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 1-7 of 7 results.

A160811 Numbers not dividing 24.

Original entry on oeis.org

5, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77
Offset: 1

Views

Author

Omar E. Pol, Jun 19 2009, Jun 28 2009

Keywords

Comments

These terms m > 5 can be called "triphile" or "3-phile" numbers, because there are 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 "triphile" is called "triphobe" or "3-phobe" (A019532). The smallest triphile number is 7 = 1 + 2 + 4 and the largest triphobe is 24. See A348517 for more explanations and link. - Bernard Schott, Oct 21 2021

Links

Crossrefs

Complement of A018253.
Cf. A000005, A000720, A161205.
Cf. A019532, A348517.

Programs

Formula

a(n) = n + 8 for n > 16. [Charles R Greathouse IV, Oct 26 2011]

Extensions

Definition corrected by Omar E. Pol, Nov 17 2009

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

Views

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

A348518 Positive integers m with the property that there are 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 m = b_1 + b_2 + b_3 + b_4 + b_5.

Original entry on oeis.org

31, 39, 43, 45, 46, 47, 55, 57, 58, 59, 61, 62, 63, 64, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 99, 100, 101, 103, 105, 106, 107, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 127, 128
Offset: 1

Views

Author

Bernard Schott, Oct 21 2021

Keywords

Comments

The idea for this sequence comes from the French website Diophante (see link) where these numbers are called “pentaphile” or “5-phile”. A number that is not pentaphile is called “pentaphobe” or “5-phobe”.
It is possible to generalize for “k-phile” or “k-phobe” numbers (see Crossrefs).
Some results:
The smallest pentaphile number is 31 = 1 + 2 + 4 + 8 + 16 and the largest pentaphobe number is 240, so, this sequence is infinite since all integers >= 241 are terms.
Every term m = r * (1+s*t) with r > 0, s > 1 and t is a tetraphile number (A348517).
Odd numbers equal to 1 + 2*t where t is tetraphile (A348517) are pentaphile numbers, so odd numbers >= 99 are pentaphile.
If m is pentaphile, q* m, q > 1, is another pentaphile number.
There exist 68 pentaphobe numbers.

Examples

			As 43 = 1 + 2 + 4 + 12 + 24, 43 is a term.
As 89 = 1 + 4 + 12 + 24 + 48, 89 is another term.

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Range@100,Select[Select[IntegerPartitions[#,{5}],Length@Union@#==5&],And@@(IntegerQ/@Divide@@@Partition[#,2,1])&]!={}&] (* Giorgos Kalogeropoulos, Oct 22 2021 *)

A348519 Tetraphobe or 4-phobe numbers: integers that are not tetraphile numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 20, 24, 25, 26, 32, 48
Offset: 1

Views

Author

Bernard Schott, Oct 23 2021

Keywords

Comments

Tetraphile numbers are described in A348517.
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, the smallest one is always 1; here, there exist 23 tetraphobe numbers and the largest one is 48.

Examples

			There are no 4 positive integers b_1 < b_2 < b_3 < b_4 such that b_1 divides b_2, b_2 divides b_3, b_3 divides b_4, and 17 = b_1 + b_2 + b_3 + b_4, hence 17 is a term.

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Range@48,Select[Select[IntegerPartitions[#,{4}],Length@Union@#==4&],And@@(IntegerQ/@Divide@@@Partition[#,2,1])&]=={}&] (* Giorgos Kalogeropoulos, Oct 24 2021 *)
  • PARI
    isok(k) = forpart(p=k, if (#Set(p) == 4, if (!(p[2] % p[1]) && !(p[3] % p[2]) && !(p[4] % p[3]), return(0))), , [4, 4]); return(1); \\ Michel Marcus, Nov 14 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 1-7 of 7 results.

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