A347105 a(n) is the greatest sum of the digital roots of the individual factorizations of n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 7, 2, 8, 4, 9, 8, 10, 8, 11, 1, 9, 10, 4, 5, 11, 10, 8, 12, 11, 2, 11, 4, 12, 6, 10, 12, 13, 1, 3, 7, 13, 5, 13, 7, 8, 14, 7, 2, 14, 14, 12, 11, 10, 8, 15, 7, 15, 4, 4, 5, 13, 7, 8, 16, 16, 9, 8, 4, 12, 8, 14, 8, 17, 1, 3, 13, 5, 9, 11, 7, 15, 18, 7, 2, 15, 13, 9, 6, 10, 8, 16, 11, 9

Offset

1,2

Links

Table of n, a(n) for n=1..92.

Project Euler, Digital root sums of factorisations, Problem 159.

Example

a(24) = 11:
  Factorization     Sum of Digital Roots
   2 *  2 * 2 * 3   2 + 2 + 2 + 3 =  9
   2 *  3 * 4           2 + 3 + 4 =  9
   2 *  2 * 6           2 + 2 + 6 = 10
   4 *  6                   4 + 6 = 10
   3 *  8                   3 + 8 = 11
   2 * 12               2 + 1 + 2 =  5
  24                        2 + 4 =  6
Result is 11.

Prog

(PARI) fcnt(n, m) = {my(d = divisors(n), v = List(select(x->((x>1) && (x<=m)), d)), list = List()); for (k=1, #v, my(x = fcnt(n/v[k], v[k])); if (#x==0, listput(list, [v[k]]), for (j=1, #x, listput(list, concat(v[k], x[j])))); ); list; }
factoriz(n) = {if (n==1, return ([[1]])); my(list = fcnt(n, n), res = List()); for (i=1, #list, my(vi=list[i]); if (vecprod(vi) == n, listput(res, vi)); ); Vec(res); } \\ A162247
a(n) = {my(v = factoriz(n)); vecmax(vector(#v, k, vecsum(apply(x->(x-1)%9+1, v[k])))); } \\ Michel Marcus, Jun 06 2022
(Python)
def drs(x, m):
  if x == 0:
    return 0
  return ((x - 1) % m) + 1
def a(n):
  DRS = [drs(x, 9) for x in range(0, n + 1)]
  for a in range(2, n):
    da = DRS[a]
    for b in range(2, 1 + (n // a)):
      ab = a * b
      if ab > n:
        break
      else:
        x = da + DRS[b]
        if DRS[ab] < x:
          DRS[ab] = x
  return DRS[1: n + 1][-1]

Crossrefs

Cf. A007953, A001055, A162247.

Sequence in context: A081597 A351578 A328469 * A245351 A028902 A081598

Adjacent sequences: A347102 A347103 A347104 * A347106 A347107 A347108

Keyword

nonn,base

Author

Darío Clavijo, May 02 2022

Status

approved