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 A269347 With a(1) = 1, a(n) is the sum of all 0 < m < n for which a(m) divides n. 8
 1, 1, 3, 3, 3, 15, 3, 3, 30, 3, 3, 51, 3, 3, 84, 3, 3, 111, 3, 3, 150, 3, 3, 195, 3, 3, 246, 3, 3, 318, 3, 3, 366, 3, 3, 435, 3, 3, 510, 3, 3, 591, 3, 3, 684, 3, 3, 771, 3, 3, 882, 3, 3, 975, 3, 3, 1086, 3, 3, 1218, 3, 3, 1326, 3, 3, 1455 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS For n > 2, I can prove that a(n) = 3 if 3 does not divide n, and in general, 3 divides a(n). The base case is a(3) = 3. Suppose that the results hold for a(n) over 3 < n < k; we will show that the results hold for a(k) also. In the case that 3 does not divide k, then a(k) = 3, since a(1) and a(2) divide k but no other previous term can. This proves the first claim. Otherwise, if 3 does divide k, then a(m) divides k for each 0 < m < k not divisible by 3; these numbers can be divided into k/3 pairs so that the sum of each pair is congruent to 0 modulo 3 (for instance, 1 + 2 == 4 + 5 == 7 + 8 == ... == 0 (mod 3)). If a(m) divides k for some 0 < m < k divisible by 3, this m does not change the congruence class of the sum that forms a(k). Thus, a(k) == 0 (mod 3) as required to prove the second claim. LINKS Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from Alec Jones) EXAMPLE a(1) = 1; a(2) = 1 because a(1) divides 2; a(3) = 3 because a(1) and a(2) divide 3: 1+2=3; a(4) = 3 because a(1) and a(2) divide 4: 1+2=3; a(5) = 3 because a(1) and a(2) divide 5: 1+2=3; a(6) = 15 because a(1), a(2), a(3), a(4), and a(5) divide 6: 1+2+3+4+5=15. MATHEMATICA a = {1}; Do[AppendTo[a, Total@ Select[Range[n - 1], Divisible[n, a[[#]]] &]], {n, 2, 66}]; a (* Michael De Vlieger, Mar 24 2016 *) PROG (Java) int[] terms = new int; terms = 1; for (int i = 1; i < 1000; i++) { int count = 0; for (int j = 0; j < i; j++) { if ((i + 1) % terms[j] == 0) { count = count + (j + 1); } } terms[i] = count; } (PARI) lista(nn) = {va = vector(nn); va = 1; for (n=2, nn, va[n] = sum(k=1, n-1, k*((n % va[k])==0)); ); va; } \\ Michel Marcus, Feb 24 2016 (Ruby) def a(n) seq =  (2..Float::INFINITY).each do |i| return seq.last[0...n].last if seq.length > n indices = seq.each_index.select { |j| i % seq[j] == 0 } seq << indices.map(&:next).reduce(:+) end end # Peter Kagey, Feb 25 2016 (Haskell) a269347 1 = 1 a269347 n = genericIndex a269347_list (n - 1) a269347_list = map a [1..] where a n = sum \$ filter ((==) 0 . mod n . a269347) [1..n-1] -- Peter Kagey, Jun 17 2016 (Python) from itertools import count, islice from sympy import divisors def A269347_gen(): # generator of terms A268347_dict = {1:1} yield 1 for n in count(2): yield (s:=sum(A268347_dict.get(d, 0) for d in divisors(n, generator=True))) A268347_dict[s] = A268347_dict.get(s, 0) + n A269347_list = list(islice(A269347_gen(), 40)) # Chai Wah Wu, Nov 17 2022 CROSSREFS Cf. A088167 which gives the number of m < n for which a(m) divides n. Cf. A271326, A271328. Sequence in context: A147823 A341211 A335518 * A183554 A229847 A196529 Adjacent sequences: A269344 A269345 A269346 * A269348 A269349 A269350 KEYWORD easy,nonn,nice AUTHOR Alec Jones, Feb 24 2016 STATUS approved

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Last modified June 1 18:23 EDT 2023. Contains 363076 sequences. (Running on oeis4.)