A224748 -id:A224748 - cp's OEIS frontend

A225216 Let p = n-th prime. Then a(n) = number of primes generated by prepending to the digits of p the digits of q, where q is any prime less than p.

0, 1, 0, 1, 2, 1, 2, 2, 4, 2, 2, 2, 4, 1, 4, 5, 4, 3, 4, 5, 6, 4, 5, 5, 6, 5, 6, 5, 3, 8, 4, 6, 8, 7, 8, 7, 5, 6, 8, 8, 4, 9, 7, 5, 10, 5, 9, 5, 8, 8, 10, 8, 8, 14, 10, 7, 14, 8, 8, 11, 10, 13, 8, 10, 10, 10, 11, 12, 13, 8, 11, 14, 12, 11, 13, 13, 13, 16

Offset: 1

Jayanta Basu, May 02 2013

The graph makes it apparent that there are fewer primes generated when the prime p increases its length from 3 to 4 and 4 to 5 digits. - T. D. Noe, May 03 2013

			a(2)=1 since second prime 3 generates 23. Also a(7)=2 since for the seventh prime 17 we have two primes 317 and 1117.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A224748, A224908.

con[x_,y_] := FromDigits[Join[IntegerDigits[Prime[x]], IntegerDigits[Prime[y]]]]; t={}; Do[c=0; Do[If[PrimeQ[con[i,n]], c=c+1], {i,n}]; AppendTo[t,c], {n,78}]; t

A224793 Least prime p which generates exactly n primes of the form p+q-1 where q < p is prime, or 0 if (conjecturally) no such p exists.

Original entry on oeis.org

2, 5, 11, 13, 47, 41, 31, 107, 43, 73, 131, 61, 191, 97, 293, 139, 353, 127, 163, 151, 0, 229, 283, 223, 659, 181, 929, 313, 241, 211, 367, 701, 271, 397, 379, 457, 337, 1031, 1259, 607, 331, 463, 643, 613, 1409, 733, 911, 1091, 541, 1997, 421, 727, 709, 673

Offset: 0

Jayanta Basu, Apr 18 2013

nonn

a(n) = 0 for n = 20, 165, 467, ... . Do there exist infinitely many such values of n?

These values of 0 are all conjectural. - Robert Israel, Apr 28 2021

			a(1) = 5 since 5 is the least prime that generates exactly one prime 7=5+3-1 of the given form. Again a(3) = 13 since 13 generates exactly 3 primes 17=13+5-1, 19=13+7-1 and 23=13+11-1 of the given form.

Cf. A072669, A224748.

Cn[n_] := Module[{c}, p = Prime[n]; c = 0; i = 1; While[i < n, If[PrimeQ[p + Prime[i] - 1], c = c + 1] i++]; c]; t = {};
Do[p = 0; j = 0; While[++j < 2000 && p != 1, If[Cn[j] == k, AppendTo[t, Prime[j]]; p = 1, p = 0]]; If[p == 0, AppendTo[t, 0]], {k, 0, 200}]; t

Definition clarified by Robert Israel, Apr 28 2021

A224961 a(n) = number of primes of the form p * q + 2 where p is the prime(n) and q is any prime < p.

Original entry on oeis.org

0, 0, 1, 2, 1, 2, 2, 3, 3, 2, 1, 4, 0, 4, 4, 4, 5, 4, 4, 3, 2, 4, 4, 3, 5, 3, 4, 4, 6, 4, 7, 4, 4, 7, 5, 5, 6, 5, 6, 8, 5, 7, 7, 6, 3, 9, 5, 8, 5, 8, 7, 10, 9, 7, 8, 8, 5, 8, 8, 9, 8, 8, 10, 7, 11, 13, 8, 10, 10, 10, 11, 9, 12, 9, 13, 11, 9, 12, 7, 11

Offset: 1

Jayanta Basu, Apr 21 2013

nonn

			For n=4, p=7, there are a(4)=2 solutions from 7*3+2=23 and 7*5+2=37.

Cf. A103960, A224748, A210481.

Table[p = Prime[n]; c = 0; i = 1; While[i < n, If[PrimeQ[p*Prime[i] + 2], c = c + 1]; i++]; c, {n, 80}]
Table[Count[Prime[n]Prime[Range[n-1]]+2,?PrimeQ],{n,80}] (* _Harvey P. Dale, Feb 28 2023 *)

cp's OEIS Frontend

A225216 Let p = n-th prime. Then a(n) = number of primes generated by prepending to the digits of p the digits of q, where q is any prime less than p.

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A224793 Least prime p which generates exactly n primes of the form p+q-1 where q < p is prime, or 0 if (conjecturally) no such p exists.

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A224961 a(n) = number of primes of the form p * q + 2 where p is the prime(n) and q is any prime < p.

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