A235727 -id:A235727 - cp's OEIS frontend

A235728 a(n) = |{0 < k < n - 2: 2m + 1, m(m+1) - prime(m) and m*(m+1) + prime(m) are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.

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

Offset: 1

Zhi-Wei Sun, Jan 15 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 5, and a(n) = 1 only for n = 191.
(ii) If n > 8 is not equal to 32, then there is a positive integer k < n - 2 such that 2*m + 1, m*(m+1) - prime(m) and m*(m+1) + prime(m) are all prime, where m = sigma(k) + phi(n-k)/2, and sigma(k) is the sum of all positive divisors of k.
(iii) If n > 444, then there is a positive integer k < n such that 2*m + 1, m^2 - prime(m) and m^2 + prime(m) are all prime, where m = sigma(k) + phi(n-k).
Clearly, part (i) of the conjecture implies that there are infinitely many odd primes p = 2*m + 1 with m*(m+1) - prime(m) = (p^2-1)/4 - prime((p-1)/2) and m*(m+1) + prime(m) = (p^2-1)/4 + prime((p-1)/2) both prime.

			a(6) = 2 since phi(1) + phi(5)/2 = phi(3) + phi(3)/2 = 3 with 2*3 + 1 = 7, 3*4 - prime(3) = 7 and 3*4 + prime(3) = 17 all prime.
a(191) = 1 since phi(153) + phi(38)/2 = 105 with 2*105 + 1 = 211, 105*106 - prime(105) = 11130 - 571 = 10559 and 105*106 + prime(105) = 11130 + 571 = 11701 all prime.

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

Cf. A000010, A000040, A235508, A235592, A235682, A235703, A235727.

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

A235806 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, 29, 41, 43, 53, 59, 89, 109, 139, 179, 181, 229, 379, 401, 421, 431, 541, 587, 659, 811, 991, 1069, 1103, 1117, 1231, 1259, 1459, 1471, 1619, 1709, 1831, 1951, 2179, 2791, 2797, 2833, 2851, 3001, 3391, 3571, 3617, 3631, 3637, 3671, 3793, 3863, 3929, 3967

Offset: 1

Zhi-Wei Sun, Jan 16 2014

nonn

By the conjecture in A235805, this sequence should have infinitely many terms.

			a(1) = 7 since neither (3^2-1)/4 - prime((3-1)/2) = 0 nor (5^2-1)/4 - prime((5+1)/2) = 1 is prime, but (7^2-1)/4 - prime((7-1)/2) = 12 - 5 = 7 and (7^2-1)/4 - prime((7+1)/2) = 12 - 7 = 5 are both prime.

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

Cf. A000040, A235592, A235727, A235805.

q[n_]:=PrimeQ[n(n+1)-Prime[n]]&&PrimeQ[n(n+1)-Prime[n+1]]
n=0;Do[If[q[(Prime[k]-1)/2],n=n+1;Print[n," ",Prime[k]]],{k,2,1000}]
Select[Prime[Range[2,600]],AllTrue[(#^2-1)/4-{Prime[(#-1)/2],Prime[ (#+1)/2]},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 05 2020 *)

A235914 Odd primes p = 2m + 1 with m(m-1) - prime(m) and m*(m+1) - prime(m) both prime.

Original entry on oeis.org

13, 17, 23, 29, 31, 43, 73, 89, 181, 229, 313, 367, 379, 557, 631, 683, 1021, 1069, 1093, 1151, 1303, 1459, 1471, 1663, 1733, 1831, 1871, 2411, 2473, 2791, 2843, 2887, 3673, 3691, 3793, 3797, 3863, 4001, 4139, 4261, 5261, 5431, 6091, 6301, 6661, 6737, 6883, 7489, 7523, 7873

Offset: 1

Zhi-Wei Sun, Jan 16 2014

nonn

By the conjecture in A235912, this sequence should have infinitely many terms.

			a(1) = 13 since none of 1*2 - prime(1) = 0, 1*2 - prime(2) = -1, 2*3 - prime(3) = 1 and 2*4 + 1 = 9 = 4*5 - prime(5) is prime, but 2*6 + 1 = 13, 5*6 - prime(6) = 30 - 13 = 17 and 6*7 - prime(6) = 42 - 13 = 29 are all prime.

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

Cf. A000040, A235727, A235912.

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

A235920 Primes p with prime(p) - p + 1 and (p^2 - 1)/4 - prime(p) both prime.

Original entry on oeis.org

17, 31, 41, 43, 61, 71, 83, 103, 109, 173, 181, 211, 271, 349, 353, 541, 661, 673, 743, 811, 911, 953, 971, 1171, 1429, 1471, 1483, 1723, 1787, 2053, 2203, 2579, 2749, 3019, 3299, 3391, 3433, 3463, 3727, 3917, 4003, 4021, 4049, 4243, 4447, 4567, 4657, 4729, 4801, 4993

Offset: 1

Zhi-Wei Sun, Jan 17 2014

nonn

By the conjecture in A235919, this sequence should have infinitely many terms.

			a(1) = 17 with prime(17) - 17 + 1 =   59 - 16 = 43 and (17^2 - 1)/4 - prime(17) = 72 - 59 = 13 both prime.

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

Cf. A000040, A234695, A235727, A235806, A235914, A235919.

PQ[n_]:=n>0&&PrimeQ[n]
p[n_]:=PrimeQ[Prime[n]-n+1]&&PQ[(n^2-1)/4-Prime[n]]
n=0;Do[If[p[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,1000}]

cp's OEIS Frontend

A235728 a(n) = |{0 < k < n - 2: 2*m + 1, m*(m+1) - prime(m) and m*(m+1) + prime(m) are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.

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A235806 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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A235914 Odd primes p = 2*m + 1 with m*(m-1) - prime(m) and m*(m+1) - prime(m) both prime.

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A235920 Primes p with prime(p) - p + 1 and (p^2 - 1)/4 - prime(p) both prime.

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A235728 a(n) = |{0 < k < n - 2: 2m + 1, m(m+1) - prime(m) and m*(m+1) + prime(m) are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.

A235914 Odd primes p = 2m + 1 with m(m-1) - prime(m) and m*(m+1) - prime(m) both prime.