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 1-8 of 8 results.

A220431 Number of ways to write n=x+y (x>0, y>0) with 3x-1, 3x+1 and xy-1 all prime.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 2, 6, 1, 3, 6, 4, 3, 3, 2, 3, 4, 3, 4, 2, 3, 3, 5, 4, 4, 7, 1, 2, 5, 1, 5, 7, 4, 2, 3, 7, 4, 7, 2, 4, 7, 4, 4, 5, 2, 5, 8, 4, 3, 3, 5, 2, 8, 5, 4, 3, 10, 7, 8, 2, 3, 5, 5, 3, 6, 3, 3, 14, 4, 3, 12, 3, 7, 7, 5, 6, 8, 7, 5, 9, 9, 4, 4, 3, 6, 10, 8
Offset: 1

Views

Author

Zhi-Wei Sun, Dec 14 2012

Keywords

Comments

Conjecture: a(n)>0 for all n>3.
This has been verified for n up to 10^8, and it is stronger than A. Murthy's conjecture related to A109909.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Jul 26 2023
The conjecture implies the twin prime conjecture for the following reason: If x_1<...
Zhi-Wei Sun also made some similar conjectures. For example, any integer n>2 not equal to 63 can be written as x+y (x>0, y>0) with 2x-1, 2x+1 and 2xy+1 all prime.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Jul 26 2023

Examples

			a(22)=1 since 22=4+18 with 3*4-1, 3*4+1 and 4*18-1 all prime.

Links

Crossrefs

Cf. A001359, A006512, A220419, A220413, A173587, A220272, A219842, A219864, A219923.

Programs

  • Mathematica
    a[n_]:=a[n]=Sum[If[PrimeQ[3k-1]==True&&PrimeQ[3k+1]==True&&PrimeQ[k(n-k)-1]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,1000}]

A227909 Number of ways to write 2*n = p + q with p, q and (p-1)*(q+1) - 1 all prime.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 3, 3, 1, 2, 5, 2, 3, 2, 3, 3, 5, 3, 1, 5, 4, 5, 4, 3, 4, 7, 4, 4, 2, 1, 4, 9, 2, 4, 11, 4, 2, 6, 2, 6, 11, 6, 4, 3, 3, 5, 6, 4, 3, 6, 2, 4, 10, 3, 10, 12, 7, 1, 6, 6, 5, 11, 4, 5, 6, 4, 3, 11, 2, 10, 13, 4, 6, 5, 2, 14, 13, 2, 2, 5, 5, 9, 15, 5, 3, 7, 8, 5, 3, 5, 7, 15, 3, 1, 8, 5, 7, 11, 4
Offset: 1

Views

Author

Olivier Gérard and Zhi-Wei Sun, Oct 13 2013

Keywords

Comments

Conjecture: a(n) > 0 for all n > 1.
This is stronger than Goldbach's conjecture for even numbers. It also implies A. Murthy's conjecture (cf. A109909) for even numbers.
We have verified the conjecture for n up to 2*10^7.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Jul 26 2023

Examples

			a(6) = 1 since 2*6 = 5 + 7, and (5-1)*(7+1)-1 = 31 is prime.
a(10) = 1 since 2*10 = 7 + 13, and (7-1)*(13+1)-1 = 83 is prime.
a(20) = 1 since 2*20 = 17 + 23, and (17-1)*(23+1)-1 = 383 is prime.

Links

Crossrefs

Cf. A002375, A109909, A218754, A219052, A220431, A220455, A220554, A227908, A230230.

Programs

  • Mathematica
    a[n_]:=Sum[If[PrimeQ[2n-Prime[i]]&&PrimeQ[(Prime[i]-1)(2n-Prime[i]+1)-1],1,0],{i,1,PrimePi[2n-2]}]
Table[a[n],{n,1,100}]

A230241 Number of ways to write n = p + q with p, 3*p - 10 and (p-1)*q - 1 all prime, where q is a positive integer.

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Oct 13 2013

Keywords

Comments

Conjecture: a(n) > 0 for all n > 5.
This implies A. Murthy's conjecture mentioned in A109909.
We have verified the conjecture for n up to 10^8.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Jul 29 2023

Examples

			a(9) = 1 since 9 = 7 + 2 with 7, 3*7-10 = 11, (7-1)*2-1 = 11 all prime.
a(27) = 1 since 27 = 13 + 14, and the three numbers 13, 3*13-10 = 29, (13-1)*14-1 = 167 are prime.

Links

Crossrefs

Cf. A109909, A227908, A227909, A230227, A230230.

Programs

  • Mathematica
    a[n_]:=Sum[If[PrimeQ[3Prime[i]-10]&&PrimeQ[(Prime[i]-1)(n-Prime[i])-1],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

A109907 Beginning with 7, a(n+1) = greatest prime of the form k*{a(n)-k}-1. If no prime is obtained the sequence ends at that point.

Original entry on oeis.org

7, 11, 29, 197, 9689, 23469167, 137700449916401, 4740353476794815041972197893, 5617737771240172767652929826457529708578746492288409399, 7889744416604625924469156192031986939513870147397674409917489724005347434748024264638497225986334149357868007
Offset: 0

Views

Author

Amarnath Murthy, Jul 15 2005

Keywords

Comments

Conjecture: The sequence is infinite.

Crossrefs

Cf. A109904, A109905, A026728, A109908, A109909.

Programs

  • PARI
    { b(n) = forstep(k=n\2,1,-1,if(isprime(k*(n-k)-1),return(k*(n-k)-1)));return(0) }
my(s=7); while(1,print1(s,", ");s=b(s)) \\ Max Alekseyev, Oct 04 2005

Extensions

More terms from Max Alekseyev, Oct 04 2005

A109908 a(n) = greatest prime of the form k*(n-k)-1, or 0 if no such prime exists.

Original entry on oeis.org

0, 0, 0, 3, 5, 7, 11, 11, 19, 23, 29, 31, 41, 47, 53, 59, 71, 79, 89, 83, 109, 71, 131, 139, 149, 167, 181, 191, 197, 223, 239, 251, 271, 263, 293, 307, 311, 359, 379, 383, 419, 439, 461, 479, 503, 503, 521, 571, 599, 599, 647, 659, 701, 727, 743, 719, 811, 839
Offset: 1

Views

Author

Amarnath Murthy, Jul 15 2005

Keywords

Comments

Conjecture: a(n) > 0 for n > 3.
Conjecture verified up to 10^9. - Mauro Fiorentini, Jul 23 2023

Crossrefs

Cf. A109904, A109905, A026728, A109907, A109909.

Programs

  • PARI
    { a(n)=forstep(k=n\2,1,-1,if(isprime(k*(n-k)-1),return(k*(n-k)-1)));return(0) } \\ Max Alekseyev, Oct 04 2005

Extensions

More terms from Max Alekseyev, Oct 04 2005
Definition corrected by David Wasserman, Oct 28 2008

A231633 Number of ways to write n = x + y (x, y > 0) with x^2 * y - 1 prime.

Original entry on oeis.org

0, 0, 1, 2, 3, 1, 3, 2, 5, 2, 4, 2, 7, 2, 5, 3, 5, 3, 10, 4, 5, 3, 8, 3, 14, 6, 5, 4, 11, 5, 11, 3, 11, 9, 4, 5, 10, 5, 11, 9, 12, 3, 19, 7, 11, 6, 12, 9, 11, 7, 17, 7, 13, 5, 22, 3, 3, 15, 16, 5, 25, 4, 9, 11, 13, 5, 19, 6, 22, 6, 11, 6, 39, 6, 24, 7, 7, 6, 25, 8, 21, 11, 24, 7, 31, 7, 19, 11, 33, 10, 14, 8, 15, 27, 18, 9, 21, 4, 27, 9
Offset: 1

Views

Author

Zhi-Wei Sun, Nov 12 2013

Keywords

Comments

Conjectures:
(i) a(n) > 0 for all n > 2. Also, any integer n > 4 can be written as x + y (x, y > 0) with x^2 * y + 1 prime.
(ii) Each n = 2, 3, ... can be expressed as x + y (x, y > 0) with (x*y)^2 + x*y + 1 prime.
(iii) Also, any integer n > 2 can be written as x + y (x, y > 0) with 2*(x*y)^2 - 1 (or (x*y)^2 + x*y - 1) prime.
From Mauro Fiorentini, Jul 31 2023: (Start)
Both parts of conjecture (i) verified for n up to 10^9.
Conjecture (ii) and both parts of conjecture (iii) verified for n up to 10^7. (End)

Examples

			a(6) = 1 since 6 = 4 + 2 with 4^2*2 - 1 = 31 prime.

Links

Crossrefs

Cf. A000040, A000290, A109909, A219842, A219864, A220272, A228424, A231201, A231155, A231561, A231557, A231631.

Programs

  • Mathematica
    a[n_]:=Sum[If[PrimeQ[x^2*(n-x)-1],1,0],{x,1,n-1}]
Table[a[n],{n,1,100}]

A231635 Number of ways to write n = x + y with 0 < x <= y such that lcm(x, y) + 1 is prime.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 3, 2, 1, 1, 4, 2, 6, 1, 3, 2, 8, 4, 5, 4, 3, 2, 7, 5, 6, 2, 3, 2, 8, 5, 10, 6, 3, 1, 8, 3, 9, 4, 4, 4, 14, 6, 16, 7, 7, 2, 12, 6, 8, 4, 5, 5, 21, 5, 8, 6, 4, 8, 11, 7, 12, 5, 6, 4, 10, 8, 22, 6, 10, 6, 17, 9, 23, 7, 11, 12, 18, 10, 19, 10, 10, 7, 23, 8, 15, 4, 7, 8, 14, 11, 19, 9, 2, 4, 11, 10, 35, 6, 10, 10
Offset: 1

Views

Author

Zhi-Wei Sun, Nov 12 2013

Keywords

Comments

Conjecture: (i) a(n) > 0 for all n > 1. Also, any integer n > 3 can be written as x + y (x, y > 0) with lcm(x, y) - 1 prime.
(ii) Each n = 2, 3, ... can be expressed as x + y (x, y > 0) with lcm(x, y)^2 + lcm(x, y) + 1 prime. Also, any integer n > 1 not equal to 10 can be written as x + y (x, y > 0) with lcm(x, y)^2 + 1 prime.
From Mauro Fiorentini, Aug 02 2023: (Start)
Both parts of conjecture (i) verified for n up to 10^9.
Both parts of conjecture (ii) verified for n up to 10^6. (End)

Examples

			a(9) = 1 since 9 = 3 + 6 with lcm(3, 6) + 1 = 7 prime.
a(10) = 1 since 10 = 4 + 6 with lcm(4, 6) + 1 = 13 prime.

Links

Crossrefs

Cf. A109909, A219842, A219864, A220272, A231201, A231516, A231555, A231561, A231633.

Programs

  • Mathematica
    a[n_]:=Sum[If[PrimeQ[LCM[x,n-x]+1],1,0],{x,1,n/2}]
Table[a[n],{n,1,100}]

A335984 Numbers m such that more than half the distinct positive terms of the sequence -k^2 + m*k - 1 are primes.

Original entry on oeis.org

4, 5, 7, 9, 11, 19, 21, 31, 33, 39, 49, 51, 81, 99, 101, 123, 129, 159, 171, 177, 189, 231, 291, 441, 879, 1011, 2751
Offset: 1

Views

Author

Robert Israel, Jul 03 2020

Keywords

Comments

Numbers m such that more than half the distinct terms in row m-2 of the triangular array A059036 are prime.
All positive terms of the sequence are prime for m = 1, 2, 4, 5, 9 and 21.
There are no more terms below 200000. - Pontus von Brömssen, Jul 06 2020
Numbers m such that A109909(m) > m/4. - Pontus von Brömssen, May 09 2021

Examples

			7 is in the sequence because with g(k) = -k^2+7*k-1, the positive terms of the sequence g(k) are 5=g(1), 9=g(2) and 11=g(3), and two out of the three (5 and 9) are prime.

Crossrefs

Cf. A059036, A109909.

Programs

  • Maple
    filter:= n -> nops(select(isprime, [seq(n*x-x^2-1,x=1..n/2)])) > 1/2*floor(n/2):
select(filter, [$1..10000]);
Showing 1-8 of 8 results.

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