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.

User: Marek Wolf

Marek Wolf's wiki page.

Marek Wolf has authored 7 sequences.

A226496 The number of primes of the form i^2 + j^4 (A028916) <= 2^n, counted with multiplicity.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 9, 13, 21, 34, 50, 77, 121, 191, 292, 458, 727, 1164, 1840, 2904, 4650, 7429, 11869, 19087, 30760, 49474, 79971, 129226, 209823, 340347, 552722, 898655, 1461698, 2381041, 3883079, 6338935, 10357549, 16935173, 27712338, 45381521, 51559329
Offset: 1

Views

Author

Marek Wolf and Robert G. Wilson v, Jun 09 2013

Keywords

Comments

Iwaniec and Friedlander have proved there is infinity of the primes of the form i^2 + j^4.
Counted with double representations. If we do not count doubles, the sequence is A226498.

Examples

			2 = 1^2+1^4, 5 = 2^2+1^4, 17 = 4^2+1^4 = 1^2+2^4, ..., 97 = 9^2+2^4 = 4^2+3^4, etc.

Crossrefs

Cf. A028916, A226495, A226497, A226498.

Programs

  • Mathematica
    mx = 2^40; lst = {};  Do[a = i^2 + j^4; If[ PrimeQ[a], AppendTo[ lst, a]], {i, Sqrt[mx]}, {j, Sqrt[ Sqrt[mx - i^2]]}]; Table[ Length@ Select[lst, # <2^n &], {n, 40}]

A226498 The number of primes of the form i^2 + j^4 (A028916) <= 2^n.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 7, 11, 17, 28, 43, 67, 108, 173, 272, 434, 690, 1115, 1772, 2815, 4528, 7267, 11646, 18799, 30378, 48956, 79270, 128267, 208509, 338533, 550262, 895284, 1457111, 2374753, 3874445, 6327042
Offset: 1

Views

Author

Marek Wolf and Robert G. Wilson v, Jun 09 2013

Keywords

Comments

Iwaniec and Friedlander proved there are infinity of the primes of the form i^2+j^4, and hence a(n) increases without bound.
Does not count double representations.

Links

Crossrefs

Cf. A028916, A226495, A226496, A226497.

Programs

  • Mathematica
    mx = 2^40; lst = {};  Do[a = i^2 + j^4; If[ PrimeQ[a], AppendTo[lst, a]], {i, Sqrt[mx]}, {j, Sqrt[ Sqrt[mx - i^2]]}]; Table[ Length@ Select[ Union@ lst, # < 2^n &], {n, 40}]
  • PARI
    a(n)=my(N=2^n,v=List(),t);for(a=1,sqrt(N),forstep(b=a%2+1, sqrtint(sqrtint(N-a^2)), 2, t=a^2+b^4;if(isprime(t),listput(v,t)))); 1+#vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jun 12 2013

A226495 The number of primes of the form i^2+j^4 (A028916) <= 10^n, counted with multiplicity.

Original entry on oeis.org

2, 8, 34, 134, 615, 2813, 13415, 65162, 323858, 1626844, 8268241, 42417710
Offset: 1

Views

Author

Marek Wolf and Robert G. Wilson v, Jun 09 2013

Keywords

Comments

Iwaniec and Friedlander have proved there is infinity of the primes of the form i^2+j^4.
Primes with more than one representation are counted multiple times.
If we do not count repetitions, the sequence is A226497: 2, 6, 28, 121, 583, 2724, 13175, 64551, ..., .

Examples

			2 = 1^2+1^4, 5 = 2^2+1^4, 17 = 4^2+1^4 = 1^2+2^4, …, 97 = 9^2+2^4 = 4^2+3^4, etc.

Crossrefs

Cf. A028916, A226496, A226497, A226498.

Programs

  • Mathematica
    mx = 10^12; lst = {};  Do[a = i^2 + j^4; If[ PrimeQ[a], AppendTo[ lst, a]], {i, Sqrt[mx]}, {j, Sqrt[ Sqrt[mx - i^2]]}]; Table[ Length@ Select[lst, # < 10^n &], {n, 12}]

A226497 The number of primes of the form i^2+j^4 (A028916) <= 10^n.

Original entry on oeis.org

2, 6, 28, 121, 583, 2724, 13175, 64551, 322110, 1621929, 8254127
Offset: 1

Views

Author

Marek Wolf and Robert G. Wilson v, Jun 09 2013

Keywords

Comments

Even if a prime has more than one representation in the form i^2+j^4, it is only counted once.
Iwaniec and Friedlander have proved there is infinity of the primes of the form i^2+j^4.

Crossrefs

Cf. A028916, A226495, A226496 & A226498.

Programs

  • Mathematica
    mx = 2^40; lst = {};  Do[a = i^2 + j^4; If[ PrimeQ[a], AppendTo[lst, a]], {i, Sqrt[mx]}, {j, Sqrt[Sqrt[mx - i^2]]}]; Table[ Length@ Select[ Union@ lst, # < 10^n &], {n, 12}]

A218014 Location of the n-th prime in its Andrica ranking.

Original entry on oeis.org

27, 6, 13, 1, 31, 4, 54, 8, 3, 100, 5, 25, 155, 28, 9, 16, 243, 19, 49, 288, 21, 62, 24, 12, 75, 422, 81, 444, 84, 2, 112, 37, 580, 11, 634, 47, 53, 150, 57, 60, 788, 20, 840, 183, 872, 10, 14, 218, 1029, 228, 80, 1074, 26, 87, 92, 99, 1237, 103, 281, 1319, 29, 15, 314, 1498, 323
Offset: 1

Views

Author

Marek Wolf and Robert G. Wilson v, Oct 18 2012

Keywords

Comments

For each consecutive prime pair p < q, d = sqrt(q) - sqrt(p) is unique. Place d in order from greatest to least and specify p.
Last appearance by prime index: 1, 5, 7, 10, 13, 17, 20, 26, 28, 33, 35, 41, 43, 45, 49, ..., .
Last appearance of a minimum prime by Andrica ranking: 2, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, ..., .
As expected, this sequence is the lesser of the twin primes beginning with the second term, 11. See A001359.

Examples

			a(1)=27 since the first prime, 2, does not show up in the ranking until the 27th term. See A218013.
a(4)=1 since the fourth prime, 7, has the maximum A_n value, see A218012; i.e., sqrt(p_n)-sqrt(p_n+1) is at a maximum.

Links

Crossrefs

Cf. A079296, A218012, A218015.

Programs

  • Mathematica
    lst = {}; p = 2; q = 3; While[p < 1600000, If[ Sqrt[q] - Sqrt[p] > 1/20, AppendTo[lst, {p, Sqrt[q] - Sqrt[p]}]]; p = q; q = NextPrime[q]]; lsu = First@ Transpose@ Sort[lst, #1[[2]] > #2[[2]] &]; Table[ Position[lsu, p, 1, 1], {p, Prime@ Range@ 65}] // Flatten

A218015 Number of primes p such that sqrt(q) - sqrt(p) > 1/n, where q is the prime after p.

Original entry on oeis.org

0, 6, 22, 41, 75, 132, 186, 258, 330, 416, 511, 613, 724, 860, 1001, 1163, 1372, 1563, 1751, 1965, 2179, 2412, 2685, 2945, 3258, 3581, 3885, 4194, 4525, 4857, 5246, 5644, 6024, 6402, 6767, 7229, 7695, 8177, 8666, 9156, 9674, 10185, 10740, 11283, 11824
Offset: 1

Views

Author

Marek Wolf and Robert G. Wilson v, Oct 18 2012

Keywords

Comments

Also, the number of terms by Andrica ranking which are greater than 1/n.

Examples

			a(1) = 6 because only the primes 3, 7, 13, 23, 31 and 113 satisfy the criterion.
As an example, - sqrt(3) + sqrt(5) ~= 0.50401717 which is greater than 1/2.

Links

Crossrefs

Cf. A079296, A218012, A218014.

Programs

  • Mathematica
    lst = {}; p = 2; q = 3; While[p < 10^8, If[ Sqrt[q] - Sqrt[p] > 1/50, AppendTo[lst, {p, Sqrt[q] - Sqrt[p]}]]; p = q; q = NextPrime[q]]; Table[ Length@ Select[ lst, #[[2]] > 1/n &], {n, 50}]
nn = 50; t = Table[0, {nn}]; p = 2; q = 3; While[p < 10^8, n = Floor[1/(Sqrt[q] - Sqrt[p])]; If[n <= nn, t[[n]]++]; p = q; q = NextPrime[q]]; Join[{0}, Accumulate[t]] (* T. D. Noe, Oct 18 2012 *)

A218012 Decimal expansion of -sqrt(7) + sqrt(11), Andrica's Maximum A_n.

Original entry on oeis.org

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

Views

Author

Marek Wolf and Robert G. Wilson v, Oct 18 2012

Keywords

Comments

For each consecutive prime pair, p and q with p < q, d = -sqrt(p) + sqrt(q) is unique. Place d in order from greatest to least and specify p. This is the maximum d.

Examples

			0.670873479290809258613316983031426258216829362506903416690347686915...

Links

Crossrefs

Cf. A079296, A218014, A218015.

Programs

  • Mathematica
    RealDigits[- Sqrt[7] + Sqrt[11], 10, 111][[1]]

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