nov 23 2025

Example

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.

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 *)

Prog

(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
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1] + 2*(2 - f[i, 1] % 4))^f[i, 2]); } \\ Amiram Eldar, Nov 23 2025

Crossrefs

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

For a(n) = n see A072011.

Cf. A072014, A072015.

Cf. A027746.

Sequence in context: A225468 A048787 A030102 * A255588 A328034 A123360

Adjacent sequences: A072007 A072008 A072009 * A072011 A072012 A072013

Keyword

nonn,mult,nice

Author

Reinhard Zumkeller, Jun 05 2002

Status

approved