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A292375
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a(1) = 1, and for n > 1, a(n) = a(A252463(n)) + [n == 1 (mod 4)].
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11
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1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 1, 1, 4, 2, 4, 2, 3, 2, 4, 1, 3, 3, 1, 2, 5, 1, 5, 1, 3, 4, 1, 2, 6, 4, 2, 2, 7, 3, 7, 2, 2, 4, 7, 1, 4, 3, 3, 3, 8, 1, 3, 2, 5, 5, 8, 1, 9, 5, 2, 1, 4, 3, 9, 4, 5, 1, 9, 2, 10, 6, 2, 4, 2, 2, 10, 2, 2, 7, 10, 3, 3, 7, 4, 2, 11, 2, 3, 4, 6, 7, 3, 1, 12, 4, 2, 3, 13, 3, 13, 3, 2
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OFFSET
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1,5
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COMMENTS
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For numbers > 1, iterate the map x -> A252463(x) which divides even numbers by 2, and shifts every prime in the prime factorization of odd n one index step towards smaller primes. a(n) counts the numbers of the form 4k+1 encountered until 1 has been reached, which is also included in the count. The count includes also n itself if it is of the form 4k+1 (A016813), thus a(1) = 1.
In other words, locate the node which contains n in binary tree A005940 and traverse from that node towards the root, counting all numbers of the form 4k+1 that occur on the path.
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LINKS
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FORMULA
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a(1) = 1, a(2n) = a(n), and for odd numbers n > 1, a(n) = a(A064989(n)) + [n == 1 (mod 4)].
Other identities and observations. For n >= 1:
(End)
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MATHEMATICA
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a[1] = 1; a[n_] := a[n] = a[Which[n == 1, 1, EvenQ@n, n/2, True, Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n]] + Boole[Mod[n, 4] == 1]; Array[a, 105] (* Michael De Vlieger, Sep 17 2017 *)
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PROG
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(PARI)
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};
(Scheme, with memoization-macro definec)
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CROSSREFS
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Cf. A000120, A001248, A005940, A061395, A064989, A252463, A292374, A292377, A292378, A292381, A292583.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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