A003960 - OEIS

%I #20 Sep 03 2023 08:45:18

%S 1,1,2,1,3,2,4,1,4,3,6,2,7,4,6,1,9,4,10,3,8,6,12,2,9,7,8,4,15,6,16,1,

%T 12,9,12,4,19,10,14,3,21,8,22,6,12,12,24,2,16,9,18,7,27,8,18,4,20,15,

%U 30,6,31,16,16,1,21,12,34,9,24,12,36,4,37,19,18,10,24,14,40,3,16,21,42,8,27

%N Fully multiplicative with a(p) = [ (p+1)/2 ] for prime p.

%H Peter Kagey, <a href="/A003960/b003960.txt">Table of n, a(n) for n = 1..10000</a>

%F If n = Product p(k)^e(k) then a(n) = Product [ (p(k)+1)/2 ]^e(k).

%F Multiplicative with a(p^e) = [(p+1)/2]^e. - _David W. Wilson_, Aug 01 2001

%F a(n) = 1 iff n is a power of 2 (A000079). - _Michel Marcus_, Nov 16 2017

%t f[p_, e_] := Floor[(p+1)/2]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 03 2023 *)

%o (PARI) { A003960(n) = my(f); f=factor(n/2^valuation(n, 2)); prod(i=1, matsize(f)[1], ((f[i, 1]+1)/2)^f[i, 2] ); } \\ _Max Alekseyev_, Jul 27 2017

%Y Cf. A003970.

%K nonn,easy,mult

%O 1,3

%A _Marc LeBrun_