A293442 - OEIS

%I #28 Apr 29 2021 04:10:15

%S 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,

%T 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,

%U 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

%N Multiplicative with a(p^e) = A019565(e).

%C From _Peter Munn_, Apr 06 2021: (Start)

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

%C 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).

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

%C (End)

%H Antti Karttunen, <a href="/A293442/b293442.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%H <a href="/index/Pri#prime_signature">Index entries for sequences related to prime signature</a>

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

%F Other identities. For all n >= 1:

%F a(a(n)) = A293444(n).

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

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

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

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

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

%F From _Peter Munn_, Mar 14 2021: (Start)

%F 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).

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

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

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

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

%F (End)

%t 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 *)

%o (Scheme, with memoization-macro definec)

%o (definec (A293442 n) (if (= 1 n) n (* (A019565 (A067029 n)) (A293442 (A028234 n)))))

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

%Y Sequences with related definitions: A225546, A293443, A293444.

%Y Cf. also A293214.

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

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

%K nonn,mult

%O 1,2

%A _Antti Karttunen_, Oct 31 2017