A160102 Multiplicative function, one-to-one and onto the squarefree numbers.

1, 2, 3, 5, 7, 6, 11, 10, 13, 14, 17, 15, 19, 22, 21, 23, 29, 26, 31, 35, 33, 34, 37, 30, 41, 38, 39, 55, 43, 42, 47, 46, 51, 58, 77, 65, 53, 62, 57, 70, 59, 66, 61, 85, 91, 74, 67, 69, 71, 82, 87, 95, 73, 78, 119, 110, 93, 86, 79, 105, 83, 94, 143, 115, 133, 102, 89, 145

Offset: 1

Author

Franklin T. Adams-Watters, May 01 2009

Keywords

• mult
• nonn

Comments

Multiplicative with a(A050376(m)) = Prime(m) = A000040(m). If k = 2^{i_1} + ... + 2^{i_j} is the binary representation of k, a(p^k) = a(p^2^{i_1}) * ... * a(p^2^{i_j}). [edited by Peter Munn, Jan 07 2020]

Equivalently, a(A050376(m)) = A000040(m); a(A059897(n,k)) = A059897(a(n), a(k)). - Peter Munn, Dec 30 2019

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000

Crossrefs

Sequences used in definitions of this sequence: A000040, A019565, A050376, A052331, A059897.

Cf. A005117 (range of values), A052330.

Programs

PARI

al(n)={local(v,k,fm,m,p);
v=vector(n);v[1]=1;p=1;
for(k=2,n,fm=factor(k);
if(matsize(fm)[1]>1,m=fm[1,1]^fm[1,2];v[k]=v[m]*v[k/m],
m=2^valuation(fm[1,2],2);
if(m==fm[1,2],p=nextprime(p+1);v[k]=p,
m=fm[1,1]^m;v[k]=v[m]*v[k/m])));
v}

Formula

From Peter Munn, Dec 30 2019: (Start)

a(n) = A019565(A052331(n)).

a(A052330(k)) = A019565(k).

(End)