A232353 -id:A232353 - cp's OEIS frontend

A235661 Primes p with p*(p+1) - prime(p) prime.

2, 3, 5, 11, 19, 29, 37, 41, 53, 61, 71, 89, 131, 137, 149, 157, 233, 263, 271, 281, 293, 331, 337, 359, 389, 431, 433, 439, 457, 487, 499, 571, 617, 631, 659, 701, 739, 751, 761, 809, 859, 877, 907, 911, 1009, 1019, 1031, 1033, 1087, 1093

Offset: 1

Zhi-Wei Sun, Jan 13 2014

nonn

This sequence is a subsequence of A235592.

By the conjecture in A232353, this sequence should have infinitely many terms.

			2 is a term because 2*3 - prime(2) = 3 is prime.
3 is a term because 3*4 - prime(3) = 7 is prime.
5 is a term because 5*6 - prime(5) = 19 is prime.

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

Cf. A000040, A232353, A234695, A235592, A235613, A235614.

PQ[n_]:=PrimeQ[n(n+1)-Prime[n]]
n=0;Do[If[PQ[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,1000}]
Select[Prime[Range[200]],PrimeQ[#(#+1)-Prime[#]]&] (* Harvey P. Dale, Feb 26 2025 *)

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}]

A235727 Odd primes p with (p^2 - 1)/4 - prime((p - 1)/2) and (p^2 - 1)/4 + prime((p - 1)/2) both prime.

Original entry on oeis.org

7, 11, 19, 23, 41, 73, 83, 109, 197, 211, 229, 271, 379, 461, 541, 631, 641, 659, 859, 991, 1031, 1049, 1051, 1093, 1103, 1217, 1429, 1451, 1879, 2063, 2131, 2287, 2341, 2411, 3019, 3257, 3461, 3659, 3673, 3691, 3709, 3917, 3967, 4409, 4463, 4519, 5279, 5303, 5471, 5477

Offset: 1

Zhi-Wei Sun, Jan 15 2014

nonn

By the conjecture in A235728, this sequence should have infinitely many terms.

			a(1) = 7 since neither (3^2-1)/4 - prime((3-1)/2) = 1 nor (5^2-1)/4 + prime((5-1)/2) = 9 is prime, but (7^2-1)/4 - prime((7-1)/2) = 12 - 5 = 7 and (7^2-1)/4 + prime((7-1)/2) = 12 + 5 = 17 are both prime.

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

Cf. A000040, A232353, A235592, A235661, A235681, A235682, A235703, A235728.

q[n_]:=q[n]=PrimeQ[n(n+1)-Prime[n]]&&PrimeQ[n(n+1)+Prime[n]]
n=0;Do[If[q[(Prime[k]-1)/2],n=n+1;Print[n," ",Prime[k]]],{k,2,1000}]

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}]

cp's OEIS Frontend

A235661 Primes p with p*(p+1) - prime(p) prime.

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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.

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A235727 Odd primes p with (p^2 - 1)/4 - prime((p - 1)/2) and (p^2 - 1)/4 + prime((p - 1)/2) both prime.

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Mathematica

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

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica