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 1-9 of 9 results.

A091183 Index of first occurrence of n in A091182.

Original entry on oeis.org

1, 4, 5, 13, 17, 71, 31, 53, 109, 67, 97, 143, 197, 101, 253, 223, 137, 193, 277, 319, 461, 269, 547, 367, 473, 353, 283, 529, 361, 563, 889, 431, 577, 809, 661, 739, 857, 599, 913, 1081, 899, 973, 811, 1109, 613, 647, 1061, 781, 1369, 907, 1373, 1039, 1381
Offset: 0

Views

Author

Ray Chandler, Dec 27 2003

Keywords

Crossrefs

Cf. A014574, A090695, A091182.

A219842 Number of ways to write n as x+y (0

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 4, 2, 2, 4, 2, 2, 7, 4, 1, 3, 4, 5, 5, 4, 2, 8, 2, 7, 4, 2, 8, 11, 5, 3, 8, 7, 5, 14, 7, 5, 10, 8, 7, 8, 4, 8, 9, 5, 4, 11, 6, 11, 14, 5, 3, 19, 12, 7, 11, 6, 9, 12, 13, 8, 9, 10, 12, 16, 5, 6, 22, 8, 11, 11, 5, 10, 26, 15, 5, 11, 15, 10
Offset: 1

Views

Author

Zhi-Wei Sun, Nov 29 2012

Keywords

Comments

Conjecture: a(n)>0 for all n>1. Moreover, any integer n>357 can be written as x+y (x>0, y>0) with 2x*y+1 and 2x*y-1 twin primes.
This conjecture has been verified for n up to 10^8, and it implies the twin prime conjecture.
Zhi-Wei Sun also made the following general conjecture: For each positive odd integer m, any sufficiently large integer n can be written as x+y, where x and y are positive integers with 2x*y+m and 2x*y-m both prime.
For example, when m=3,5,7,9,11 it suffices to require that n is greater than 5090, 222, 1785, 548, 603 respectively.

Examples

			a(10)=2 since 10=1+9=3+7 with 2*1*9+1=19 and 2*3*7+1=43 both prime.

Links

Crossrefs

Cf. A091182, A219838, A219782.

Programs

  • Mathematica
    a[n_] := a[n] = Sum[If[PrimeQ[2k(n-k)+1] == True, 1, 0], {k, n/2}]; Do[Print[n," ", a[n]], {n, 100}]

A219782 Number of ways to write n=x+y (0

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Nov 27 2012

Keywords

Comments

Conjecture: a(n)>0 if n is not among 1, 8, 10, 18, 20, 41, 46, 58, 78, 116, 440.
Zhi-Wei Sun also made the following general conjecture:
For any k=0,1,2,4,5,6,... and positive odd integer m, each sufficiently large integer n can be written as x+y (0
For example, if n>6 is different from 24 then n can be written as x+y with x,y positive, and xy-n and xy+n both prime; if n>308 then n can be written as x+y with x,y positive, and 3n^2-xy and 3n^2+xy both prime.

Examples

			a(9)=2 since 9=1+8=4+5 with 9^2+1*8, 9^2-1*8, 9^2+4*5, 9^2-4*5 all prime.

Links

Crossrefs

Cf. A091182, A218585, A218654.

Programs

  • Mathematica
    a[n_]:=a[n]=Sum[If[PrimeQ[n^2-k(n-k)]==True&&PrimeQ[n^2+k(n-k)]==True,1,0],{k,1,n/2}]
Do[Print[n," ",a[n]],{n,1,10000}]

A090695 Integers which are not the sum of 2 integers A and B with AB+1 and AB-1 both primes. In other words, the product cannot be the middle integer of a Twin Prime pair.

Original entry on oeis.org

1, 2, 3, 6, 10, 12, 14, 15, 20, 26, 30, 40, 45, 54, 60, 66, 75, 80, 90, 100, 105, 117, 120, 150, 180, 250, 270, 280, 290, 315, 320, 342, 360, 390, 410, 432, 440, 450, 455, 480, 495, 510, 540, 560, 590, 630, 645, 765, 810, 980, 1080, 1170, 1220, 1305, 1430, 1530, 1860, 2235, 2310, 2670, 3120
Offset: 1

Views

Author

William F. Sindelar (w_sindelar(AT)juno.com), Dec 19 2003

Keywords

Comments

Sequence may be finite. Next term after 3120 if it exists must be greater than 867750.
If the sequence can be proved to be finite, then one may surmise that there are infinitely many twin primes and that every integer greater than 3120 and every prime, except 2 and 3, is the sum of 2 integers whose product is the middle number of a twin prime pair.
If there is a term after 3120, it is larger than 4*10^9. - Giovanni Resta, Oct 31 2017

Examples

			15 is a member: 15 is the sum of these pairs of integers: (2+13) (3+12) (4+11) (5+10) (6+9) (7+8). Their products (2*13) (3*12), etc. plus and minus 1 are not primes and therefore the products cannot be the middle integers of Twin Prime sets.

Crossrefs

Cf. A014574, A091182, A091183.

Programs

  • PARI
    isok(n) = {for (a=1, n\2, ab = a*(n-a); if (isprime(ab+1) && isprime(ab-1), return (0));); return (1);} \\ Michel Marcus, Jul 12 2013

A219791 Number of ways to write n=x+y (0

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Nov 28 2012

Keywords

Comments

Conjecture: a(n)>0 if n is different from 1, 6, 16, 24.
This conjecture has been verified for n up to 10^7. It implies that there are infinitely many primes of the form x^2+1.
Zhi-Wei Sun also made the following general conjecture: For any positive integer k, each sufficiently large integer n cna be written as x+y (x>0, y>0) with (xy)^{2^k}+1 prime.
For example, for k=2,3,4 it suffices to require that n is greater than 22, 386, 748 respectively.

Examples

			a(8)=1 since 8=4+4 with (4*4)^2+1=257 prime.
a(9)=2 since 9=2+7=4+5, and (2*7)^2+1=197 and (4*5)^2+1=401 are prime.

Links

Crossrefs

Cf. A091182, A219782.

Programs

  • Mathematica
    a[n_] := a[n] = Sum[If[PrimeQ[(k(n-k))^2+1] == True, 1, 0], {k, n/2}]; Do[Print[n, " ", a[n]], {n, 100}]

A219838 Number of ways to write n as x + y with 0 < x <= y and (xy)^2 + xy + 1 prime.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 2, 3, 2, 1, 2, 1, 2, 3, 1, 1, 3, 2, 2, 6, 4, 1, 3, 3, 4, 4, 4, 4, 4, 2, 2, 5, 4, 2, 4, 2, 4, 3, 5, 5, 8, 1, 2, 6, 2, 4, 13, 1, 8, 8, 3, 3, 9, 5, 4, 8, 5, 3, 9, 5, 4, 17, 9, 2, 6, 5, 5, 9, 10, 7, 13, 5, 3, 6, 12, 8, 10, 6, 5, 8, 10, 11, 12, 9, 10, 8, 6, 6, 11, 7, 11, 5, 5, 4, 15, 14, 12, 14, 9
Offset: 1

Views

Author

Zhi-Wei Sun, Nov 29 2012

Keywords

Comments

Conjecture: a(n) > 0 for all n > 1.
This has been verified for n up to 10^8. It implies that there are infinitely many primes of the form x^2 + x + 1.
The author also guesses that any integer n > 1157 can be written as x + y with x and y positive integers, and (x*y)^2 + x*y + 1 and (x*y)^2 + x*y - 1 twin primes.
Zhi-Wei Sun has made the following general conjecture: For each prime p, any sufficiently large integer n can be written as x + y, where x and y are positive integers with ((x*y)^p - 1)/(x*y - 1) prime. (For p = 5, 7 it suffices to require n > 28 and n > 46 respectively.)
Compare this with Sun's another conjecture related to A219791.

Examples

			a(49) = 1 since 49 = 3 + 46 with (3*46)^2 + 3*46 + 1 = 19183 prime.

Links

Crossrefs

Cf. A219791, A091182, A219782.

Programs

  • Mathematica
    a[n_] := a[n] = Sum[If[PrimeQ[k(n - k)(k(n - k) + 1) + 1] == True, 1, 0], {k, 1, n/2}]; Do[Print[n, " ", a[n]], {n, 1, 10000}]

A091417 Integers n which are not the difference of 2 integers A and B where A is less than or equal to n such that AB+1 and AB-1 are both primes. In other words, the product A*B cannot be the middle integer of a twin prime pair.

Original entry on oeis.org

8, 9, 10, 15, 18, 20, 30, 40, 51, 58, 60, 65, 75, 96, 100, 110, 120, 170, 195, 240, 250, 260, 315, 330, 345, 390, 540, 630, 750, 780, 910, 1020, 1720, 1890
Offset: 1

Views

Author

William F. Sindelar (w_sindelar(AT)juno.com), Jan 05 2004

Keywords

Comments

Sequence may be finite. Next term after 1890 if it exists must be greater than 867750.

Examples

			8 is a member: 8 is the difference between these pairs of integers: (9-1) (10-2) (11-3) (12-4) (13-5) (14-6) (15-7) (16-8). Their products (9*1) (10*2), etc. plus and minus 1 are not primes and therefore the products cannot be the middle integers of twin prime sets.

Crossrefs

Cf. A090695, A091182, A091183. A091418.

A091418 Integers n which are not the sum or difference of 2 integers A and B where A is less than or equal to n such that AB+1 and AB-1are both primes. In other words, the product A*B cannot be the middle integer of a twin prime pair.

Original entry on oeis.org

10, 15, 20, 30, 40, 60, 75, 100, 120, 250, 315, 390, 540, 630
Offset: 1

Views

Author

William F. Sindelar (w_sindelar(AT)juno.com), Jan 05 2004

Keywords

Comments

Sequence may be finite. Next term after 630 if it exists must be greater than 867750.
The terms in this sequence are those which are common to A090695 and A091417.

Examples

			See A090695 and A091417 for examples.

Crossrefs

Cf. A090695, A091417, A091182, A091183.

A219789 Least prime in the form x*y-1 with x > 0, y > 0 and x + y = n > 3.

Original entry on oeis.org

2, 3, 7, 5, 11, 7, 23, 17, 19, 11, 23, 13, 47, 29, 31, 17, 83, 19, 71, 41, 43, 23, 47, 71, 131, 53, 103, 29, 59, 31, 167, 149, 67, 101, 71, 37, 383, 113, 79, 41, 83, 43, 167, 89, 251, 47, 263, 97, 191, 101, 103, 53, 107, 109, 311, 113, 223, 59, 431, 61, 239, 353
Offset: 4

Views

Author

Zhi-Wei Sun, Nov 28 2012

Keywords

Comments

See comments in A091182, A219782 and A219791.

Examples

			a(6)=7 since 6=2+4 with 2*4-1=7 prime.

Links

Crossrefs

Cf. A091182, A219782, A219791.

Programs

  • Mathematica
    Do[Do[If[PrimeQ[k(n-k)-1]==True, Print[n," ",k(n-k)-1]; Goto[aa]], {k,1,n/2}]; Print[n," ",counterexample]; Label[aa]; Continue, {n,4,100}]
Table[k = 1; While[p = k*(n - k) - 1; k <= n/2 && ! PrimeQ[p], k++]; If[k > n/2, Print["counterexample at n=", n]; p = 0]; p, {n, 4, 100}] (* T. D. Noe, Nov 28 2012 *)
Showing 1-9 of 9 results.

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