A240231 Number of factors needed in the unique factorization of positive integers into terms of A186285 where any term is used at most twice.

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

Offset

1,4

Comments

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n.

Formula

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.

From Antti Karttunen, Aug 12 2017: (Start)

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

Example

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.

Mathematica

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 *)

Prog

(Scheme)
(define (A240231 n) (if (= 1 n) n (A240231with_a1_0 n)))
(definec (A240231with_a1_0 n) (if (= 1 n) 0 (+ (A053735 (A067029 n)) (A240231with_a1_0 (A028234 n)))))
;; Antti Karttunen, Aug 12 2017

Crossrefs

Cf. A053735, A077761,A186285, A240230.

Sequence in context: A353426 A140192 A324905 * A366075 A065373 A047895

Adjacent sequences: A240228 A240229 A240230 * A240232 A240233 A240234

Keyword

nonn,easy

Author

Wolfdieter Lang, May 15 2014

Extensions

Description clarified and more terms added by Antti Karttunen, Aug 12 2017

Status

approved