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.

User: Zizheng Fang

Zizheng Fang's wiki page.

Zizheng Fang has authored 3 sequences.

A334354 The number of optimal sets of n distinct integers less than or equal to k with this property: the sum of every two members of this set divides the product of all the members of this set. An optimal set with this property is one whose greatest member is the least possible.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 2, 6, 1, 2, 5, 4, 1, 1, 1, 196, 115, 34, 4, 674, 303, 86, 14, 1, 852, 164, 19, 1, 1, 176, 19, 1, 1, 44503, 11396
Offset: 1

Author

Zizheng Fang, Apr 24 2020

Keywords

Examples

			n=1:
1
n=2:
3, 6
3*6 = 18
3+6 divides 18
n=3:
3, 12, 15
3*12*15 = 540
3+12 divides 540
3+15 divides 540
12+15 divides 540
n=4:
2, 6, 10, 14
2*6*10*14 = 1680
2+6 divides 1680
2+10 divides 1680
2+14 divides 1680
6+10 divides 1680
6+14 divides 1680
10+14 divides 1680
n=5:
2, 6, 10, 14, 18
n=6:
2, 6, 10, 14, 18, 22
n=7:
2, 4, 10, 14, 18, 22, 26
2, 6, 10, 14, 18, 22, 26
4, 6, 10, 14, 18, 22, 26
n=8:
2, 4, 6, 10, 14, 18, 22, 26
n=9:
2, 6, 8, 10, 14, 18, 22, 26, 34
2, 6, 10, 14, 18, 22, 26, 30, 34
		

Crossrefs

Cf. A334352.

A334352 The least positive integer k such that there exists a set of n distinct integers less than or equal to k with this property: the sum of every two members of this set divides the product of all the members of this set.

Original entry on oeis.org

1, 6, 15, 14, 18, 22, 26, 26, 34, 38, 38, 29, 29, 29, 29, 37, 41, 43, 43, 43, 43, 47, 47, 47, 47, 47, 59, 59, 59, 59, 61, 71, 71, 71, 77, 79, 79
Offset: 1

Author

Zizheng Fang, Apr 24 2020

Keywords

Comments

Upper bound: For every n != 3, a(n) <= 4n-2. Proof: For every n >= 5, we have the set {2, 6, 10, ..., 4n-2}, which obviously possesses the desired property. It happens to also work for n = 1, 2, 4.

Examples

			n=1:
1
n=2:
3, 6
3*6 = 18
3+6 divides 18
n=3:
3, 12, 15
3*12*15 = 540
3+12 divides 540
3+15 divides 540
12+15 divides 540
n=4:
2, 6, 10, 14
2*6*10*14 = 1680
2+6 divides 1680
2+10 divides 1680
2+14 divides 1680
6+10 divides 1680
6+14 divides 1680
10+14 divides 1680
n=5:
2, 6, 10, 14, 18
n=6:
2, 6, 10, 14, 18, 22
n=7:
2, 4, 10, 14, 18, 22, 26
2, 6, 10, 14, 18, 22, 26
4, 6, 10, 14, 18, 22, 26
n=8:
2, 4, 6, 10, 14, 18, 22, 26
n=9:
2, 6, 8, 10, 14, 18, 22, 26, 34
2, 6, 10, 14, 18, 22, 26, 30, 34
		

Crossrefs

Cf. A334354.

A331828 Numbers k such that the divisors of k form an addition chain.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 30, 32, 36, 40, 42, 48, 54, 60, 64, 72, 80, 84, 90, 96, 100, 108, 120, 126, 128, 140, 144, 150, 156, 160, 162, 168, 180, 192, 198, 200, 210, 216, 220, 240, 252, 256, 264, 270, 272, 280, 288, 294, 300, 312, 320, 324, 330, 336, 342, 360
Offset: 1

Author

Zizheng Fang, Jan 27 2020

Keywords

Comments

Every divisor of a term, except 1, can be expressed as the sum of two other divisors.
This sequence is a subsequence of A308115. Numbers that are in A308115 but not in this sequence include 462, 1300, 3234, etc.

Examples

			1: divisors -- 1;
2: divisors -- 1, 2 = 1 + 1;
4: divisors -- 1, 2 = 1 + 1, 4 = 2 + 2;
6: divisors -- 1, 2 = 1 + 1, 3 = 1 + 2, 6 = 3 + 3;
8: divisors -- 1, 2 = 1 + 1, 4 = 2 + 2, 8 = 4 + 4;
12: divisors -- 1, 2 = 1 + 1, 3 = 1 + 2, 4 = 1 + 3, 6 = 2 + 4, 12 = 6 + 6.
		

Crossrefs

Subsequence of A308115.
Supersequence of A060765.

Programs

  • Maple
    q:= n-> (s-> andmap(x-> x=1 or ormap(y-> yAlois P. Heinz, Jan 30 2020
  • PARI
    isokd(k, d) = {for (j=1, k-1, if (vecsearch(d, d[k] - d[j]), return (1));); return (0);}
    isok(k) = {my(d=divisors(k)); for (j=2, #d, if (! isokd(j, d), return(0));); return (1);} \\ Michel Marcus, Jan 30 2020