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

1,2

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

Formula

From Peter Munn, Dec 30 2019: (Start)

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

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

(End)

Prog

(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}

Crossrefs

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

Cf. A005117 (range of values), A052330.

Sequence in context: A126890 A122637 A076229 * A338224 A355701 A353955

Adjacent sequences: A160099 A160100 A160101 * A160103 A160104 A160105

Keyword

mult,nonn

Author

Franklin T. Adams-Watters, May 01 2009

Status

approved