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A240231 Number of factors needed in the unique factorization of positive integers into terms of A186285 where any term is used at most twice. 3
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 (list; graph; refs; listen; history; text; internal format)
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)

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 *)

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, A186285, A240230.

Sequence in context: A093921 A140192 A324905 * A065373 A047895 A307322

Adjacent sequences:  A240228 A240229 A240230 * A240232 A240233 A240234

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, May 15 2014

EXTENSIONS

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

STATUS

approved

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Last modified September 22 19:36 EDT 2021. Contains 347608 sequences. (Running on oeis4.)