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

Author

Wolfdieter Lang, May 15 2014

Keywords

• nonn
• easy

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.

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A053735, A077761,A186285, A240230.

Programs

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 *)

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

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

Extensions

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