A234695 -id:A234695 - cp's OEIS frontend

A234850 Primes in A014692, i.e., of the form prime(k)-k+1, for some k.

2, 2, 3, 7, 11, 29, 43, 53, 61, 73, 97, 139, 149, 179, 223, 283, 313, 349, 373, 461, 467, 491, 541, 599, 619, 659, 727, 787, 859, 907, 911, 919, 941, 1019, 1087, 1091, 1187, 1223, 1249, 1283, 1301, 1321, 1433, 1471, 1481, 1511, 1523, 1543

Offset: 1

M. F. Hasler, Dec 31 2013

nonn

The first term is listed twice because A014692(1) = 2-1+1 = A014692(2) = 3-2+1 = 2 both are prime; thereafter the sequence A014692 is strictly increasing, so there is no other duplicate value.

Cf. A234695.

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

a(n) = prime(A234851(n)), prime = A000040.

A234852 Indices k of primes p=prime(k) such that prime(p)-p+1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 11, 13, 14, 18, 20, 23, 24, 27, 29, 34, 36, 40, 42, 46, 47, 48, 53, 58, 59, 60, 62, 63, 64, 66, 68, 70, 71, 73, 90, 95, 100, 103, 105, 110, 121, 122, 127, 128, 132, 133, 134, 141, 143, 144, 153, 155, 156, 162

Offset: 1

M. F. Hasler, Dec 31 2013

nonn

Cf. A000040, A000720, A014692, A234850, A234851.

Select[Range[200],PrimeQ[Prime[Prime[#]]-Prime[#]+1]&] (* Harvey P. Dale, Jul 14 2024 *)

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

a(n) = PrimePi(A234695(n)), PrimePi = A000720.

A235508 Number of ways to write 2n = p + q with q > 0 such that p, p(p+1) - prime(p) and prime(q) - q + 1 are all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 14 2014

nonn

Conjecture: a(n) > 0 for all n > 1.

			a(7) = 1 since 2*7 = 11 + 3 with 11, 11*12 - prime(11) = 101 and prime(3) - 3 + 1 = 3 all prime.
a(19) = 1 since 2*19 = 37 + 1 with 37, 37*38 - prime(37) = 1249 and prime(1) - 1 + 1 = 2 all prime.
a(98) = 1 since 2*98 = 11 + 185 with 11, 11*12 - prime(11) = 101 and prime(185) - 185 + 1 = 919 all prime.

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

Cf. A000040, A234695, A234851, A235189, A235330, A235661, A235681.

p[k_]:=PrimeQ[Prime[k](Prime[k]+1)-Prime[Prime[k]]]
q[m_]:=PrimeQ[Prime[m]-m+1]
a[n_]:=Sum[If[p[k]&&q[2n-Prime[k]],1,0],{k,1,PrimePi[2n-1]}]
Table[a[n],{n,1,100}]

A235935 Primes p with f(p), f(f(p)), f(f(f(p))), f(f(f(f(p)))) all prime, where f(n) = prime(n) - n + 1.

Original entry on oeis.org

2, 3, 2861, 8753, 56821, 83449, 162787, 165883, 167197, 186397, 217309, 261721, 275939, 309493, 355571, 382351, 467293, 501187, 539303, 560029, 602839, 640307, 657299, 708959, 879859, 919129, 973813, 1057741, 1085779, 1115899, 1156031, 1302667, 1366297, 1396427, 1516279, 1580461, 1760419, 1829797, 1867249, 1870021

Offset: 1

Zhi-Wei Sun, Jan 17 2014

nonn

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

			a(3) = 2861 with 2861, f(2861) = 23143, f(23143) = 240769 and f(240769) = 3117791 and f(3117791) =  48951967 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A234695, A235925, A235934.

f[n_]:=Prime[n]-n+1
p[k_]:=PrimeQ[f[Prime[k]]]&&PrimeQ[f[f[Prime[k]]]]&&PrimeQ[f[f[f[Prime[k]]]]]&&PrimeQ[f[f[f[f[Prime[k]]]]]]
n=0;Do[If[p[k],n=n+1;Print[n," ",Prime[k]]],{k,1,100000}]

A238756 Number of ordered ways to write n = k + m (k > 0 and m > 0) such that 2*k + 1, prime(prime(k)) - prime(k) + 1 and prime(prime(m)) - prime(m) + 1 are all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 05 2014

nonn

Conjecture: a(n) > 0 for all n > 1.
We have verified this for n up to 10^7.
The conjecture suggests that there are infinitely many primes p with 2*pi(p) + 1 and prime(p) - p + 1 both prime.

			a(6) = 2 since 6 = 2 + 4 with 2*2 + 1 = 5, prime(prime(2)) - prime(2) + 1 = prime(3) - 3 + 1 = 3 and prime(prime(4)) - prime(4) + 1 = prime(7) - 7 + 1 = 17 - 6 = 11 all prime, and 6 = 3 + 3 with 2*3 + 1 = 7 and prime(prime(3)) - prime(3) + 1 = prime(5) - 5 + 1 = 11 - 4 = 7 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, A218829, A234694, A234695, A235189, A236832, A237413, A238134.

p[k_]:=PrimeQ[Prime[Prime[k]]-Prime[k]+1]
a[n_]:=Sum[If[PrimeQ[2k+1]&&p[k]&&p[n-k],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]

A238814 Primes p with prime(p) - p + 1 and prime(q) - q + 1 both prime, where q is the first prime after p.

Original entry on oeis.org

2, 3, 5, 13, 41, 83, 199, 211, 271, 277, 293, 307, 349, 661, 709, 743, 751, 823, 907, 1117, 1447, 1451, 1741, 1747, 2203, 2371, 2803, 2819, 2861, 2971, 3011, 3251, 3299, 3329, 3331, 3691, 3877, 4021, 4027, 4049, 4051, 4093, 4129, 4157, 4447, 4513, 4549, 4561, 4751, 4801, 5179, 5479, 5519, 5657, 5813, 6007, 6011, 6571, 7057, 7129

Offset: 1

Zhi-Wei Sun, Mar 05 2014

nonn

Conjecture: The sequence is infinite, in other words, A234695 contains infinitely many consecutive prime pairs prime(k) and prime(k+1).

This is motivated by the comments in A238766 and A238776, and the sequence is a subsequence of A234695.

			a(1) = 2 since prime(2) - 2 + 1 = 3 - 1 = 2 and prime(3) - 3 + 1 = 5 - 2 = 3 are both prime.
a(2) = 3 since prime(3) - 3 + 1 = 5 - 2 = 3 and 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, A234694, A234695, A238766, A238776.

p[k_]:=PrimeQ[Prime[Prime[k]]-Prime[k]+1]
n=0
Do[If[p[k]&&p[k+1],n=n+1;Print[n," ",Prime[k]]],{k,1,914}]
Select[Prime[Range[1000]],AllTrue[{Prime[#]-#+1,Prime[NextPrime[#]]-NextPrime[ #]+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 24 2019 *)

step(p,k)=k++;while(k--,p=nextprime(p+1)); p
p=0;forprime(r=2,1e6,if(isprime(p++) && isprime(r-p+1), q=nextprime(p+1); if(isprime(step(r,q-p)-q+1), print1(p", ")))) \\ Charles R Greathouse IV, Mar 06 2014

A235919 a(n) = |{0 < k < n - 2: p = prime(k) + phi(n-k)/2, prime(p) - p + 1 and (p^2 - 1)/4 - prime(p) are all prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 3, 0, 1, 2, 3, 1, 0, 3, 3, 2, 1, 2, 1, 3, 2, 4, 2, 1, 6, 2, 6, 2, 3, 2, 3, 6, 2, 1, 7, 2, 5, 4, 3, 4, 3, 6, 4, 5, 4, 2, 1, 2, 8, 2, 4, 5, 5, 6, 4, 5, 4, 6, 3, 3, 5, 6, 5, 3, 4, 8, 2, 3, 7, 7, 8, 5, 5, 3, 3, 7, 9, 3, 8, 2, 4, 4, 4, 9, 2, 5, 8, 5, 5

Offset: 1

Zhi-Wei Sun, Jan 17 2014

nonn

Conjecture: a(n) > 0 for all n > 24.

This implies that there are infinitely many primes p with prime(p) - p + 1 and (p^2 - 1)/4 - prime(p) both prime.

			 a(30) = 1 since prime(6) + phi(24)/2 = 13 + 4 = 17, prime(17) - 16 = 59 - 16 = 43 and (17^2 - 1)/4 - prime(17) = 72 - 59 = 13 are all prime.
  a(35) = 1 since prime(19) + phi(16)/2 = 67 + 4 = 71, prime(71) - 70 = 353 - 70 = 283 and (71^2 - 1)/4 - prime(71) = 1260 - 353 = 907 are all prime.

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

Cf. A000010, A000040, A234695, A235917.

PQ[n_]:=PQ[n]=n>0&&PrimeQ[n]
p[n_]:=p[n]=PrimeQ[n]&&PrimeQ[Prime[n]-n+1]&&PQ[(n^2-1)/4-Prime[n]]
f[n_,k_]:=f[n,k]=Prime[k]+EulerPhi[n-k]/2
a[n_]:=a[n]=Sum[If[p[f[n,k]],1,0],{k,1,n-3}]
Table[a[n],{n,1,100}]

A235984 Primes p with f(p), f(f(p)), f(f(f(p))), f(f(f(f(p)))), f(f(f(f(f(p))))) all prime, where f(n) = prime(n) - n + 1.

Original entry on oeis.org

2, 3, 501187, 560029, 2076881, 2836003, 2907011, 8254787, 8822347, 10322189, 11329181, 11354641, 12307693, 14528069, 15801601, 17757427, 19023091, 24995669, 25871971

Offset: 1

Zhi-Wei Sun, Jan 17 2014

nonn

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

			a(3) = 501187 with 501187, f(501187) = 6886357, f(6886357) = 113948711, f(113948711) = 2224096873, f(2224096873) =  50351471977 and f(50351471977) = 1303792228393 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..19
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A234694, A234695, A235924, A235925, A235934, A235935.

f[n_]:=Prime[n]-n+1
p[k_]:=PrimeQ[f[Prime[k]]]&&PrimeQ[f[f[Prime[k]]]]&&PrimeQ[f[f[f[Prime[k]]]]]&&PrimeQ[f[f[f[f[Prime[k]]]]]]&&PrimeQ[f[f[f[f[f[Prime[k]]]]]]]
n=0;Do[If[p[k],n=n+1;Print[n," ",Prime[k]]],{k,1,10^7}]

A236066 Primes p with g(p), g(g(p)), g(g(g(p))), g(g(g(g(p)))), g(g(g(g(g(p))))) all prime, where g(n) = prime(n) - n - 1.

Original entry on oeis.org

5, 98893, 1110709, 4231849, 5319707, 6763349, 7904087, 10823431, 13893109, 15323939, 15544079, 15716713, 17642899, 18978439, 20126237

Offset: 1

Zhi-Wei Sun, Jan 18 2014

nonn

Conjecture: For any integer m > 1, there are infinitely many chains p(1) < ... < p(m) of m primes with p(k+1) = prime(p(k)) - p(k) - 1 for all 0 < k < m.

This is similar to the conjecture in A235925.

			a(1) = 5 since neither g(2) = prime(2) - 2 - 1 = 0 nor g(3) = prime(3) - 3 - 1 = 1 is prime, but 5 = g(5) = g(g(5)) =  g(g(g(5))) = g(g(g(g(5)))) = g(g(g(g(g(5))))) is prime.
a(2) = 98893 with 98893, g(98893) = 1185113, g(1185113) = 17381209, g(17381209) = 304696943, g(304696943) = 6262760333, g(6262760333) = 148561011217 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..15

Cf. A000040, A234695, A235925, A235934, A235935, A235984.

g[n_]:=Prime[n]-n-1
p[k_]:=PrimeQ[g[Prime[k]]]&&PrimeQ[g[g[Prime[k]]]]&&PrimeQ[g[g[g[Prime[k]]]]]&&PrimeQ[g[g[g[g[Prime[k]]]]]]&&PrimeQ[g[g[g[g[g[Prime[k]]]]]]]
n=0;Do[If[p[k],n=n+1;Print[n," ",Prime[k]]],{k,1,10^6}]

A236143 Odd primes p with prime(p-1) - (p-1) and prime(p-1) - 2*prime((p-1)/2) both prime.

Original entry on oeis.org

7, 11, 31, 67, 179, 193, 197, 281, 347, 349, 563, 599, 757, 1123, 1453, 1543, 1933, 1987, 2083, 2531, 2971, 3037, 3259, 3547, 3583, 3701, 3919, 4027, 4483, 5023, 5581, 5591, 5647, 5981, 6449, 7207, 7297, 7603, 8291, 9049

Offset: 1

Zhi-Wei Sun, Jan 19 2014

nonn

By part (i) of the conjecture in A236138, this sequence should have infinitely many terms.

			a(1) = 7 with prime(6) - 6 = 13 - 6 = 7 and prime(6) - 2*prime(3) = 13 - 2*5 = 3 both prime.

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

Cf. A000040, A234695, A235925, A236075, A236119, A236138.

PQ[n_]:=n>0&&PrimeQ[n]
p[n_]:=PrimeQ[Prime[n-1]-(n-1)]&&PQ[Prime[n-1]-2*Prime[(n-1)/2]]
n=0;Do[If[p[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,2,10^5}]

s=[]; forprime(p=3, 10000, if(isprime(prime(p-1)-(p-1)) && isprime(prime(p-1)-2*prime((p-1)/2)), s=concat(s, p))); s \\ Colin Barker, Jan 19 2014

cp's OEIS Frontend

A234850 Primes in A014692, i.e., of the form prime(k)-k+1, for some k.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

A234852 Indices k of primes p=prime(k) such that prime(p)-p+1 is prime.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A235508 Number of ways to write 2*n = p + q with q > 0 such that p, p*(p+1) - prime(p) and prime(q) - q + 1 are all prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A235935 Primes p with f(p), f(f(p)), f(f(f(p))), f(f(f(f(p)))) all prime, where f(n) = prime(n) - n + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A238756 Number of ordered ways to write n = k + m (k > 0 and m > 0) such that 2*k + 1, prime(prime(k)) - prime(k) + 1 and prime(prime(m)) - prime(m) + 1 are all prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A238814 Primes p with prime(p) - p + 1 and prime(q) - q + 1 both prime, where q is the first prime after p.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A235919 a(n) = |{0 < k < n - 2: p = prime(k) + phi(n-k)/2, prime(p) - p + 1 and (p^2 - 1)/4 - prime(p) are all prime}|, where phi(.) is Euler's totient function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A235984 Primes p with f(p), f(f(p)), f(f(f(p))), f(f(f(f(p)))), f(f(f(f(f(p))))) all prime, where f(n) = prime(n) - n + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A235508 Number of ways to write 2n = p + q with q > 0 such that p, p(p+1) - prime(p) and prime(q) - q + 1 are all prime.