A279067 -id:A279067 - cp's OEIS frontend

A179234 a(n) is the smallest prime q such that, for the previous prime p and the following prime r, the fraction (r-q)/(q-p) has denominator n in lowest terms.

3, 11, 29, 367, 149, 521, 127, 1847, 1087, 1657, 1151, 4201, 2503, 2999, 5779, 10831, 1361, 9587, 30631, 19373, 16183, 36433, 81509, 28277, 31957, 25523, 40343, 82129, 44351, 102761, 34123, 89753, 282559, 134581, 173429, 705389, 404671, 212777, 371027, 1060861, 265703, 461801, 156007, 544367, 576881, 927961, 1101071, 1904407, 604171, 396833

Offset: 1

Vladimir Shevelev, Jan 05 2011

nonn

The conjecture that a(n) exists for every n is a weaker conjecture than a related one in the comment to A179210.

			For q=3 we have (r-q)/(q-p)=2/1. Therefore, a(1)=3.
For q=5: (r-q)/(q-p) = 1/1; for q = 7: (r-q)/(q-p) = 2/1; for q = 11: (r-q)/(q-p) = 1/2. Therefore, a(2)=11.

Robert G. Wilson v, Table of n, a(n) for n = 1..123

Cf. A001223, A168253, A179210, A279067.

f[n_] := Block[{p = 2, q = 3, r = 5}, While[ Denominator[(r - q)/(q - p)] != n, p = q; q = r; r = NextPrime@ r]; q]; Array[f, 50]
p = 2; q = 3; r = 5; t[] = 0; While[q < 100000000, If[ t[ Denominator[(r - q)/(q - p)]] == 0, t[ Denominator[(r - q)/(q - p)]] = q]; p = q; q = r; r = NextPrime@ r]; t@# & /@ Range@100 (* _Robert G. Wilson v, Dec 11 2016 *)

a(n)=my(p=2,q=3);forprime(r=5,default(primelimit),if(denominator((r-q)/(q-p))==n,return(q));p=q;q=r)

Revised definition, new program, and terms past a(5) from Charles R Greathouse IV, Jan 12 2011

A279066 Least prime q such that (q-p)/(r-q), where pA038566/A038567.

Original entry on oeis.org

5, 3, 31, 23, 8123, 89, 139, 7963, 337, 409, 199, 797, 45439, 113, 953, 88547, 293, 2633, 1933, 3643, 137029, 13381, 523, 2861, 1381, 1259, 7621, 7433, 156157, 3089, 546781, 30911, 1951, 294563, 1129, 3229, 285871, 10369, 14221, 3651341, 25819, 3967, 1669, 6173, 23473, 51383

Offset: 1

Andres Cicuttin and Robert G. Wilson v, Dec 05 2016

Almost a bisection of A275785 with only the term 5 being in both A279066 & A279067.
The union of A279066 & A279067 is A275785 with only 5 as a common term.
1/n = A179210(n).
Records: 5, 31, 8123, 45439, 88547, 137029, 156157, 546781, 3651341, 11931613, 16613347, 54636251, 72510257, 102626747, 148379059, 290018137, 847428851, 1165527283, 8232085373, 32592174133, 40113962921, ..., .

			Row 1:        1/1                                       5
Row 2:        1/2                                       3
Row 3:     1/3  2/3                                 31      23
Row 4:     1/4  3/4                               8123      89
Row 5: 1/5 2/5  3/5 4/5                      139  7963     337    409
Row 6:     1/6  5/6                                199     797
Row 7:    1/7 .. 6/7                   45439 113   953   88547    293   2633
Row 8: 1/8 3/8  5/8 7/8                     1933  3643  137029  13381
etc.

Andres Cicuttin and Robert G. Wilson v, Table of n, a(n) for n = 1..322
Andres Cicuttin and Robert G. Wilson v, Table of n, Farey fraction = A038566/A038567 and a(n) for n=1..1000, or 0 if no such value is known.

Cf. A275785, A279067.

f[n_] := Block[{p = 2, q = 3, r = 5}, While[q != n(r - q) + p, p = q; q = r; r = NextPrime@ r]; q]; Farey[n_] := Union@ Flatten@ Table[a/b, {b, n}, {a, 0, b}]; ff = Rest@ Reverse@ Sort[ Farey[25], Denominator[#2] < Denominator[#1] &]; f@# & /@ ff

cp's OEIS Frontend

A179234 a(n) is the smallest prime q such that, for the previous prime p and the following prime r, the fraction (r-q)/(q-p) has denominator n in lowest terms.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions