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 1-6 of 6 results.

A079017 Suppose p and q = p+14 are primes. Define the difference pattern of (p,q) to be the successive differences of the primes in the range p to q. There are 15 possible difference patterns, namely [14], [2,12], [6,8], [8,6], [12,2], [2,4,8], [2,6,6], [2,10,2], [6,2,6], [6,6,2], [8,4,2], [2,4,6,2], [2,6,4,2], [2,2,4,2,4], [2,4,2,4,2]. Sequence gives smallest value of p for each difference pattern, sorted by magnitude.

Original entry on oeis.org

3, 5, 17, 23, 29, 47, 83, 89, 113, 137, 149, 197, 359, 509, 1997
Offset: 1

Views

Author

Labos Elemer, Jan 24 2003

Keywords

Examples

			p=1997, q=2011 has difference pattern [2,4,8] and {1997,1999,2003,2011} is the corresponding consecutive prime 4-tuple.

Crossrefs

A022006(1)=5, A022007(1)=7, A078847(1)=17, A078851(1)=19, A078946(1)=17, A078854(1)=23, A078948(1)=29, A078857(1)=47, A031932(1)=113, A078849(1)=149.
Cf. A079016-A079024.

A079021 Suppose p and q = p+22 are primes. Define the difference pattern of (p,q) to be the successive differences of the primes in the range p to q. There are 51 possible difference patterns, shown in the Comments line. Sequence gives smallest value of p for each difference pattern, sorted by magnitude.

Original entry on oeis.org

7, 19, 31, 37, 61, 67, 79, 109, 127, 151, 157, 211, 241, 271, 331, 337, 397, 409, 421, 457, 487, 499, 541, 619, 661, 739, 751, 787, 919, 991, 1069, 1129, 1471, 1531, 1597, 1867, 2221, 2287, 2671, 2707, 2797, 2857, 3187, 3301, 3391, 3637, 4651, 6547, 12637, 17011, 90001
Offset: 1

Views

Author

Labos Elemer, Jan 24 2003

Keywords

Comments

The 51 difference patterns are [22], [4,18], [6,16], [10,12], [12,10], [16,6], [18,4], [4,2,16], [4,6,12], [4,8,10], [4,12,6], [4,14,4], [6,4,12], [6,6,10], [6,10,6], [6,12,4], [10,2,10], [10,6,6], [10,8,4], [12,4,6], [12,6,4], [16,2,4], [4,2,4,12], [4,2,6,10], [4,2,10,6], [4,6,2,10], [4,6,6,6], [4,6,8,4], [4,8,4,6], [4,8,6,4], [6,4,2,10], [6,4,6,6], [6,4,8,4], [6,6,4,6], [6,6,6,4], [6,10,2,4], [10,2,4,6], [10,2,6,4], [10,6,2,4], [12,4,2,4], [4,2,4,2,10], [4,2,4,6,6], [4,2,6,4,6], [4,6,2,4,6], [4,6,2,6,4], [6,4,2,4,6], [6,4,2,6,4], [6,4,6,2,4], [6,6,4,2,4], [10,2,4,2,4], [4,2,4,2,4,6].
Certain patterns are singular, i.e. occur only once like [4,2,4,2,4,6].

Examples

			p=6547, q=6569 has difference pattern [4,2,10,6] and {6547,6551,6553,6563,6569} is the corresponding consecutive prime 5-tuple.

Crossrefs

A078957(1)=12637, A078964(1)=157, A078967(1)=151, A078969(1)=3301, A000230(11)=1129. Cf. A079016-A079024.

A079018 Suppose p and q = p+16 are primes. Define the difference pattern of (p,q) to be the successive differences of the primes in the range p to q. There are 17 possible difference patterns, namely [16], [4,12], [6,10], [10,6], [12,4], [4,2,10], [4,6,6], [4,8,4], [6,4,6], [6,6,4], [10,2,4], [4,2,4,6], [4,2,6,4], [4,6,2,4], [6,4,2,4], [4,2,4,2,4], [2,2,4,2,4,2]. Sequence gives smallest value of p for each difference pattern, sorted by magnitude.

Original entry on oeis.org

3, 7, 13, 31, 43, 67, 73, 151, 181, 211, 241, 277, 331, 463, 487, 1597, 1831
Offset: 1

Views

Author

Labos Elemer, Jan 24 2003

Keywords

Examples

			p=181, q=197 has difference pattern [10,2,4] and {181,191,193,197} is the corresponding consecutive prime 4-tuple.

Crossrefs

A022008(1)=7, A078952(1)=13, A078852(1)=73, A078953(1)=67, A078954(1)=1597, A078961(1)=31, A078856(1)=73, A078858(1)=151, A031934(1)=A000230(8)=1831.
Cf. A079016-A079024.

A079019 Suppose p and q = p+18 are primes. Define the difference pattern of (p,q) to be the successive differences of the primes in the range p to q. There are 49 possible difference patterns, shown in the Comments line. Sequence gives smallest value of p for each difference pattern, sorted by magnitude.

Original entry on oeis.org

5, 11, 13, 19, 23, 29, 41, 43, 61, 71, 79, 83, 89, 109, 113, 131, 139, 149, 179, 181, 193, 211, 239, 251, 331, 401, 461, 491, 503, 523, 569, 601, 659, 691, 733, 739, 743, 821, 1303, 1531, 1601, 1861, 1931, 1933, 1993, 2069, 3313, 4201, 18043
Offset: 1

Views

Author

Labos Elemer, Jan 24 2003

Keywords

Comments

Difference patterns are [18], [2,16], [4,14], [6,12], [8,10], [10,8], [12,6], [14,4], [16,2], [2,4,12], [2,6,10], [2,10,6], [2,12,4], [4,2,12], [4,6,8], [4,8,6], [4,12,2], [6,2,10], [6,4,8], [6,6,6], [6,8,4], [6,10,2], [8,4,6], [8,6,4], [10,2,6], [10,6,2], [12,2,4], [12,4,2], [2,4,2,10], [2,4,6,6], [2,6,4,6], [2,6,6,4], [2,10,2,4], [4,2,4,8], [4,2,10,2], [4,6,2,6], [4,6,6,2], [6,2,4,6], [6,2,6,4], [6,4,2,6], [6,4,6,2], [6,6,4,2], [8,4,2,4], [10,2,4,2], [2,4,2,4,6], [2,6,4,2,4], [4,2,4,6,2], [6,4,2,4,2], [2,4,2,4,2,4].

Examples

			p=18043, q=18061 has difference pattern [4,2,10,2] and {18043,18047,18049,18059,18061} is the corresponding consecutive prime 5-tuple.

Crossrefs

A078947[1]=41, A078949[1]=71, A078950[1]=149, A078955[1]=19, A078956[1]=43, A078959[1]=23, A078962[1]=61, A078966[1]=601, A078958[1]=1601, A078963[1]=3313, A031936[1]=A000230[9]=523.
Cf. A079016-A079024.

Extensions

Corrected by Rick L. Shepherd, Aug 30 2003

A079020 Suppose p and q = p+20 are primes. Define the difference pattern of (p,q) to be the successive differences of the primes in the range p to q. There are 56 possible difference patterns, shown in the Comments line. Sequence gives smallest value of p for each difference pattern, sorted by magnitude.

Original entry on oeis.org

3, 11, 17, 23, 41, 47, 59, 83, 89, 107, 131, 137, 179, 191, 251, 293, 317, 347, 353, 359, 389, 401, 467, 503, 521, 593, 599, 653, 887, 947, 971, 1031, 1151, 1193, 1229, 1259, 1301, 1307, 1439, 1601, 1931, 1979, 1997, 2069, 2531, 3167, 3299, 4241, 5261, 5639, 5849, 8081, 10091, 17189, 18041, 19421
Offset: 1

Views

Author

Labos Elemer, Jan 24 2003

Keywords

Comments

The 56 difference patterns are [20], [2,18], [6,14], [8,12], [12,8], [14,6], [18,2], [2,4,14], [2,6,12], [2,10,8], [2,12,6], [2,16,2], [6,2,12], [6,6,8], [6,8,6], [6,12,2], [8,4,8], [8,6,6], [8,10,2], [12,2,6], [12,6,2], [14,4,2], [2,4,2,12], [2,4,6,8], [2,4,8,6], [2,4,12,2], [2,6,4,8], [2,6,6,6], [2,6,10,2], [2,10,2,6], [2,10,6,2], [2,12,4,2], [6,2,4,8], [6,2,6,6], [6,2,10,2], [6,6,2,6], [6,6,6,2], [6,8,4,2], [8,4,2,6], [8,4,6,2], [8,6,4,2], [12,2,4,2], [2,4,2,4,8], [2,4,2,10,2], [2,4,6,2,6], [2,4,6,6,2], [2,6,4,2,6], [2,6,4,6,2], [2,6,6,4,2], [2,10,2,4,2], [6,2,4,6,2], [6,2,6,4,2], [8,4,2,4,2], [2,4,2,4,6,2], [2,6,4,2,4,2], [2,2,4,2,4,2,4].
Certain patterns are singular, i.e. occur only once like [2,2,4,2,4,2,4]. Impossible patterns are [2,14,4] or [10,10] etc.

Examples

			p=10091, q=10111 has difference pattern [2, 6, 4, 8] and {10091, 10093, 10099, 10103, 10111} is the corresponding consecutive prime 5-tuple.

Crossrefs

A000230(10)=A031938(1)=887, A078951(1)=3299, A078965(1)=47, A078968(1)=251.
Cf. A079016-A079024.

Extensions

Edited by Rick L. Shepherd, Sep 10 2003

A079022 Suppose p and q = p + 2*n are primes. Define the difference pattern of (p, q) to be the successive differences of the primes in the range p to q. There are a(n) possible difference patterns.

Original entry on oeis.org

1, 2, 3, 5, 5, 14, 15, 17, 49, 56, 51, 175, 150, 148, 666, 581, 561, 1922, 1449
Offset: 1

Views

Author

Labos Elemer, Jan 24 2003

Keywords

Examples

			n=4, d=8: there are five difference patterns: [8], [6,2], [2,6], [2,4,2], [2,2,4]. The last pattern is singular with prime 4-tuple {p=3,5,7,11=q}.

Crossrefs

See A079016, A079017, A079018, A079019, A079020, A079021 for cases n=6 through 11.
Cf. A000230, A079023, A079024.

Programs

  • Mathematica
    t[x_] := Table[Length[FactorInteger[x+j]], {j, 0, d}]; p[x_] := Flatten[Position[Table[PrimeQ[x+2*j], {j, 0, d/2}], True]]; dp[x_] := Delete[RotateLeft[p[x]]-p[x], -1]; k=0; d=12; {n1=2, n2=2000, h0=PrimePi[n1], h=PrimePi[n2]}; t1={}; Do[s=Prime[n]; If[PrimeQ[s + d], k=k+1; Print[{k, s, pt=2*dp[s]}]; t1=Union[t1, {2*dp[s]}], 1], {n, h0, h}]; {d, n1, n2, Length[t1], t1} (* program for d=12; partition list is enlargable if t1={} is replaced with already obtained set *)

Extensions

a(14)-a(17) and a(19) from David A. Corneth, Aug 30 2019
a(18) from Jinyuan Wang, Feb 16 2021
Showing 1-6 of 6 results.

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