A210481 -id:A210481 - cp's OEIS frontend

A224748 Given n-th prime p, a(n) = number of primes of the form p + q - 1 where q is any prime < p.

0, 0, 1, 1, 2, 3, 3, 3, 2, 3, 6, 6, 5, 8, 4, 4, 4, 11, 11, 8, 9, 11, 4, 5, 13, 9, 11, 7, 13, 5, 17, 10, 9, 15, 7, 19, 17, 18, 10, 8, 8, 25, 12, 17, 12, 18, 29, 23, 12, 21, 12, 15, 28, 18, 11, 11, 12, 32, 25, 19, 22, 14, 29, 17, 27, 14, 40, 36, 18, 29, 16, 13

Offset: 1

Jayanta Basu, Apr 17 2013

nonn

Conjecture: a(n)>0 for all n>2. - Dmitry Kamenetsky, Feb 09 2017

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A072669, A210481, A224908.

a:= n-> add(`if`(isprime(ithprime(n)+ithprime(i)-1), 1, 0), i=1..n-1):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 18 2013

Table[p = Prime[n]; c = 0; i = 1; While[i < n, If[PrimeQ[p + Prime[i] - 1], c = c + 1]; i++]; c, {n, 72}]
Table[Count[Prime[n]+Prime[Range[n-1]]-1,?PrimeQ],{n,80}] (* _Harvey P. Dale, Nov 04 2020 *)

A224908 Given n-th prime p, a(n)=number of primes of the form p+q+1 where q is any prime < p.

Original entry on oeis.org

0, 0, 0, 2, 2, 2, 3, 3, 5, 5, 3, 4, 7, 4, 7, 8, 11, 5, 6, 9, 4, 7, 12, 14, 8, 11, 7, 13, 10, 12, 9, 15, 15, 12, 19, 9, 8, 8, 20, 19, 24, 11, 16, 11, 18, 15, 9, 13, 21, 14, 24, 27, 11, 26, 24, 26, 32, 13, 12, 21, 14, 28, 19, 27, 14, 26, 14, 14, 29, 24, 26, 39

Offset: 1

Jayanta Basu, Apr 19 2013

nonn

Conjecture: a(n)>0 for all n>3. - Dmitry Kamenetsky, Feb 09 2017

			For n=5, p=11, there are a(5)=2 solutions from 11+5+1=17 and 11+7+1=19.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A210481, A224748.

Table[p = Prime[n]; c = 0; i = 1; While[i < n, If[PrimeQ[p + Prime[i] + 1], c = c + 1]; i++]; c, {n, 72}]
Table[p = Prime[n] + 1; Sum[If[PrimeQ[p + Prime[i]], 1, 0], {i, 1, n - 1}], {n, 72}] (* Zak Seidov, Apr 19 2013 *)
Table[Count[Prime[n]+Prime[Range[n-1]]+1,?PrimeQ],{n,80}] (* _Harvey P. Dale, Mar 03 2024 *)

for(n = 1,72, q = prime (n) + 1; print1 (sum (i = 1, n - 1, isprime (q + prime (i))) ","))\\ Zak Seidov, Apr 19 2013

A137291 Numbers m such that prime(m)^2-2 is prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 14, 15, 18, 20, 24, 27, 28, 31, 32, 34, 40, 43, 47, 48, 51, 52, 55, 62, 65, 68, 72, 82, 86, 87, 91, 94, 99, 100, 104, 107, 111, 119, 123, 128, 129, 130, 132, 133, 134, 135, 139, 141, 150, 152, 170, 172, 177, 180, 182, 191, 200, 202, 209, 211

Offset: 1

Ctibor O. Zizka, Apr 05 2008

For m>=1, for these and only these numbers m, A242719(m) = prime(m)^2 + 1. Since A242719(m) >= prime(m)^2 + 1, then the equality is obtained on this and only this sequence. - Vladimir Shevelev, Sep 04 2014

			prime(24)*prime(24)-2 = 89*89-2 = 7919 is prime, so n=24 belongs to the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A062326, A242719, A242720, A246824, A010051, A049001, A103960, A210481.

a137291 n = a137291_list !! (n-1)
a137291_list = filter ((== 1) . a010051' . a049001) [1..]
-- Reinhard Zumkeller, Jul 30 2015

Select[Range[211],PrimeQ[Prime[#]^2-2]&] (* James C. McMahon, May 28 2025 *)

is(n,p=prime(n))=isprime(p^2-2) \\ Charles R Greathouse IV, Feb 17 2017

A103960(a(n)) - A210481(a(n)) = 1. - Reinhard Zumkeller, Jul 30 2015

a(n) = A049084(A049002(n)). - R. J. Mathar, Apr 09 2008

More terms from R. J. Mathar, Apr 09 2008

Offset corrected by Reinhard Zumkeller, Jul 30 2015

A103960 Number of primes p such that prime(n)*p - 2 is prime and p <= prime(n).

Original entry on oeis.org

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

Offset: 1

Lei Zhou, Feb 22 2005

Conjecture: All items of this sequence are greater than or equal to 1. Tested to prime(1000000).

a(A137291(n)) = A210481(A137291(n)) + 1. - Reinhard Zumkeller, Jul 30 2015

			Prime(1)*2-2 = 2, so a(1)=1;
Prime(3) = 5, 5*3-2 = 13, 5*5-2 = 23, so a(3)=2;

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A103959.

Cf. A010051, A000040, A137291, A210481.

a103960 n = sum [a010051' $ p * q - 2 |
                 let p = a000040 n, q <- takeWhile (<= p) a000040_list]
-- Reinhard Zumkeller, Jul 30 2015

Table[p=Prime[n]; ct=0; Do[pk=Prime[k]; If[PrimeQ[p*pk-2], ct=ct+1], {k, n}]; ct, {n, 100}]

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

A224748 Given n-th prime p, a(n) = number of primes of the form p + q - 1 where q is any prime < p.

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A224908 Given n-th prime p, a(n)=number of primes of the form p+q+1 where q is any prime < p.

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A137291 Numbers m such that prime(m)^2-2 is prime.

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A103960 Number of primes p such that prime(n)*p - 2 is prime and p <= prime(n).

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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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