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.

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A238458 Number of primes p < n with 2*P(n-p) + 1 prime, where P(.) is the partition function (A000041).

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Feb 27 2014

Keywords

Comments

Conjecture: a(n) > 0 for all n > 2. Also, for each n > 3 there is a prime p < n with 2*P(n-p) - 1 prime.
We have verified the conjecture for n up to 10^5.
See also A238459 for a similar conjecture involving the strict partition function.

Examples

			a(3) = 1 since 2 and 2*P(3-2) + 1 = 2*1 + 1 = 3 are both prime.
a(41) = 1 since 37 and 2*P(41-37) + 1 = 2*5 + 1 = 11 are both prime.

Links

Crossrefs

Cf. A000040, A000041, A237705, A237768, A237769, A238457, A238459.

Programs

  • Mathematica
    p[n_,k_]:=PrimeQ[2*PartitionsP[n-Prime[k]]+1]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

A238459 Number of primes p < n with q(n-p) + 1 prime, where q(.) is the strict partition function (A000009).

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Feb 27 2014

Keywords

Comments

Conjecture: a(n) > 0 for all n > 2. Also, for each n > 6 there is a prime p < n with q(n-p) - 1 prime.
We have verified the conjecture for n up to 10^5.
See also A238458 for a similar conjecture involving the partition function p(n).

Examples

			a(3) = 1 since 2 and q(3-2) + 1 = 1 + 1 = 2 are both prime.
a(28) = 1 since 17 and q(28-17) + 1 = q(11) + 1 = 12 + 1 = 13 are both prime.

Links

Crossrefs

Cf. A000009, A000040, A237705, A237768, A237769, A238457, A238458.

Programs

  • Mathematica
    q[n_,k_]:=PrimeQ[PartitionsQ[n-Prime[k]]+1]
a[n_]:=Sum[If[q[n,k],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

A238646 Number of primes p < n such that the number of squarefree numbers among 1, ..., n-p is prime.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 2, 2, 3, 1, 2, 1, 3, 1, 3, 1, 4, 2, 3, 2, 5, 4, 5, 1, 3, 3, 4, 2, 5, 3, 4, 5, 8, 3, 5, 1, 5, 5, 7, 3, 5, 2, 6, 3, 6, 6, 9, 4, 8, 7, 7, 6, 7, 4, 6, 7, 8, 5, 6, 4, 7, 8, 9, 6, 6, 6, 9, 5, 7, 4, 8, 6, 10, 6, 5, 8, 11, 7, 10, 6
Offset: 1

Views

Author

Zhi-Wei Sun, Mar 02 2014

Keywords

Comments

Conjecture: a(n) > 0 for all n > 3, and a(n) = 1 only for n = 4, 10, 12, 14, 16, 24, 36.
This is analog of the conjecture in A237705 for squarefree numbers.
We have verified the conjecture for n up to 60000.

Examples

			a(10) = 1 since 7 and 3 are both prime, and there are exactly 3 squarefree numbers among 1, ..., 10-7.
a(36) = 1 since 17 and 13 are both prime, and there are exactly 13 squarefree numbers among 1, ..., 36-17 (namely, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19).

Links

Crossrefs

Cf. A000040, A005117, A013928, A237705, A237768, A237769, A238645.

Programs

  • Mathematica
    s[n_]:=Sum[If[SquareFreeQ[k],1,0],{k,1,n}]
a[n_]:=Sum[If[PrimeQ[s[n-Prime[k]]],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,80}]
Previous Showing 11-13 of 13 results.

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