cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A309264 Numbers k such that s + t = k with 0 < s < t where t and t - s are both prime.

Original entry on oeis.org

4, 7, 8, 9, 11, 12, 15, 17, 19, 20, 21, 23, 24, 25, 27, 29, 31, 32, 33, 35, 36, 39, 41, 43, 44, 45, 47, 49, 51, 53, 55, 56, 57, 59, 60, 61, 63, 65, 67, 69, 71, 72, 73, 75, 77, 79, 80, 81, 83, 84, 85, 87, 89, 91, 92, 93, 95, 97, 99, 101, 103, 104, 105, 107
Offset: 1

Views

Author

Wesley Ivan Hurt, Jul 19 2019

Keywords

Examples

			4 is in the sequence since there are numbers, s=1 and t=3, that satisfy s + t = 4, where s < t, t = 3 (prime) and t - s = 3 - 1 = 2 (prime).
7 is in the sequence since there are numbers, s=2 and t=5 that satisfy s + t = 7, where s < t, t = 5 (prime) and t - s = 5 - 2 = 3 (prime).

Crossrefs

Cf. A309152, A309265.

Programs

  • Mathematica
    Flatten[Table[If[Sum[(PrimePi[n - i] - PrimePi[n - i - 1]) (PrimePi[n - 2 i] - PrimePi[n - 2 i - 1]), {i, Floor[n/2]}] > 0, n, {}], {n, 100}]]
  • PARI
    isok(k) = {forprime (t=1, k, if (((s = k - t) < t) && (s > 0) && isprime(t-s), return (1)););} \\ Michel Marcus, Jul 20 2019

A309265 Numbers k such that s + t = k with 0 < s < t where s and t-s are both prime.

Original entry on oeis.org

6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 21, 23, 24, 25, 27, 28, 29, 31, 33, 35, 36, 37, 39, 40, 41, 43, 45, 47, 48, 49, 51, 53, 55, 57, 59, 60, 61, 63, 64, 65, 67, 69, 71, 73, 75, 76, 77, 79, 81, 83, 84, 85, 87, 88, 89, 91, 93, 95, 96, 97, 99, 101, 103, 105
Offset: 1

Views

Author

Wesley Ivan Hurt, Jul 19 2019

Keywords

Comments

Essentially the same as A210147 with s=p, t-s=q. - R. J. Mathar, Aug 09 2019

Examples

			6 is in the sequence since there are numbers s=2 and t=4 such that s + t = 6 with s < t, and where s=2 and t-s = 4-2 = 2 are both prime.
7 is in the sequence since there are numbers s=3 and t=5 such that s + t = 7 with s < t and where s=3 and t-s = 5-3 = 2 are both prime.

Crossrefs

Cf. A309152, A309264.

Programs

  • Mathematica
    Flatten[Table[If[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - 2 i] - PrimePi[n - 2 i - 1]), {i, Floor[n/2]}] > 0, n, {}], {n, 100}]]
  • PARI
    isok(k) = {forprime (s=1, k, if (((t = k - s) > s) && isprime(t-s), return (1)););} \\ Michel Marcus, Jul 20 2019

A309277 Sums of two primes whose difference is squarefree.

Original entry on oeis.org

5, 7, 8, 9, 12, 15, 16, 19, 20, 21, 24, 25, 28, 32, 33, 36, 39, 40, 43, 44, 45, 48, 52, 55, 56, 60, 61, 63, 64, 68, 69, 72, 73, 75, 76, 80, 81, 84, 88, 91, 92, 96, 99, 100, 104, 105, 108, 109, 111, 112, 115, 116, 120, 124, 128, 132, 133, 136, 140, 141, 144
Offset: 1

Views

Author

Wesley Ivan Hurt, Jul 20 2019

Keywords

Examples

			5 is in the sequence since 5 = 2 + 3 (both prime) and 3 - 2 = 1 is squarefree.
8 is in the sequence since 8 = 5 + 3 (both prime) and 5 - 3 = 2 is squarefree.

Crossrefs

Cf. A309152.

Programs

  • Mathematica
    Flatten[Table[If[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i] - PrimePi[n - i - 1]) MoebiusMu[n - 2 i]^2, {i, Floor[(n - 1)/2]}] > 0, n, {}], {n, 150}]]

A309304 Sums of two primes whose difference is semiprime.

Original entry on oeis.org

10, 13, 16, 18, 19, 20, 24, 25, 28, 30, 32, 36, 39, 40, 42, 43, 44, 48, 52, 55, 60, 61, 64, 68, 69, 72, 73, 78, 80, 81, 84, 88, 90, 91, 92, 96, 99, 100, 108, 112, 115, 120, 128, 132, 133, 138, 140, 144, 152, 156, 159, 162, 165, 168, 172, 180, 181, 184, 192
Offset: 1

Views

Author

Wesley Ivan Hurt, Jul 21 2019

Keywords

Examples

			10 is in the sequence since 10 = 3 + 7 (both prime) and 7 - 3 = 4 is semiprime.
13 is in the sequence since 13 = 2 + 11 (both prime) and 11 - 2 = 9 is semiprime.
16 is in the sequence since 16 = 3 + 13 (both prime) and 13 - 3 = 10 is semiprime.
18 is in the sequence since 18 = 7 + 11 (both prime) and 11 - 7 = 4 is semiprime.

Crossrefs

Cf. A309152.

Programs

  • Mathematica
    Flatten[Table[If[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i] - PrimePi[n - i - 1]) (KroneckerDelta[PrimeOmega[n - 2 i], 2]), {i, Floor[(n - 1)/2]}] > 0, n, {}], {n, 200}]]

A309305 Sums of two primes whose difference is a nonzero square.

Original entry on oeis.org

5, 10, 13, 18, 22, 30, 42, 46, 50, 58, 70, 78, 82, 85, 90, 98, 102, 106, 110, 114, 122, 126, 130, 138, 142, 150, 154, 158, 162, 170, 174, 178, 182, 190, 198, 202, 210, 218, 222, 229, 234, 238, 242, 246, 250, 258, 262, 270, 278, 282, 290, 294, 298, 302, 310
Offset: 1

Views

Author

Wesley Ivan Hurt, Jul 21 2019

Keywords

Examples

			5 is in the sequence since 5 = 2 + 3 (both prime) and 3 - 2 = 1 is a nonzero square.
10 is in the sequence since 10 = 3 + 7 (both prime) and 7 - 3 = 4 is a nonzero square.
13 is in the sequence since 13 = 2 + 11 (both prime) and 11 - 2 = 9 is a nonzero square.
18 is in the sequence since 7 + 11 (both prime) and 11 - 7 = 4 is a nonzero square.

Crossrefs

Cf. A000040, A000290, A309152.

Programs

  • Mathematica
    Flatten[Table[If[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i] - PrimePi[n - i - 1]) (Floor[Sqrt[n - 2 i]] - Floor[Sqrt[n - 2 i - 1]]), {i, Floor[(n - 1)/2]}] > 0, n, {}], {n, 300}]]
Showing 1-5 of 5 results.

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