A234695 -id:A234695 - cp's OEIS frontend

A234851 Indices of primes in A014692, i.e., numbers k such that prime(k)-k+1 is prime.

1, 2, 3, 5, 7, 13, 17, 21, 23, 25, 31, 41, 43, 49, 61, 71, 77, 83, 89, 103, 105, 109, 121, 129, 133, 139, 151, 161, 173, 181, 183, 185, 189, 199, 211, 213, 223, 231, 235, 241, 243, 247, 265, 271, 273, 277, 279, 281, 285, 293, 301, 303, 307, 311, 317

Offset: 1

M. F. Hasler, Dec 31 2013

nonn

Sequence A234695 lists primes in this sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A000720, A014692, A234695, A234850, A234852.

select(k -> isprime(ithprime(k)-k+1), [$1..1000]); # Robert Israel, Feb 19 2021

for(k=1,999,isprime(prime(k)-k+1)&&print1(k","))

PARI
```
is_A234851(n)=isprime(prime(k)-k+1)
```

a(n) = PrimePi(A234850(n)), PrimePi = A000720.

A235330 Number of ways to write 2*n = p + q with p, q, prime(p) - p + 1 and prime(q) + q + 1 all prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 3, 1, 1, 2, 3, 0, 1, 2, 0, 3, 1, 0, 2, 2, 0, 0, 1, 1, 2, 3, 0, 1, 3, 0, 2, 0, 0, 2, 1, 0, 1, 2, 0, 3, 0, 0, 4, 2, 1, 1, 1, 1, 3, 4, 1, 1, 3, 1, 0, 2, 1, 1, 3, 0, 0, 2, 3, 3, 3, 1, 1, 3, 3, 2, 3, 1, 1, 5, 0, 1, 4, 2, 1, 1

Offset: 1

Zhi-Wei Sun, Jan 05 2014

nonn

Conjecture: (i) a(n) > 0 for all n >= 2480.
(ii) If n > 4368 then 2*n+1 can be written as 2*p + q with p and q terms of the sequence A234695.
Parts (i) and (ii) are stronger than Goldbach's conjecture  (A045917) and Lemoine's conjecture (A046927) respectively.

			a(8) = 1 since 2*8 = 5 + 11 with 5, 11, prime(5) - 5 + 1 = 7 and prime(11) + 11 + 1 = 43 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A045917, A046927, A234695, A235187, A235189.

p[n_] := PrimeQ[n] && PrimeQ[Prime[n] - n + 1];
q[n_] := PrimeQ[n] && PrimeQ[Prime[n] + n + 1];
a[n_] := Sum[If[p[k] && q[2 n - k], 1, 0], {k, 1, 2 n - 1}];
Table[a[n], {n, 1, 100}]

A235682 Number of ways to write n = k + m with k > 0 and m > 2 such that p = phi(k) + phi(m)/2 + 1, prime(p) - p + 1 and p*(p+1) - prime(p) are all prime, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 1, 3, 2, 1, 3, 1, 1, 2, 2, 3, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 3, 0, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 6, 3, 6, 0, 6, 4, 5, 3, 1, 3, 4, 2, 3, 4, 1, 8, 6, 4, 8, 8

Offset: 1

Zhi-Wei Sun, Jan 13 2014

nonn

Conjecture: a(n) > 0 for all n > 84.

Clearly, this implies that there are infinitely many primes p with prime(p) - p + 1 and p*(p+1) - prime(p) both prime.

			a(10) = 1 since 10 = 1 + 9 with phi(1) + phi(9)/2 + 1 = 5, prime(5) - 5 + 1 = 7 and 5*6 - prime(5) = 19 all prime.
a(95) = 1 since 95 = 62 + 33 with phi(62) + phi(33)/2 + 1 = 41, prime(41) - 41 + 1 = 139 and 41*42 - prime(41) = 1543 all prime.
a(421) = 1 since 421 = 289 + 132 with phi(289) + phi(132)/2 + 1 = 293, prime(293) - 293 + 1 = 1621 and 293*294 - prime(293) = 84229 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A232353, A234694, A234695, A235189, A235330, A235613, A235614, A235661, A235681.

PQ[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n+1]&&PrimeQ[n(n+1)-Prime[n]]
f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/2+1
a[n_]:=Sum[If[PQ[f[n,k]],1,0],{k,1,n-3}]
Table[a[n],{n,1,100}]

A236687 Primes p such that prime(p^2) - 2 is also prime.

Original entry on oeis.org

2, 11, 17, 47, 61, 137, 163, 229, 239, 263, 317, 389, 419, 449, 467, 557, 571, 617, 619, 653, 709, 937, 953, 1009, 1033, 1087, 1123, 1129, 1181, 1249, 1481, 1831, 1987, 2003, 2099, 2207, 2381, 2441, 2579, 2663, 2707, 3109, 3457, 3833, 4013, 4463, 4519, 4783

Offset: 1

K. D. Bajpai, Jan 29 2014

nonn

			17 is prime and appears in the sequence because prime(17^2) = 1879 and 1879 - 2 = 1877, which is also prime.
47 is prime and appears in the sequence because prime(47^2) = 19471 and 19471 - 2 = 19469, which is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..564

Cf. A000040, A234695, A236119.

KD := proc() local a,b,d; a:=ithprime(n); b:=ithprime(a^2)-2; if isprime (b) then RETURN (a);fi; end: seq(KD(), n=1..500);

default(primelimit,2^31)
s=[]; forprime(p=2, 5000, if(isprime(prime(p^2)-2), s=concat(s, p))); s \\ Colin Barker, Jan 30 2014

A236688 Primes p such that prime(p^2) + 2 is also prime.

Original entry on oeis.org

7, 53, 83, 107, 149, 223, 367, 509, 701, 769, 853, 971, 1039, 1229, 1283, 1327, 1373, 1381, 1439, 1447, 1459, 1783, 1873, 1973, 2237, 2243, 2269, 2339, 2347, 2437, 2459, 2521, 2531, 2797, 2857, 3001, 3391, 3413, 3461, 3583, 3593, 3631, 3659, 3769, 3889, 3947

Offset: 1

K. D. Bajpai, Jan 29 2014

nonn

			7 is prime and appears in the sequence: prime(7^2) = 227 and 227+2 = 229, which is also prime.
53 is prime and appears in the sequence: prime(53^2) = 25469 and 25469+2 = 25471, which is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..1119

Cf. A000040, A234695, A236119.

KD := proc() local a,b; a:=ithprime(n); b:=ithprime(a^2)+2; if isprime (b) then RETURN (a);fi; end: seq(KD(), n=1..700);

Select[Prime[Range[600]],PrimeQ[Prime[#^2]+2]&] (* Harvey P. Dale, Aug 29 2021 *)

default(primelimit,2^31)
s=[]; forprime(p=2, 4000, if(isprime(prime(p^2)+2), s=concat(s, p))); s \\ Colin Barker, Jan 30 2014

A238776 Primes p with prime(p) - p + 1 and prime(q) - q + 1 both prime, where q = prime(2*pi(p)+1) with pi(.) given by A000720.

Original entry on oeis.org

2, 5, 7, 13, 31, 41, 43, 83, 109, 151, 211, 281, 307, 317, 349, 353, 499, 601, 709, 757, 883, 911, 971, 1447, 1453, 1483, 1531, 1801, 2053, 2281, 2819, 2833, 3163, 3329, 3331, 3881, 3907, 4051, 4243, 4447, 4451, 4703, 4751, 5483, 5659, 5701, 5737, 6011, 6271, 6311, 6361, 6379, 6427, 6571, 6827, 6841, 6983, 7159, 7879, 8209

Offset: 1

Zhi-Wei Sun, Mar 05 2014

nonn

Conjecture: The sequence has infinitely many terms.

This is motivated by the conjecture in A238766. Note that the sequence is a subsequence of A234695.

			a(1) = 2 since prime(2) - 2 + 1 = 2 and prime(prime(2*pi(2)+1)) - prime(2*pi(2)+1) + 1 = prime(5) - 5 + 1 = 11 - 4 = 7 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Cf. A000040, A000720, A234695, A238766.

p[k_]:=PrimeQ[Prime[Prime[k]]-Prime[k]+1]
n=0;Do[If[p[k]&&p[2k+1],n=n+1;Print[n," ",Prime[k]]],{k,1,1029}]

A235681 Primes p with prime(p) - p + 1 and p*(p+1) - prime(p) both prime.

Original entry on oeis.org

2, 3, 5, 41, 61, 71, 89, 271, 281, 293, 337, 499, 571, 751, 907, 911, 1093, 1531, 2027, 2341, 2707, 2861, 3011, 3359, 3391, 3511, 4133, 5179, 5189, 5483, 5573, 5657, 5867, 6577, 6827, 7159, 7411, 7753, 7879, 8179, 8467, 9209, 9391, 9419, 9433, 10259, 10303, 10859, 10993, 11287

Offset: 1

Zhi-Wei Sun, Jan 13 2014

nonn

This is the intersection of A234695 and A235661. For any prime p in this sequence, p^2 + 1 is the sum of the two primes prime(p) - p + 1 and p*(p+1) - prime(p).

By the conjecture in A235682, this sequence should have infinitely many terms.

			a(1) = 2 since prime(2) - 2 + 1 = 2 and 2*3 - prime(2) = 3 are both prime.
a(2) = 3 since prime(3) - 3 + 1 = 3 and 3*4 - prime(3) = 7 are both prime.
a(3) = 5 since prime(5) - 5 + 1 = 7 and 5*6 - prime(5) = 19 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A234694, A234695, A232353, A235189, A235330, A235613, A235614, A235661, A235682.

PQ[n_]:=PrimeQ[Prime[n]-n+1]&&PrimeQ[n(n+1)-Prime[n]]
n=0;Do[If[PQ[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,1000}]

A235934 Primes p with f(p), f(f(p)) and f(f(f(p))) all prime, where f(n) = prime(n) - n + 1.

Original entry on oeis.org

2, 3, 23, 311, 1777, 2341, 2861, 3329, 3833, 4051, 8753, 9007, 11587, 13093, 13309, 14551, 16001, 19687, 23143, 26993, 37309, 41981, 44131, 45491, 54623, 56431, 56821, 57991, 60223, 61643, 66413, 66883, 67511, 68767, 69029, 70003, 75743, 76261, 76819, 80021

Offset: 1

Zhi-Wei Sun, Jan 17 2014

nonn

By the general conjecture in A235925, this sequence should have infinitely many terms.

			a(3) = 23 with 23, f(23) = 61, f(61) = 223 and f(223) = 1187 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A234694, A234695, A235924, A235925.

f[n_]:=Prime[n]-n+1
p[k_]:=PrimeQ[f[Prime[k]]]&&PrimeQ[f[f[Prime[k]]]]&&PrimeQ[f[f[f[Prime[k]]]]]
n=0;Do[If[p[k],n=n+1;Print[n," ",Prime[k]]],{k,1,10000}]

A238134 Number of primes p < n with q = floor((n-p)/4) and prime(q) - q + 1 both prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 6, 5, 5, 5, 3, 4, 6, 6, 7, 6, 4, 4, 4, 4, 5, 5, 5, 5, 4, 4, 4, 4, 3, 3, 4, 4, 6, 6, 4, 5, 5, 5, 7, 6, 6, 6, 5, 5, 4, 4, 5, 5, 5, 5, 5, 6, 8, 8, 8, 7, 7, 7, 4, 4, 4, 4

Offset: 1

Zhi-Wei Sun, Mar 03 2014

nonn

Conjecture: Let m > 0 and n > 2*m + 1 be integers. If m = 1 and 2 | n, or m = 3 and n is not congruent to 1 modulo 6, or m = 2, 4, 5, ..., then there is a prime p < n such that q = floor((n-p)/m) and prime(q) - q + 1 are both prime.
In the cases m = 1, 2, this gives refinements of Goldbach's conjecture and Lemoine's conjecture (see also A235189). For m > 2, the conjecture is completely new.
See also A238701 for a similar conjecture involving primes q with q^2 - 2 also prime.

			 a(29) = 3 since 7, floor((29-7)/4) = 5 and prime(5) - 5 + 1 = 11 - 4 = 7 are all prime; 17, floor((29-17)/4) = 3 and prime(3) - 3 + 1 = 5 - 2 = 3 are all prime; 19, floor((29-19)/4) = 2 and prime(2) - 2 + 1 = 3 - 1 = 2 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Cf. A000040, A045917, A046927, A234694, A234695, A235189, A237705, A238701.

PQ[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n+1]
a[n_]:=Sum[If[PQ[Floor[(n-Prime[k])/4]],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,80}]

A238878 a(n) = |{0 < k <= n: prime(prime(k)) - prime(k) + 1 and prime(prime(kn)) - prime(kn) + 1 are both prime}|.

Original entry on oeis.org

1, 2, 3, 1, 1, 4, 3, 2, 5, 5, 3, 4, 2, 2, 3, 3, 5, 3, 1, 3, 4, 4, 2, 5, 2, 2, 7, 3, 2, 4, 4, 7, 4, 4, 4, 4, 4, 3, 4, 4, 4, 2, 4, 3, 7, 4, 9, 6, 3, 4, 5, 4, 2, 4, 4, 4, 3, 4, 5, 6, 10, 4, 4, 8, 9, 6, 5, 6, 5, 7, 8, 9, 5, 2, 5, 7, 1, 7, 4, 5

Offset: 1

Zhi-Wei Sun, Mar 06 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 4, 5, 19, 77.

(ii) For any integer n > 0, there is a number k among 1, ..., n such that 2*k + 1 and prime(prime(k^2*n)) - prime(k^2*n) + 1 are both prime.

			a(5) = 1 since prime(prime(4)) - prime(4) + 1 = prime(7) - 7 + 1 = 17 - 6 = 11 and prime(prime(4*5)) - prime(4*5) + 1 = prime(71) - 71 + 1 = 353 - 70 = 283 are both prime.
a(77) = 1 since prime(prime(3)) - prime(3) + 1 = prime(5) - 5 + 1 = 11 - 4 = 7 and prime(prime(3*77)) - prime(3*77) + 1 = prime(1453) - 1453 + 1 = 12143 - 1452 = 10691 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Cf. A000040, A234695, A237715, A238573, A238576, A238756, A238766, A238776, A238814, A238881.

PQ[n_]:=PrimeQ[Prime[n]-n+1]
p[k_,n_]:=PQ[Prime[k]]&&PQ[Prime[k*n]]
a[n_]:=Sum[If[p[k,n],1,0],{k,1,n}]
Table[a[n],{n,1,80}]

cp's OEIS Frontend

A234851 Indices of primes in A014692, i.e., numbers k such that prime(k)-k+1 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

PARI

Formula

A235330 Number of ways to write 2*n = p + q with p, q, prime(p) - p + 1 and prime(q) + q + 1 all prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A235682 Number of ways to write n = k + m with k > 0 and m > 2 such that p = phi(k) + phi(m)/2 + 1, prime(p) - p + 1 and p*(p+1) - prime(p) are all prime, where phi(.) is Euler's totient function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A236687 Primes p such that prime(p^2) - 2 is also prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

A236688 Primes p such that prime(p^2) + 2 is also prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A238776 Primes p with prime(p) - p + 1 and prime(q) - q + 1 both prime, where q = prime(2*pi(p)+1) with pi(.) given by A000720.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A235681 Primes p with prime(p) - p + 1 and p*(p+1) - prime(p) both prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A235934 Primes p with f(p), f(f(p)) and f(f(f(p))) all prime, where f(n) = prime(n) - n + 1.

Views

Author

Keywords

Comments

Examples

A238878 a(n) = |{0 < k <= n: prime(prime(k)) - prime(k) + 1 and prime(prime(kn)) - prime(kn) + 1 are both prime}|.