A293442 Multiplicative with a(p^e) = A019565(e).

1, 2, 2, 3, 2, 4, 2, 6, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 12, 3, 4, 6, 6, 2, 8, 2, 10, 4, 4, 4, 9, 2, 4, 4, 12, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 12, 4, 12, 4, 4, 2, 12, 2, 4, 6, 15, 4, 8, 2, 6, 4, 8, 2, 18, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4, 4, 12, 2, 12, 4, 6, 4, 4, 4, 20, 2, 6, 6, 9, 2, 8, 2, 12, 8

Offset: 1

Author

Antti Karttunen, Oct 31 2017

Keywords

• nonn
• mult

Comments

From Peter Munn, Apr 06 2021: (Start)

a(n) is determined by the prime signature of n.

Compare with the multiplicative, self-inverse A225546, which also maps 2^e to the squarefree number A019565(e). However, this sequence maps p^e to the same squarefree number for every prime p, whereas A225546 maps the e-th power of progressively larger primes to progressively greater powers of A019565(e).

Both sequences map powers of squarefree numbers to powers of squarefree numbers.

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from exponents in factorization of n
• Index entries for sequences related to prime signature

Crossrefs

Sequences used in a definition of this sequence: A000188, A003961, A019565, A028234, A059895, A067029, A162642.

Sequences with related definitions: A225546, A293443, A293444.

Cf. also A293214.

Sequences used to express relationship between terms of this sequence: A006519, A007913, A008833, A064989, A334747.

Sequences related via this sequence: (A001222, A048675, A064547), (A007814, A162642), (A087207, A267116), (A248663, A268387).

Programs

Mathematica

f[n_] := If[n == 1, 1, Apply[Times, Prime@ Flatten@ Position[Reverse@ IntegerDigits[Last@ #, 2], 1]] * f[n/Apply[Power, #]] &@ FactorInteger[n][[1]]]; Array[f, 105] (* Michael De Vlieger, Oct 31 2017 *)

Formula

a(1) = 1; for n > 1, a(n) = A019565(A067029(n)) * a(A028234(n)).

Other identities. For all n >= 1:

a(a(n)) = A293444(n).

A048675(a(n)) = A001222(n).

A001222(a(n)) = A064547(n) = A048675(A293444(n)).

A007814(a(n)) = A162642(n).

A087207(a(n)) = A267116(n).

A248663(a(n)) = A268387(n).

From Peter Munn, Mar 14 2021: (Start)

Alternative definition: a(1) = 1; a(2) = 2; a(n^2) = A003961(a(n)); a(A003961(n)) = a(n); if A059895(n, k) = 1, a(n*k) = a(n) * a(k).

For n >= 3, a(n) < n.

a(2n) = A334747(a(A006519(n))) * a(n/A006519(n)), where A006519(n) is the largest power of 2 dividing n.

a(2n+1) = a(A064989(2n+1)).

a(n) = a(A007913(n)) * a(A008833(n)) = 2^A162642(n) * A003961(a(A000188(n))).

(End)