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.

A019532 Numbers k such that Fibonacci(k) divides k!.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 12, 24
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

These terms m are called “triphobe” or “3-phobe” numbers, by the French website Diophante (see link), because there are no 3 positive integers b_1 < b_2 < b_3 such that b_1 divides b_2, b_2 divides b_3, and m = b_1 + b_2 + b_3. A number that is not “triphobe” is called “triphile” or “3-phile” (A160811). The set of k-phobe numbers is always finite, there exist 9 triphobe numbers and the largest one is 24. - Bernard Schott, Oct 23 2021

References

  • Posting to math-fun(AT)cs.arizona.edu by R. W. Gosper Nov 06 1996.

Links

Crossrefs

k-phobe numbers: this sequence (k=3), A348519 (k=4), A348520 (k=5).
k-phile numbers: A160811 \ {5} (k=3), A348517 (k=4), A348518 (k=5).
Cf. A000045, A000142, A160811.

Programs

  • Mathematica
    Select[Range[30],Divisible[#!,Fibonacci[#]]&] (* Harvey P. Dale, Jun 14 2020 *)

Extensions

Offset changed to 1 by David A. Corneth, Oct 27 2021

A348520 Pentaphobe or 5-phobe numbers: integers that are not pentaphile numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 44, 48, 49, 50, 51, 52, 53, 54, 56, 60, 65, 66, 68, 72, 74, 80, 84, 97, 98, 102, 104, 108, 120, 132, 144, 168, 194, 240
Offset: 1

Views

Author

Bernard Schott, Nov 02 2021

Keywords

Comments

Pentaphile numbers are described in A348518.
The idea for this sequence comes from the French website Diophante (see link).
It is possible to generalize for "k-phile" or "k-phobe" numbers (see Crossrefs).
The set of k-phobe numbers is always finite and the smallest one is always 1; here, there exist 68 pentaphobe numbers and the largest one is 240.

Examples

			There are no 5 positive integers b_1 < b_2 < b_3 < b_4 < b_5 such that b_1 divides b_2, b_2 divides b_3, b_3 divides b_4, b_4 divides b_5, and 32 = b_1 + b_2 + b_3 + b_4 + b_5, hence 32 is a term.

Links

Crossrefs

k-phile numbers: A160811 \ {5} (k=3), A348517 (k=4), A348518 (k=5).
k-phobe numbers: A019532 (k=3), A348519 (k=4), this sequence (k=5).

Programs

  • PARI
    isok(k) = forpart(p=k, if (#Set(p) == 5, if (!(p[2] % p[1]) && !(p[3] % p[2]) && !(p[4] % p[3]) && !(p[5] % p[4]), return(0))), , [5, 5]); return(1); \\ Michel Marcus, Nov 14 2021

A349188 Largest n-phobe number.

Original entry on oeis.org

2, 24, 48, 240, 1440, 2400, 7440, 25920, 72000, 234000
Offset: 2

Views

Author

Bernard Schott, Nov 09 2021

Keywords

Comments

A n-phile integer m is such that there are n positive integers b_1 < b_2 < ... < b_j < ... < b_n such that b_1 divides b_2, b_2 divides b_3, ..., b_[j-1] divides b_j, ..., b_[n-1] divides b_n, and m = b_1 + b_2 + ... + b_j + ... + b_n. A number that is not n-phile is called n-phobe.
The idea for this sequence and the words 'n-phile' and 'n-phobe' come from the French website Diophante (see link).
The number of n-phobe numbers is always finite (A349189), the smallest one is always 1, and this sequence lists the largest n-phobe numbers.
a(6) >= 720. - Michel Marcus, Nov 14 2021
From David A. Corneth, Nov 14 2021: (Start)
a(6) >= 1440. If a(6) > 1440 then a(6) > 50000.
a(7) >= 2400. If a(7) > 2400 then a(7) > 50000.
a(8) >= 7440. If a(8) > 7440 then a(8) > 100000.
a(9) >= 25920. If a(9) > 25920 then a(9) > 100000. (End)
Indeed, all these bounds are the corresponding values of a(6), a(7), a(8), a(9). Proof in link. For n >= 5, the five known terms are divisible by 240. - Bernard Schott, Nov 19 2021

Examples

			For n = 2, integers 1 and 2 are 2-phobe, then for m >= 3, every m = 1 + (m-1) with 1 < m-1 and 1 divides m-1, so, each m >= 3 is 2-phile number and a(2) = 2.

Links

Crossrefs

Cf. A349189.
k-phile numbers: A160811 \ {5} (k=3), A348517 (k=4), A348518 (k=5).
k-phobe numbers: A019532 (k=3), A348519 (k=4), A348520 (k=5).

Extensions

a(6)-a(11) from David A. Corneth, Nov 19 2021

A349189 Number of n-phobe numbers.

Original entry on oeis.org

2, 9, 23, 68, 177, 459, 1162, 2947, 7306, 18202
Offset: 2

Views

Author

Bernard Schott, Nov 14 2021

Keywords

Comments

A n-phile integer m is such that there are n positive integers b_1 < b_2 < ... < b_j < ... < b_n with the property that b_1 divides b_2, b_2 divides b_3, ..., b_[j-1] divides b_j, ..., b_[n-1] divides b_n, and m = b_1 + b_2 + ... + b_j + ... + b_n. A number that is not n-phile is called n-phobe.
The words 'n-phile' and 'n-phobe' come from the French website Diophante (see link).
The number of n-phobe numbers is always finite, the smallest one is always 1 and the largest n-phobe number is in A349188.
a(6) >= 176. - Michel Marcus, Nov 15 2021
a(6) >= 177, a(7) >= 459, a(8) >= 1162, a(9) >= 2947. - David A. Corneth, Nov 15 2021
Indeed, all these bounds are the corresponding values of a(6), a(7), a(8) and a(9). Proof comes from Proof link in A349188. - Bernard Schott, Nov 19 2021

Examples

			For n = 2, integers 1 and 2 are 2-phobe, then for m >= 3, every m = 1 + (m-1) with 1 < m-1 and 1 divides m-1, so, each m >= 3 is 2-phile number and a(2) = 2.

Links

Crossrefs

k-phile numbers: A160811 \ {5} (k=3), A348517 (k=4), A348518 (k=5).
k-phobe numbers: A019532 (k=3), A348519 (k=4), A348520 (k=5).
Cf. A349188.

Extensions

a(6)..a(11) from David A. Corneth, Nov 19 2021
Showing 1-4 of 4 results.

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

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