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.

Previous Showing 11-16 of 16 results.

A234647 Primes of the form q(p) - 1, where p is a prime and q(.) is the strict partition function (A000009).

Original entry on oeis.org

2, 11, 17, 37, 53, 103, 1259, 1609, 5119, 9791, 70487, 570077, 20792119, 281138047, 23515017983, 35692320959, 48626519093, 3626048321047, 27077619952639, 1651411233432319, 10743948315198451, 13378670620050079, 39413984631175423, 58553713102334907283, 145464242180631569963, 25408177717067357968543, 1374387931601409538722802926765483199, 20557774525717988142856527912112710143, 326033386646595458662191828888146112979, 27403889354101748193301659902924397784656229
Offset: 1

Views

Author

Zhi-Wei Sun, Dec 29 2013

Keywords

Comments

Though the primes in this sequence are very rare, by the conjecture in A234615 there should be infinitely many such primes.
See A234644 for a list of known primes p with q(p) - 1 prime.

Links

Crossrefs

Cf. A000009, A000040, A234366, A234470, A234475, A234514, A234530, A234567, A234569, A234572, A234615, A234644

Programs

  • Maple
    a(1) = 2 since 2 = q(5) - 1 with 2 and 5 both prime.
  • Mathematica
    p[n_]:=A234615(n)
Table[PartitionsQ[p[n]]-1,{n,1,30}]

Formula

a(n) = A000009(A234615(n)) - 1.

A234808 a(n) = |{0 < k < n: p = k + phi(n-k) and 2*n - p are both prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 0, 3, 1, 2, 5, 2, 1, 5, 1, 2, 7, 2, 1, 4, 1, 2, 1, 4, 1, 4, 2, 4, 11, 4, 2, 3, 1, 5, 2, 3, 2, 6, 1, 5, 15, 4, 2, 9, 1, 6, 2, 5, 4, 6, 4, 4, 3, 8, 3, 6, 4, 7, 21, 2, 4, 7, 1, 7, 4, 6, 4, 6, 4, 8, 22, 7, 3, 13, 1, 10, 5, 3, 5, 7, 4, 9, 5, 10, 5, 8, 7, 7, 6, 8, 5, 6, 3, 8, 6, 7, 4, 8, 4
Offset: 1

Views

Author

Zhi-Wei Sun, Dec 30 2013

Keywords

Comments

Conjecture: a(n) > 0 except for n = 1, 8.
Clearly, this implies Goldbach's conjecture.

Examples

			a(3) = 1 since 2 + phi(1) = 3 and 2*3 - 3 = 3 are both prime.
a(20) = 1 since 11 + phi(9) = 17 and 2*20 - 17 = 23 are both prime.
a(22) = 1 since 1 + phi(21) = 13 and 2*22 - 13 = 31 are both prime.
a(24) = 1 since 9 + phi(15) = 17 and 2*24 - 17 = 31 are both prime.
a(76) = 1 since 67 + phi(9) = 73 and 2*76 - 73 = 79 are both prime.

Links

Crossrefs

Cf. A000010, A000040, A002372, A002375, A233547, A233918, A234200, A234246, A234360, A234470, A234475, A234514, A234567, A234615, A234694

Programs

  • Mathematica
    f[n_,k_]:=k+EulerPhi[n-k]
p[n_,k_]:=PrimeQ[f[n,k]]&&PrimeQ[2n-f[n,k]]
a[n_]:=a[n]=Sum[If[p[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A234963 Number of ways to write n = k + m with k > 0 and m > 2 such that C(2*sigma(k) + phi(m), sigma(k) + phi(m)/2) - 1 is prime, where sigma(k) is the sum of all positive divisors of k and phi(.) is Euler's totient function.

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Jan 01 2014

Keywords

Comments

Conjecture: a(n) > 0 for all n >= 180.
Clearly, this implies that there are infinitely many primes of the form C(2*n,n) - 1. We have verified the conjecture for n up to 10000.
Note that every n = 400, ..., 9123 can be written as k + m with k > 0 and m > 0 such that f(k, m) = sigma(k) + phi(m) is even and C(f(k, m) + 2, f(k, m)/2 + 1) + 1 is prime, but this fails for n = 9124.

Examples

			a(5) = 1 since 5 = 1 + 4 with C(2*sigma(1) + phi(4), sigma(1) + phi(4)/2) - 1 = C(4, 2) - 1 = 5 prime.
a(28) = 1 since 28 = 2 + 26 with C(2*sigma(2) + phi(26), sigma(2) + phi(26)/2) - 1 = C(18, 9) - 1 = 48619 prime.

Links

Crossrefs

Cf. A000010, A000040, A000203, A000984, A066726, A066699, A232270, A233544, A234451, A234470, A234503, A234514, A234567, A234615.

Programs

  • Mathematica
    sigma[n_] := DivisorSigma[1, n];
f[n_,k_] := Binomial[2*sigma[k] + EulerPhi[n-k], sigma[k] + EulerPhi[n-k]/2] - 1;
a[n_] := Sum[If[PrimeQ[f[n,k]], 1, 0], {k, 1, n-3}];
Table[a[n], {n, 1, 100}]

A239232 a(n) = |{0 < k <= n: p(n+k) + 1 is prime}|, where p(.) is the partition function (A000041).

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Mar 13 2014

Keywords

Comments

Conjecture: (i) a(n) > 0 for all n > 3.
(ii) If n > 15, then p(n+k) - 1 is prime for some k = 1, ..., n.
(iii) If n > 38, then p(n+k) is prime for some k = 1, ..., n.
The conjecture implies that there are infinitely many positive integers m with p(m) + 1 (or p(m) - 1, or p(m)) prime.

Examples

			a(4) = 1 since p(4+4) + 1 = 22 + 1 = 23 is prime.
a(8) = 2 since p(8+1) + 1 = 31 and p(8+2) + 1 = 43 are both prime.
a(11) = 1 since p(11+8) + 1 = 491 is prime.

Links

Crossrefs

Cf. A000040, A000041, A049575, A234470, A234569, A238393, A238457, A238509, A238516, A239207, A239209, A239214.

Programs

  • Mathematica
    p[n_]:=PartitionsP[n]
a[n_]:=Sum[If[PrimeQ[p[n+k]+1],1,0],{k,1,n}]
Table[a[n],{n,1,80}]

A234809 a(n) = |{0 < k < n: p = k + phi(n-k) and 2*(n-p) + 1 are both prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 1, 2, 1, 3, 1, 4, 1, 1, 1, 5, 3, 7, 3, 1, 1, 7, 5, 9, 4, 2, 1, 9, 5, 2, 4, 3, 1, 10, 5, 14, 2, 2, 2, 1, 6, 14, 5, 4, 1, 15, 5, 16, 5, 5, 3, 17, 8, 4, 5, 6, 3, 17, 7, 5, 2, 6, 6, 17, 11, 25, 3, 5, 3, 1, 11, 25, 4, 4, 4, 22, 10, 26, 6, 7, 8, 3, 9, 26, 7, 9, 6, 25, 8, 3, 7, 9, 10, 25, 15, 6, 2, 9, 9, 2, 13, 29, 3, 7
Offset: 1

Views

Author

Zhi-Wei Sun, Dec 30 2013

Keywords

Comments

Conjecture: a(n) > 0 for all n > 2.
Clearly, this implies Lemoine's conjecture which states that any odd number 2*n + 1 > 5 can be written as 2*p + q with p and q both prime.
See also A234808 for a similar conjecture.

Examples

			a(5) = 1 since 1 + phi(4) = 3 and 2*(5-3) + 1 = 5 are both prime.
a(16) = 1 since 7 + phi(9) = 13 and 2*(16-13) + 1 = 7 are both prime.
a(41) = 1 since 7 +phi(34) = 23 and 2*(41-23) + 1 = 37 are both prime.
a(156) = 1 since 131 + phi(25) = 151 and 2*(156-151) + 1 = 11 are both prime.

Links

Crossrefs

Cf. A000010, A000040, A046927, A234470, A234475, A234514, A234567, A234615, A234694, A234808

Programs

  • Mathematica
    f[n_,k_]:=k+EulerPhi[n-k]
p[n_,k_]:=PrimeQ[f[n,k]]&&PrimeQ[2*(n-f[n,k])+1]
a[n_]:=a[n]=Sum[If[p[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A234900 Primes p with P(p+1) also prime, where P(.) is the partition function (A000041).

Original entry on oeis.org

2, 3, 5, 131, 167, 211, 439, 2731, 3167, 3541, 4261, 7457, 8447, 18289, 22669, 23201, 23557, 35401, 44507, 76781, 88721, 108131, 126097, 127079, 136319, 141359, 144139, 159169, 164089, 177487, 202627, 261757, 271181, 282911, 291971, 307067, 320561, 389219, 481589, 482627, 602867, 624259, 662107, 682361, 818887, 907657, 914189, 964267, 1040191, 1061689
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 01 2014

Keywords

Comments

It seems that this sequence contains infinitely many terms.
See also A234569 for a similar sequence.

Examples

			a(1) = 2 since P(2+1) = 3 is prime.
a(2) = 3 since P(3+1) = 5 is prime.
a(3) = 5 since P(5+1) = 11 is prime.

Links

Crossrefs

Cf. A000040, A000041, A049575, A233346, A234470, A234514, A234530, A234567, A234569, A234615, A234644.

Programs

  • Mathematica
    p[k_]:=p[k]=PrimeQ[PartitionsP[Prime[k]+1]]
n=0;Do[If[p[k],n=n+1;Print[n," ",Prime[k]]],{k,1,10000}]
Previous Showing 11-16 of 16 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.