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

1,2

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

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)

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 *)

Prog

(Scheme, with memoization-macro definec)
(definec (A293442 n) (if (= 1 n) n (* (A019565 (A067029 n)) (A293442 (A028234 n)))))

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).

Sequence in context: A317943 A305800 A366293 * A318470 A175501 A144370

Adjacent sequences: A293439 A293440 A293441 * A293443 A293444 A293445

Keyword

nonn,mult

Author

Antti Karttunen, Oct 31 2017

Status

approved