A179240 -id:A179240 - cp's OEIS frontend

A179328 a(n) is the smallest prime q > a(n-1) such that, for the previous prime p and the following prime r, the fraction (q-p)/(r-q) has denominator prime(n) (or 0, if such a prime does not exist).

3, 23, 139, 293, 1129, 2477, 8467, 30593, 81463, 85933, 190409, 404597, 535399, 840353, 1100977, 2127163, 4640599, 6613631, 6958667, 10343761, 24120233, 49269581, 83751121, 101649649, 166726367, 273469741, 310845683, 568951459

Offset: 1

Vladimir Shevelev, Jan 06 2011

nonn

Conjecture: a(n) > 0 for all n.

Cf. A168253, A179210, A179234, A179240, A179256, A001223

with(numtheory):
a:= proc(n) option remember; local k, p, q, r, pn;
      pn:= ithprime(n);
      for k from `if`(n=1, 1, pi(a(n-1))) do
        p:= ithprime(k);
        q:= ithprime(k+1);
        r:= ithprime(k+2);
        if denom((q-p)/(r-q)) = pn then break fi
      od; q
    end:
seq(a(n), n=1..10);  # Alois P. Heinz, Jan 06 2011

a[n_] := a[n] = Module[{k, p, q, r, pn},
     pn = Prime[n];
     For[k = If[n == 1, 1, PrimePi[a[n - 1]]], True, k++,
     p = Prime[k];
     q = Prime[k + 1];
     r = Prime[k + 2];
     If [Denominator[(q - p)/(r - q)] == pn, Break[]]]; q];
Table[a[n], {n, 1, 10}] (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)

More terms from Alois P. Heinz, Jan 06 2011

A178942 a(1) = 3; for n >= 2, a(n) is the smallest prime q > a(n-1) such that, for the previous prime p and the following prime r, the fraction (q-p)/(r-q) has denominator equal to A001223(n)/2 (or 0, if no such prime exists).

Original entry on oeis.org

3, 5, 11, 13, 17, 19, 29, 37, 47, 53, 61, 67, 71, 79, 83, 131, 137, 151, 163, 173, 233, 277, 331, 359, 379, 397, 401, 419, 439, 773, 823, 941, 947, 1021, 1031, 1033, 1063, 1087, 1097, 1117, 1123, 1153, 1187, 1237, 1277, 1709, 1789, 1823

Offset: 1

Vladimir Shevelev, Jan 06 2011

nonn

Conjecture: a(n) > 0 for all n.

The smallest prime(k) > a(n-1) such that the denominator of A001223(k-1)/A001223(k) equals A001223(n)/2. - R. J. Mathar, Jan 07 2011

Cf. A001223, A168253, A179210, A179234, A179240, A179328.

A001223 := proc(n) ithprime(n+1)-ithprime(n) ; end proc:
A178942 := proc(n) option remember; local p,q,r ; if n = 1 then 3; else for q from procname(n-1)+1 do if isprime(q) then p := prevprime(q) ; r := nextprime(q) ; denom((q-p)/(r-q)) ; if % = A001223(n)/2 then return q; end if; end if; end do: end if; end proc: # R. J. Mathar, Jan 07 2011

A001223[n_] := Prime[n + 1] - Prime[n];
a[n_] := a[n] = Module[{p, q, r, d}, If[n == 1, 3, For[q = a[n - 1] + 1, True, q++, If [PrimeQ[q], p = NextPrime[q, -1]; r = NextPrime[q]; d = Denominator[(q - p)/(r - q)]; If[d == A001223[n]/2, Return[q]]]]]];
Array[a, 48] (* Jean-François Alcover, May 21 2020, after Maple *)

More terms from Alois P. Heinz, Jan 06 2011

A179479 a(n) is the smallest prime q > a(n-1) such that, for the previous prime p and the following prime r, the fraction (q-p)/(r-q) has denominator 2, for odd n and 1 for even n (or 0, if such a prime does not exist).

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 29, 43, 53, 67, 71, 79, 97

Offset: 1

Vladimir Shevelev, Jan 08 2011

nonn

Conjecture: a(n) > 0 for all n.

Since, as it is accepted in the OEIS, we consider the uncancelled fractions, then, by the condition, for even n, we have (r-q)|(q-p).

			If n=1, then denominator should be 2. Thus a(1)=3, since (3-2)/(5-3)=1/2. If n=2, then denominator should be 1. Thus a(2)=5, since (5-3)/(7-5)=1/1, etc.

Cf. A168253, A179210, A179234, A179240, A179256, A179328.

cp's OEIS Frontend

A179328 a(n) is the smallest prime q > a(n-1) such that, for the previous prime p and the following prime r, the fraction (q-p)/(r-q) has denominator prime(n) (or 0, if such a prime does not exist).

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A178942 a(1) = 3; for n >= 2, a(n) is the smallest prime q > a(n-1) such that, for the previous prime p and the following prime r, the fraction (q-p)/(r-q) has denominator equal to A001223(n)/2 (or 0, if no such prime exists).

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A179479 a(n) is the smallest prime q > a(n-1) such that, for the previous prime p and the following prime r, the fraction (q-p)/(r-q) has denominator 2, for odd n and 1 for even n (or 0, if such a prime does not exist).

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