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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A096830 Number of primes in neighborhood with center=n! and radius = ceiling(log(n!)).

Original entry on oeis.org

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

Views

Author

Labos Elemer, Jul 14 2004

Keywords

Examples

			n=8: 8!=40320; radius=11, a(8)=0 because there are no primes in the neighborhood.

Crossrefs

Cf. A096509-A096523, A000142.

Programs

  • Mathematica
    {ta=Table[0, {1000}], u=1}; Do[s=Count[Table[PrimeQ[n!+i], {i, -Ceiling[Log[n! ]//N], Ceiling[Log[n! ]//N]}], True]; Print[{n, s}];ta[[u]]=s;u=u+1, {n, 1, 1000}];ta

Formula

a(n) = A096509(n!) = A096509(A000142(n)).

A096840 a(n) = x is the least number such that around x^2 (the center) the number of primes is equal to n. The radius of neighborhood is ceiling(log(x^2)).

Original entry on oeis.org

1, 6, 3, 2, 14, 36, 117, 1652, 9582, 41361, 908637, 36284185
Offset: 0

Views

Author

Labos Elemer, Jul 14 2004

Keywords

Examples

			n=9: a(9) = 41361, center = 1710732321, radius = 22; the nine primes in the zone are {1710732299, 1710732307, 1710732311, 1710732313, 1710732319, 1710732323, 1710732329, 1710732337, 1710732343}.

Crossrefs

Cf. A096509-A096523, A096830-A096839.

Programs

  • Mathematica
    f[n_] := (PrimePi[n^2 + Ceiling[ Log[n^2]]] - PrimePi[n^2 - Ceiling[ Log[n^2]] - 1]); t = Table[0, {15}]; Do[a = f[n]; If[a < 15 && t[[a + 1]] == 0, t[[a + 1]] = n], {n, 10^5}] (* Robert G. Wilson v, Jul 14 2004 *)

Extensions

Offset corrected and a(11) from Donovan Johnson, Jul 11 2010

A096520 Number of primes in the neighborhood of center=2^n and radius=Ceiling[Log[2^n]].

Original entry on oeis.org

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

Views

Author

Labos Elemer, Jul 12 2004

Keywords

Examples

			First in the suitable neighborhood of 2^25 no primes occur: a[25]=0, while around 2^127 6 primes arise: a[127]=6.

Crossrefs

Cf. A096509-A096523.

Programs

  • Mathematica
    t=Table[Count[Table[PrimeQ[2^n+i], {i, -Ceiling[Log[2^n]//N], Ceiling[Log[2^n]//N]}], True], {n, 1, 256}]

A096521 Smallest exponent of 2 when the number of primes in the neighborhood of center=2^n and radius=ceiling(log(2^n)) equals n.

Original entry on oeis.org

25, 9, 1, 2, 20, 18, 127, 844, 573
Offset: 0

Views

Author

Labos Elemer, Jul 12 2004

Keywords

Examples

			First in the suitable neighborhood of 2^25 no primes occur: a[0]=25, while corresponding around 2^127 6 primes arise: a[6]=127.

Crossrefs

Cf. A096509-A096523.

Programs

  • Mathematica
    t=Table[Count[Table[PrimeQ[2^n+i], {i, -Ceiling[Log[2^n]//N], Ceiling[Log[2^n]//N]}], True], {n, 1, 256}] Table[Min[Flatten[Position[t, j]]], {j, 0, 10}]

A096831 Number of primes in the neighborhood with center = n-th primorial and radius = ceiling(log(n-th primorial)).

Original entry on oeis.org

2, 2, 2, 1, 2, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Labos Elemer, Jul 14 2004

Keywords

Comments

What is exceptional in such neighborhoods of primorials is that in most cases no primes occur, i.e., these zones are peculiarly poor or empty of primes!
Primes are scarce in these zones because log(A002110(n)) < prime(n), so A002110(n)+1 and A002110(n)-1 are the only numbers in the neighborhood that are not divisible by one of the first n primes. - David Wasserman, Nov 16 2007

Examples

			n=7: 7th primorial=510510; radius=14, a(7)=0 because there are no primes in the relevant neighborhood.
[1, 3], [4, 8], [26, 34], [2302, 2318] (around 2, 6, 30, 2310, respectively) are the only zones in which 2 primes were found.

Crossrefs

Cf. A096509-A096523, A002110.

Formula

a(n) = A096509(A002110(n)).

A096839 Number of primes in the neighborhood of n^2 with radius ceiling(log(n^2)).

Original entry on oeis.org

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

Views

Author

Labos Elemer, Jul 14 2004

Keywords

Examples

			n=10: a[10]=3 and the 3 primes in interval [95, 105] around 10^2=100 are {97,101,103}.

Crossrefs

Cf. A096509-A096523, A096830-A096840.

Programs

  • Mathematica
    f[n_] := (PrimePi[n^2 + Ceiling[Log[n^2]]] - PrimePi[n^2 - Ceiling[Log[n^2]] - 1]); Table[ f[n], {n, 105}] (* Robert G. Wilson v, Jul 14 2004 *)

A096510 a(n) is the smallest number x such that the number of prime powers (including primes, excluding 1), in the neighborhood of x with radius ceiling(log(x)), equals n.

Original entry on oeis.org

1, 54, 2, 12, 3, 8, 4792, 75991, 284736, 6561003, 51448375, 964669618, 18320500423
Offset: 0

Views

Author

Labos Elemer, Jul 12 2004

Keywords

Comments

With increasing n the radius of log(n) slowly increases, while frequency of prime-powers decreases.
No more terms < 369*10^8. - David Wasserman, Nov 16 2007

Examples

			a[8]=284736: because in [284723,284749] around a(8),
8 prime(powers) occur first,with radius=r=13;
a[0]=1;a[1]=54 means that in [50,58] only 53 is prime,r=4.

Crossrefs

Cf. A096508-A096512.
Cf. A096509-A096523.

Extensions

4 more terms from David Wasserman, Nov 16 2007

A096833 Values of n such that in the interval centered at A002110(n) = n-th primorial and of radius ceiling(log(center)) there is a single prime.

Original entry on oeis.org

4, 6, 11, 13, 24, 66, 68, 75, 167, 171, 172, 287, 310, 352, 384, 457, 564, 590, 616, 620, 643, 849, 1391, 1552, 1613, 1849, 2122, 2647, 2673, 4413
Offset: 1

Views

Author

Labos Elemer, Jul 14 2004

Keywords

Comments

Neighborhoods of most primorials (=center) are either empty or contain few primes. In the listed few cases a single prime arises if radius=ceiling(log(center)).

Examples

			n=6: around 30030 the prime in question is 30029.

Crossrefs

Cf. A096509-A096523, A096830-A096840, A002110.

Extensions

More terms from David Wasserman, Nov 16 2007
Definition revised by N. J. A. Sloane, Dec 06 2014

A096522 Solutions to A096520[x]=0, that is in neighborhood with c=2^x center and r=Ceiling[Log[2^x]] no prime occurs.

Original entry on oeis.org

25, 27, 34, 35, 45, 49, 59, 62, 65, 97, 99, 103, 106, 113, 120, 139, 142, 169, 176, 181, 183, 185, 186, 208, 215, 216, 223, 227, 232, 237, 256, 259, 267, 269, 293, 298, 302, 309, 332, 333, 334, 340, 349, 358, 361, 362, 364, 371, 373, 381, 387, 403, 426, 433
Offset: 1

Views

Author

Labos Elemer, Jul 12 2004

Keywords

Crossrefs

Cf. A096509-A096521.

Programs

  • Mathematica
    {ta=Table[0, {1000}], u=1}; Do[s=Count[Table[PrimeQ[2^n+i], {i, -Ceiling[Log[2^n]//N], Ceiling[Log[2^n]//N]}], True];If[Equal[s, 0], Print[{n, s}];ta[[u]]=n;u=u+1], {n, 1, 1000}];ta

A096832 Number of primes in enlarged neighborhood with center = n-th primorial and radius = 2*ceiling(log(n-th primorial)). So compared to A096831, the radius is doubled.

Original entry on oeis.org

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

Views

Author

Labos Elemer, Jul 14 2004

Keywords

Comments

What is exceptional in such neighborhoods of primorials is that in most cases no primes occur, i.e., these zones are peculiarly poor or empty of primes! If the radius is doubled then the density of primes appears to be "normal".

Examples

			n=7: 7th primorial = 510510; for radius=14, no primes in the relevant neighborhood; for radius=28, then one prime appears: 510529.

Crossrefs

Cf. A096509-A096523, A096830-A096840; A002110.
Previous Showing 11-20 of 26 results. Next

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