cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Jun 12 2014

Keywords

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

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.

A106807 Primes with digit sum = 67.

Original entry on oeis.org

59899999, 69899899, 69899989, 69979999, 69997999, 69999799, 77899999, 78997999, 78998989, 78999889, 78999979, 79699999, 79879999, 79889899, 79979899, 79979989, 79988899, 79989979, 79996999, 79997899, 79997989
Offset: 1

Views

Author

Zak Seidov, May 18 2005

Keywords

Comments

499999909 is the smallest term that contains 0 as a digit. - Altug Alkan, Mar 25 2018

Links

Crossrefs

Cf. A007605, A062339, A062341, A062343, A106754 - A106787, A107617, A107618.
Cf. similar sequences listed in A244018.

Programs

  • Magma
    [p: p in PrimesUpTo(90000000) | &+Intseq(p) eq 67]; // Vincenzo Librandi, Jul 09 2014
  • Maple
    F:= proc(t,d)
  if d = 1 then
     if t<=9 then return [t] else return [] fi
  fi;
  if t > 9*d then return [] fi;
  [seq(op(map(x -> a*10^(d-1)+x, procname(t-a,d-1))), a=0..min(9,t))]
end proc:
select(isprime, F(67,8)); # Robert Israel, Mar 25 2018
  • Mathematica
    Select[Prime[Range[600000]], Total[IntegerDigits[#]]==67 &] (* Vincenzo Librandi, Jul 09 2014 *)
  • PARI
    isok(n) = isprime(n) && (sumdigits(n) == 67); \\ Altug Alkan, Mar 25 2018

A244016 Numbers in A243700 sorted into increasing order.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 13, 14, 15, 16, 18, 20, 22, 24, 25, 26, 28, 29, 30, 31, 32, 33, 35, 37, 38, 40, 41, 43, 44, 45, 47, 49, 50, 52, 53, 54, 55, 56, 58, 59, 60, 62, 63, 65, 67, 68, 70, 71, 72, 74, 76, 77, 79, 80, 82, 84, 86, 87, 88, 89, 91, 93, 94, 96, 97, 98, 100
Offset: 1

Views

Author

Max Alekseyev and N. J. A. Sloane, Jun 19 2014

Keywords

Crossrefs

Cf. A243700, A243864, A244010, A244011, A244017, A244018.

A244017 The quotients when A243700 is calculated (A244011), but shown in the order corresponding to the sorted sequence A244016.

Original entry on oeis.org

1, 2, 2, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87
Offset: 1

Views

Author

Max Alekseyev and N. J. A. Sloane, Jun 19 2014

Keywords

Crossrefs

Cf. A243700, A243864, A244010, A244011, A244016, A244018.

Formula

a(n) = A244010(A244016(n))/A244016(n).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.