A243700 The lexicographically earliest sequence of distinct terms with a(1) = 1 such that a(n) divides the sum of the first a(n) terms.

1, 3, 2, 5, 9, 7, 8, 13, 15, 11, 14, 16, 26, 24, 41, 29, 18, 28, 20, 30, 22, 32, 25, 33, 43, 45, 31, 37, 50, 52, 54, 56, 58, 35, 87, 38, 55, 67, 40, 60, 72, 44, 63, 77, 79, 47, 70, 49, 121, 88, 53, 129, 94, 96, 98, 100, 59, 89, 105, 107, 62, 158, 113, 65, 102, 68, 103, 189

Offset: 1

Author

N. J. A. Sloane, Jun 12 2014

Keywords

• nonn
• nice

Comments

If the sum a(1) + a(2) + ... + a(m) is not divisible by m, then m does not belong to this sequence. Sequence A019444 gives a variant of this sequence, where every positive integer is a term. - Max Alekseyev, Jun 11 2014

Positive integers that do not appear in this sequence form A243864.

Is there any index n > 3 such that a(n) <= n? - Max Alekseyev, Jun 13 2014

Comment from Max Alekseyev, Jun 19 2014 (Start)

I've added b-files for sequences A244010 and A244011. Btw, it may be worth adding two more related sequences:

(a1) Integers that appear in the current sequence, sorted (A244016).

(a2) The corresponding ratios (similar to A244011), i.e., a2(n) = A244010(a1(n)) / a1(n). (A244011) (End)

From Bill McEachen, May 21 2024: (Start)

Conjecture: For n > 1000, a(n) falls within 1% of one of the following six values. a(n) = n, 1.576385*n, 1.788185*n, 2.576385*n, 2.788185*n, or 3.576285*n, using floor at the low bound and ceiling at the high bound, inclusive.

For example, a(1153) = 1836. This is between floor(1.576385 * 1153 * 0.99) and ceiling(1.576385 * 1153 * 1.01). About 90% of values fall in the three lower slopes. (End)

Conjectured asymptotic slopes of the 6 lines in the function graph are, from low to high: 1, sqrt(3)/3+1, sqrt(3)/6+3/2, sqrt(3)/3+2, sqrt(3)/6+5/2, sqrt(3)/3+3. - Hugo Pfoertner, Dec 19 2024

Examples

			1 divides the sum of the first 1 term  (yes:  1/1=1)
3 divides the sum of the first 3 terms (yes:  6/3=2)
2 divides the sum of the first 2 terms (yes:  4/2=2)
5 divides the sum of the first 5 terms (yes: 20/5=4)
9 divides the sum of the first 9 terms (yes: 63/9=7)
7 divides the sum of the first 7 terms (yes: 35/7=5)
8 divides the sum of the first 8 terms (yes: 48/8=6)
...
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..100000 (first 1100 terms from Jean-Marc Falcoz)
• Éric Angelini, a(n) divides the sum of the first a(n) terms of T, posting to the Sequence Fans Mailing List, Jun 11 2014 [Broken link]
• Éric Angelini, a(n) divides the sum of the first a(n) terms of T, posting to the Sequence Fans Mailing List, Jun 11 2014, Cached copy.
• Éric Angelini, Franklin Adams-Watters, Max Alekseyev, A. E. Povolotsky, N. J. A. Sloane, and R. G. Wilson v, a(n) divides the sum of the first a(n) terms of T, Various postings to the old Sequence Fans Mailing List, assembled by _N. J. A. Sloane_, Dec 24 2024
• Hugo Pfoertner, 1.73*10^6 terms, 7z compressed b-file.

Crossrefs

Cf. A019444, A243864 (complement, i.e. the missing terms), A244010 (partial sums), A244011 (the quotients), A244016 (sorted), A244017, A244018.

Programs

PARI

{ printA243700() = my( S=Set(), T=[], s=0, m=1, k); for(n=1,10^5, k=m; while( ((k==n || setsearch(S,n)) && Mod(s+k,n)) || if(kMax Alekseyev, Jun 13 2014

Extensions

First 1100 terms were computed by Jean-Marc Falcoz.

Edited by N. J. A. Sloane, Dec 18 2024, adding comments from the lost Sequence Fans Mailing List archive. Deleted an incorrect comment.