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 1-10 of 59 results. Next

A052352 Least prime in A031924 (lesser of 6-twins) such that the distance to the next 6-twin is 2*n.

Original entry on oeis.org

47, 23, 73, 61, 353, 31, 233, 131, 331, 653, 2441, 3733, 1033, 4871, 1063, 1621, 503, 607, 4211, 7823, 2287, 83, 383, 1231, 2903, 5981, 1123, 173, 11981, 11833, 1367, 2063, 4723, 19681, 2207, 2131, 2713, 9533, 6571, 1657, 23081, 15913, 7013, 14051, 9967, 22447
Offset: 3

Views

Author

Labos Elemer, Mar 07 2000

Keywords

Comments

The increment of distance of 6-twins (A053321) is 2 (not 6), the smallest distance (A052380) is 6.
The middle gap 2n-6 may include primes, e.g., n = 12, a(12) = 653 and between 659 and 659 + 2*12 - 6 = 677, two primes occur (661 and 673).
a(n) = p yields a prime quadruple [p, p+6, p+2n, p+2n+6] with difference pattern [6, 2n-6, 6].

Examples

			For n = 3, 4, 5,  the quadruples are [47, 53, 53, 59] (a triple), [23, 29, 31, 37], [73, 79, 83, 89] with 53 - 47 = 6, 31 - 23 = 8 and 83 - 73 = 10 twin distances.

Links

Crossrefs

Cf. A031924, A052380, A052381, A053321.
Cf. A052350, A052351, A052353, A052354, A052355, A052356, A052357, A052358, A052359.

Programs

  • Mathematica
    seq[m_] := Module[{p = Prime[Range[m]], d, i, pp, dd, j}, d = Differences[p]; i = Position[d, 6] // Flatten; pp = p[[i]]; dd = Differences[pp]/2 - 2; j = TakeWhile[FirstPosition[dd, #] & /@ Range[Max[dd]] // Flatten, ! MissingQ[#] &]; pp[[j]]]; seq[10000] (* Amiram Eldar, Mar 04 2025 *)
  • PARI
    list(len) = {my(s = vector(len), c = 0, p1 = 2, q1 = 0, q2, d); forprime(p2 = 11, , if(p2 == p1 + 6, q2 = p1; if(q1 > 0, d = (q2 - q1)/2 - 2; if(d <= len && s[d] == 0, c++; s[d] = q1; if(c == len, return(s)))); q1 = q2); p1 = p2);} \\ Amiram Eldar, Mar 04 2025

Extensions

Name and offset corrected by Amiram Eldar, Mar 04 2025

A053321 First differences of A031924.

Original entry on oeis.org

8, 16, 6, 8, 12, 10, 48, 20, 6, 10, 6, 60, 18, 6, 6, 8, 60, 22, 14, 6, 10, 50, 10, 60, 38, 16, 6, 8, 16, 6, 8, 6, 40, 6, 24, 50, 6, 18, 190, 6, 24, 6, 14, 22, 20, 30, 34, 6, 14, 6, 58, 6, 30, 6, 8, 52, 8, 30, 40, 6, 66, 20, 40, 50, 10, 48, 12, 8, 36, 84, 6, 6, 24, 84, 40, 6, 66, 14, 24
Offset: 1

Views

Author

Labos Elemer, Mar 06 2000

Keywords

Links

Crossrefs

Cf. A031924.
Cf. A053322, A053323, A053324, A053325, A053326, A053327, A053331.

Programs

  • GAP
    P:=Filtered([1..2100],IsPrime);;
P1:=List(Filtered([1..Length(P)-1],i->P[i+1]-P[i]=6),k->P[k]);;
a:=List([1..Length(P1)-1],i->P1[i+1]-P1[i]);; Print(a); # Muniru A Asiru, Dec 23 2018
  • Mathematica
    With[{p = Prime[Range[330]]}, Differences[p[[Position[Differences[p], 6] // Flatten]]]] (* Amiram Eldar, Mar 10 2025 *)

A052230 Primes p from A031924 such that A052180(primepi(p)) = 5.

Original entry on oeis.org

23, 31, 53, 61, 83, 151, 173, 233, 263, 271, 331, 353, 383, 443, 503, 541, 563, 571, 593, 601, 653, 751, 991, 1013, 1103, 1223, 1231, 1283, 1291, 1321, 1433, 1493, 1553, 1613, 1621, 1741, 1861, 1973, 2011, 2063, 2131, 2281, 2333, 2341, 2371, 2393, 2543
Offset: 1

Views

Author

Labos Elemer, Feb 01 2000

Keywords

Links

Crossrefs

Cf. A031924, A031925, A052180.

Programs

  • Maple
    filter:= proc(p) local t,m,flag;
  flag:= false;
  for t from p+1 to p+5 do
    m:= min(numtheory:-factorset(t));
    if m > 5 then return false
    elif m = 5 then flag:= true
    fi
  od;
  flag
end proc:
Res:= NULL: count:= 0:
q:= 1: p:= 2:
while count < 100 do
  q:= p;
  p:= nextprime(p);
  if p-q = 6 and filter(q) then
    count:= count+1; Res:= Res, q;
  fi
od:
Res; # Robert Israel, Aug 12 2018

A078861 Least positive residues [mod 210] representing those residue classes which can be smaller prime of a difference 6 taken from A031924.

Original entry on oeis.org

11, 13, 17, 23, 31, 37, 41, 47, 53, 61, 67, 73, 83, 97, 101, 103, 107, 121, 131, 137, 143, 151, 157, 163, 167, 173, 181, 187, 191, 193
Offset: 1

Views

Author

Labos Elemer, Dec 13 2002

Keywords

Crossrefs

Cf. A031924, A008364, A078859, A078860.

Programs

  • Mathematica
    t=Flatten[Position[Table[GCD[w, 210], {w, 1, 210}], 1]] t2=Intersection[t, t+6]-6

Formula

Intersection[RRS(210), 6+RRS{210)]-6. RRS[210]=reduced residue system of 210=first 48=phi[210] terms of A008364; 210k+r generates complete A031924 with suitable k and r taken from these 30 numbers.

A173198 Number of pairs of sexy consecutive primes between (A031924(n))^2 and A031924(n)*A031925(n).

Original entry on oeis.org

10, 10, 12, 8, 11, 14, 12, 15, 18, 19, 21, 21, 25, 31, 19, 23, 32, 29, 27, 28, 43, 36, 36, 35, 42, 51, 52, 46, 43, 53, 45, 55, 41, 55, 51, 46, 71, 52, 66, 60, 54, 62, 75, 66, 56, 67, 91, 65, 78, 75, 77, 97, 62, 80, 90, 81, 68, 78, 89, 99, 86, 90, 98, 98, 106, 96, 90, 84, 105, 89
Offset: 1

Views

Author

Jaspal Singh Cheema, Feb 12 2010

Keywords

Comments

If you graph a(n) versus n, a clear pattern emerges.
As you go farther along the n-axis, greater are the number of consecutive sexy primes, on average, within each interval obtained.
If one could prove that there is at least one consecutive sexy prime within each interval, this would imply that consecutive sexy primes are infinite.
I suspect all numbers in the sequence are > 0.

Examples

			The first sexy prime pair with consecutive primes is (23,29) = A031924(1) and A031925(1). Square the first term, you get 529, then take the product of the two primes, you get 667.
Between these two numbers, namely (529,667), there are ten consecutive sexy primes: (541,547), (557,563), (563,569),
(571,577), (587,593), (593,599), (601,607), (607,613), (647,653), and (653 659).
Hence the very first term of the sequence is 10.

Links

Crossrefs

Cf. A023201

Programs

  • Maple
    isA031924 := proc(p) return (isprime(p) and (nextprime(p)-p) = 6 ); end proc:
A031924 := proc(n) local p; if n = 1 then 23; else p := nextprime(procname(n-1)) ; while not isA031924(p) do p := nextprime(p) ; end do ; return p ; end if ; end proc:
A031925 := proc(n) A031924(n)+6 ; end proc:
A173198 := proc(n) local ulim,llim,a,i ; llim := A031924(n)^2 ; ulim := A031924(n)*A031925(n) ; a := 0 ; for i from llim to ulim-6 do if isA031924(i) then a := a+1 ; end if; end do ; a ; end proc:
seq(A173198(n),n=1..80) ; # R. J. Mathar, Feb 15 2010

Extensions

Comments condensed by R. J. Mathar, Feb 15 2010

A357483 Decimal expansion of sum of squares of reciprocals of primes whose distance to the next prime is equal to 6, Sum_{j>=1} 1/A031924(j)^2.

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Sep 30 2022

Keywords

Examples

			0.004757286975...

Crossrefs

Cf. A031924.
Cf. A085548, A160910, A242301, A356793, A357059.

A052229 a(n) is the first prime p from A031924 such that A052180(primepi(p)) = prime(n).

Original entry on oeis.org

23, 47, 251, 167, 727, 433, 941, 1187, 1453, 1367, 2417, 4597, 2207, 3761, 4657, 4451, 5557, 6317, 7517, 8923, 9043, 17707, 15227, 12823, 10607, 33487, 28663, 29717, 50417, 31567, 24793, 24043, 28753, 28837, 29983, 29173, 59951, 45497
Offset: 3

Views

Author

Labos Elemer, Feb 01 2000

Keywords

Crossrefs

Cf. A031924, A031925, A052158, A052180.

A052231 Primes p from A031924 such that A052180(primepi(p)) = 7.

Original entry on oeis.org

47, 73, 131, 157, 257, 367, 677, 971, 1097, 1123, 1181, 1543, 1601, 1753, 2383, 2441, 2467, 2677, 3307, 3407, 3617, 3727, 3911, 4357, 4457, 4903, 4987, 5113, 5297, 5381, 5407, 5743, 5801, 6037, 6373, 6977, 7187, 7213, 7481, 7717, 7817, 7901, 7927, 8053
Offset: 1

Views

Author

Labos Elemer, Feb 01 2000

Keywords

Crossrefs

Cf. A031924, A031925, A052180.

A052232 Primes p from A031924 such that A052180(primepi(p)) = 11.

Original entry on oeis.org

251, 647, 733, 977, 1063, 1657, 1901, 1987, 2713, 2957, 3637, 4211, 4871, 4937, 5683, 5861, 6257, 6673, 7247, 7577, 8831, 9491, 9643, 11801, 11953, 12197, 12613, 13121, 13451, 14923, 15101, 15187, 15761, 15913, 16421, 16487, 18223, 18797
Offset: 1

Views

Author

Labos Elemer, Feb 01 2000

Keywords

Crossrefs

Cf. A031924, A031925, A052180.

A052233 Primes p from A031924 such that A052180(primepi(p)) = 13.

Original entry on oeis.org

167, 373, 557, 607, 947, 1777, 2351, 2897, 4507, 5081, 5443, 5471, 6067, 7237, 8747, 9343, 9967, 10903, 11087, 12491, 12697, 13037, 14051, 15767, 15817, 16001, 16363, 16547, 16937, 16987, 17327, 19483, 21277, 24971, 26687, 26921, 30197, 30637
Offset: 1

Views

Author

Labos Elemer, Feb 01 2000

Keywords

Crossrefs

Cf. A031924, A031925, A052180.
Showing 1-10 of 59 results. Next

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