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A240231
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Number of factors needed in the unique factorization of positive integers into terms of A186285 where any term is used at most twice.
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3
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1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 2, 2, 2, 1, 3, 1, 3, 1, 3, 2, 2, 2, 4, 1, 2, 2, 2, 1, 3, 1, 3, 3, 2, 1, 3, 2, 3, 2, 3, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 3, 2, 2, 3, 1, 3, 2, 3, 1, 3, 1, 2, 3, 3, 2, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 2, 1, 4, 2, 3, 2, 2, 2, 4, 1, 3, 3, 4, 1, 3, 1, 2, 3
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OFFSET
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1,4
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COMMENTS
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The number 1 with factorization defined to be 1 has been included. See a comment on A240230.
This is the row length sequence for the table A240230.
a(n) = 1 if and only if n = 1 or n is a term of A186285.
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LINKS
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FORMULA
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a(n) is, for n >= 2, the sum of all entries in the base 3 representation of the exponents of the primes in the usual prime number factorization of n.
That is, apart from the initial term, additive with a(p^e) = A053735(e).
Define b(1) = 0; and for n > 1, b(n) = A053735(A067029(n)) + b(A028234(n)). Then a(n) = b(n) for n > 1, with a(1) = 1 by convention.
(End)
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.38090372984518844518..., where f(x) = -x + Sum_{k>=0} (x^(3^k) + 2*x^(2*3^k))/(1 + x^(3^k) + x^(2*3^k)). - Amiram Eldar, Sep 28 2023
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EXAMPLE
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a(12) = 3 because the usual prime factorization is 12 = 2^2*3^1 and (2)_3 = [2] and (1)_3 = [1], hence the sum of the base-3 representations of the exponents is 3.
a(24) = 2 as 24 = 3*8, using two factors from A186285. Note also how 3*8 = 3^1 * 2^3, and ternary representations of 1 and 3 are "1" and "10", thus their digit sum is 2. - Antti Karttunen, Aug 12 2017
a(36) = 4 from 2^2*3^2, (2)_3 = [2] and 2 + 2 = 4.
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MATHEMATICA
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Block[{nn = 105, s}, s = Select[Select[Range@ nn, PrimePowerQ], IntegerQ@ Log[3, FactorInteger[#][[1, -1]]] &]; {1}~Join~Table[Length@ Rest@ NestWhileList[Function[{k, m}, {k/#, #} &@ SelectFirst[Reverse@ TakeWhile[s, # <= k &], Divisible[k, #] &]] @@ # &, {n, 1}, First@ # > 1 &][[All, -1]], {n, 2, nn}]] (* Michael De Vlieger, Aug 14 2017 *)
a[n_] := Total[Plus @@ IntegerDigits[#, 3] & /@ (FactorInteger[n][[;; , 2]])]; Array[a, 100] (* Amiram Eldar, May 18 2023 *)
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PROG
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(Scheme)
(define (A240231 n) (if (= 1 n) n (A240231with_a1_0 n)))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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