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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A002126 Number of solutions to n=p+q where p and q are primes or zero.

Original entry on oeis.org

1, 0, 2, 2, 1, 4, 1, 4, 2, 2, 3, 2, 2, 4, 3, 2, 4, 2, 4, 4, 4, 2, 5, 2, 6, 2, 5, 0, 4, 2, 6, 4, 4, 2, 7, 0, 8, 2, 3, 2, 6, 2, 8, 4, 6, 2, 7, 2, 10, 2, 8, 0, 6, 2, 10, 2, 6, 0, 7, 2, 12, 4, 5, 2, 10, 0, 12, 2, 4, 2, 10, 2, 12, 4, 9, 2, 10, 0, 14, 2, 8, 2, 9, 2, 16, 2, 9, 0, 8, 2, 18, 2, 8, 0, 9, 0, 14
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Arises in studying the Goldbach conjecture.

References

  • P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. [Coll. Papers, Vol. II, pp. 354-382] [The sequence N_{n,2}]
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002375, A045917, A061358, A073610

Programs

  • PARI
    (a(n) = sum(k=0, n, zp(k)*zp(n-k))); {zp(n) = if( n==0, 1, isprime(n))}; /* Michael Somos, Jul 26 1999 */

Formula

G.f.: (1 + Sum_i x^prime(i))^2. [Corrected by T. D. Noe, Dec 05 2006]

Extensions

a(54) corrected by Paul Zimmermann, Mar 15 1996
Better description from Michael Somos, Jul 26 1999

A045919 Partial sum of Goldbach numbers A045917.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 10, 12, 14, 17, 20, 23, 25, 28, 30, 34, 38, 40, 43, 47, 50, 54, 59, 63, 66, 71, 74, 78, 84, 87, 92, 98, 100, 105, 111, 116, 121, 128, 132, 137, 145, 150, 154, 163, 167, 172
Offset: 1

Views

Author

Felice Russo

Keywords

Comments

If the Goldbach conjecture is false then a(n)=a(n+1) for some n.

References

  • C. Clawson, Mathematical mysteries, Plenum Press 1996, p. 241.

Links

Crossrefs

Cf. A002375, A045917, A045922.

Programs

  • Mathematica
    Table[Sum[Sum[(PrimePi[2 k - i] - PrimePi[2 k - i - 1]) (PrimePi[i] - PrimePi[i - 1]), {i, k}], {k, n}], {n, 100}] (* Wesley Ivan Hurt, Apr 07 2018 *)

Formula

a(n) = Sum_{k=1..n} Sum_{i=1..k} A010051(i) * A010051(2k-i). - Wesley Ivan Hurt, Apr 07 2018

A084562 Even numbers having no more than two partitions into two odd primes.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 28, 32, 38, 68
Offset: 1

Views

Author

Reinhard Zumkeller, May 30 2003

Keywords

Comments

A002375(a(n)/2) <= 2.
Conjecture: there are no more terms.

Links

Crossrefs

Cf. A084563, A084564.

A084563 Even numbers having no more than three partitions into two odd primes.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 38, 40, 44, 52, 56, 62, 68, 98, 128
Offset: 1

Views

Author

Reinhard Zumkeller, May 30 2003

Keywords

Comments

A002375(a(n)/2) <= 3.
There are no other terms <= 30000. Could it be that no other terms exist? - John W. Layman, Jul 28 2003
There are no further terms < 100000. - Harvey P. Dale, Apr 17 2018

Links

Crossrefs

Cf. A084562, A084564.

Programs

A084564 Even numbers having no more than four partitions into two odd primes.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 50, 52, 56, 58, 62, 68, 80, 88, 92, 98, 122, 128, 152
Offset: 1

Views

Author

Reinhard Zumkeller, May 30 2003

Keywords

Comments

A002375(a(n)/2) <= 4.
Almost certainly there are no further terms. - David Wasserman, Dec 30 2004

Links

Crossrefs

Cf. A084562, A084563.
Cf. A056636.

A109679 Smallest even number which is the unordered sum of two primes in more ways than any previous even number.

Original entry on oeis.org

2, 4, 10, 22, 34, 48, 60, 78, 84, 90, 114, 120, 168, 180, 210, 300, 330, 390, 420, 510, 630, 780, 840, 990, 1050, 1140, 1260, 1470, 1650, 1680, 1890, 2100, 2310, 2730, 3150, 3570, 3990, 4200, 4410, 4620, 5250, 5460, 6090, 6510, 6930, 7980, 8190, 9030, 9240
Offset: 1

Views

Author

Gilmar Rodriguez Pierluissi (gilmarlily(AT)yahoo.com), Aug 30 2005

Keywords

Comments

Record value of A023036 or A045917.
a(n)== 0 (mod 30) for n > 13.

Links

Crossrefs

Essentially the same as A082917. Cf. A082918, A002375, A023036, A045917, A000954.

Programs

  • Mathematica
    f[n_] := Length[ Select[n - Prime@ Range@ PrimePi[n/2], PrimeQ]]; t = {}; mxm = -1; Do[ If[ f[n] > mxm, AppendTo[t, n]; mxm = f[n]], {n, 2, 9000, 2}]; t

Extensions

Edited and extended by Robert G. Wilson v, Sep 08 2005
Changed offset from 0 to 1 by Vincenzo Librandi, Apr 18 2013

A147517 Number of pairs of primes p < q such that (p+q)/2 = A002110(n), the n-th primorial.

Original entry on oeis.org

0, 1, 6, 30, 190, 1564, 17075, 226758, 3792532, 82116003, 1975662890
Offset: 1

Views

Author

Bill McEachen, Nov 05 2008

Keywords

Comments

The sequence is infinite and illustrates the number of primes expected to be centered around a given primorial.
Given ever-increasing primorial P, one can expect to find the highest symmetrical prime just below 2P.
Using a limited dataset, the approximate relation is the quadratic Y=Ax^2+Bx+C (A,B,C)=(0.12267, 0.75758, -1.592) where Y = log(number of prime pairs) (each > the prime factors) and x is number of prime factors of the seed primorial.
Standard heuristics give a(n) ~ exp(gamma)*log(p)*p#/p^2 where p is the n-th prime and gamma is A001620. - Charles R Greathouse IV, Jul 13 2022

Examples

			There are 6 pairs centered at primorial=30: (29,31),(23,37),(19,41),(17,43),(13,47),(7,53). As they are symmetrical, each prime pair sums to twice the primorial center.

Links

Crossrefs

Cf. A002110, A002375, A116979.

Programs

Formula

a(n) = A002375(A002110(n)). - T. D. Noe, Nov 07 2008

Extensions

Better description by T. D. Noe, Nov 09 2008
Typo corrected typo by T. D. Noe, Nov 10 2008
Edited by Michel Marcus, Jul 09 2017
a(10)-a(11) from Bill McEachen, Jan 30 2018

A187758 Number of ways to write n=x+y (x,y>0) with 2x-3, 2x+3, 6y+1 and 6y+5 all prime.

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Jan 03 2013

Keywords

Comments

Conjecture: a(n)>0 for all n>4.
This has been verified for n up to 10^8. It implies that there are infinitely many cousin primes and also infinitely many sexy primes.

Examples

			a(5)=1 since 5=4+1 with 2*4-3, 2*4+3, 6*1+1 and 6*1+5 all prime.

Links

Crossrefs

Cf. A023200, A046132, A023201, A002375, A187757, A199920, A219055, A218867, A220455.

Programs

  • Mathematica
    a[n_]:=a[n]=Sum[If[PrimeQ[2k-3]==True&&PrimeQ[2k+3]==True&&PrimeQ[6(n-k)+1]==True&&PrimeQ[6(n-k)+5]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

A190353 Goldbach conjecture: number of decompositions of n into an unordered sum of two odd primes (if n even) or three odd primes (if n odd).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 4, 3, 4, 3, 5, 3, 6, 2, 7, 3, 6, 2, 8, 4, 7, 4, 9, 2, 10, 3, 10, 4, 10, 3, 11, 4, 12, 5, 12, 4, 14, 3, 16, 5, 14, 3, 16, 4, 16, 6, 16, 3, 18, 5, 20, 6, 20, 2, 20, 5, 21, 6, 21, 5, 21, 5, 27, 7, 24
Offset: 1

Views

Author

Daniel Forgues, May 09 2011

Keywords

Comments

This sequence differs from A083338 because A083338 allows 2 as a prime.

Links

Crossrefs

Cf. A002375, A007963, A083338.

Formula

a(2n) = A002375(n) and a(2n+1) = A007963(n).

A191004 Number of ways to write n = p+q+(n mod 2)q, where p is an odd prime and q<=n/2 is a prime such that JacobiSymbol[q,n]=1 if n is odd, and JacobiSymbol[(q+1)/2,n+1]=1 if n is even.

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Dec 30 2012

Keywords

Comments

Conjecture: a(n)>0 for all n>5.
We have verified this for n up to 10^9. It is stronger than Goldbach's conjecture and Lemoine's conjecture.
Zhi-Wei Sun also conjectured the following refinement: Any odd number 2n+1>64 not among 105, 247, 255, 1105 can be written as p+2q, where p and q are primes, and JacobiSymbol[q,p']=1 for any prime divisor p' of 2n+1; also, any even number 2n>8 not among 32 and 152 can be written as p+q, where p and q<=n/2 are primes, and JacobiSymbol[(q+1)/2,p']=1 for any prime divisor p' of 2n+1.

Examples

			a(19)=1 since 19=5+2*7 with JacobiSymbol[7,19]=1.
a(32)=1 since 32=29+3 with JacobiSymbol[(3+1)/2,32+1]=1.

Links

Crossrefs

Cf. A002375, A046927, A185150.

Programs

  • Mathematica
    a[n_]:=a[n]=Sum[If[(Mod[n,2]==1&&PrimeQ[n-2Prime[k]]==True&&JacobiSymbol[Prime[k],n]==1)||(Mod[n,2]==0&&n-Prime[k]>2&&PrimeQ[n-Prime[k]]==True&&JacobiSymbol[(Prime[k]+1)/2,n+1]==1),1,0],{k,1,PrimePi[n/2]}]
Do[Print[n," ",a[n]],{n,1,200}]
Previous Showing 91-100 of 172 results. Next

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

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