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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A052167 Primes at which difference pattern X2424Y (X and Y >= 6) occurs in A001223.

Original entry on oeis.org

1481, 21011, 22271, 55331, 144161, 165701, 166841, 195731, 201821, 225341, 247601, 268811, 326141, 347981, 361211, 397751, 465161, 518801, 536441, 633461, 633791, 661091, 768191, 795791, 829721, 857951, 876011, 958541, 1008851
Offset: 1

Views

Author

Labos Elemer, Jan 26 2000

Keywords

Examples

			21011 is here because 21011+{2,2+4,2+4+2,2+4+2+4}=21011+{1,6,8,12}= {21013,21013,21017,21019,21023} are consecutive primes but the primes in the immediate neighborhood (21001 and 21031) are in distance 10 and 8. Thus the d-pattern "around 21011" is {10,2,4,2,4,12}.

Crossrefs

A001223, A052160, A052162-A052168, A022008, A047078.

Programs

  • Mathematica
    patQ[n_]:=Module[{d=Differences[n]},First[d]>5&&Last[d]>5&&Most[ Rest[d]] == {2,4,2,4}]; Transpose[Select[Partition[Prime[ Range[ 80000]],7,1],patQ]] [[2]] (* Harvey P. Dale, Dec 11 2013 *)

A118359 Primes for which the weight as defined in A117078 is 7 and the gap as defined in A001223 is 6.

Original entry on oeis.org

83, 167, 251, 433, 503, 587, 601, 727, 1063, 1217, 1231, 1553, 1777, 1861, 1973, 1987, 2281, 2351, 2393, 2897, 3541, 4073, 4283, 4451, 4507, 4591, 4871, 5081, 5431, 5557, 5641, 5683
Offset: 1

Views

Author

Rémi Eismann, May 24 2006, May 04 2007

Keywords

Comments

The prime numbers in this sequence are of the form (14i-1) with i=(level(n)+1)/2, level(n) defined in A117563. level(n) is not multiple of 3.

Examples

			prime(24) = prime (23) + prime(23)mod(7) = prime (23) + prime(23)mod(77)
89 = 83 + 83mod(7) = 83 + 83mod(77)
k=7, level = 77/7 = 11

Links

Crossrefs

Cf. A117078, A117563, A001359, A074822, A118922, A118924, A119504, A119597, A119596, A119595.

A155752 Where 2's occur in the prime differences A001223.

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 16, 19, 25, 27, 32, 34, 40, 42, 44, 48, 51, 56, 59, 63, 68, 80, 82, 88, 97, 103, 108, 112, 115, 119, 139, 141, 143, 147, 151, 170, 172, 175, 177, 181, 189, 200, 205, 208, 211, 214, 224, 229, 233, 235, 252, 255, 261, 264, 267, 276, 285, 287, 293, 295, 301
Offset: 1

Views

Author

Paul Curtz, Jan 26 2009

Keywords

Comments

Starts with the same numbers as A053096.

Programs

  • Mathematica
    Flatten[Position[Differences[Prime[Range[400]]],2]]-1 (* Harvey P. Dale, Jun 21 2017 *)

Formula

A001223(1+a(n)) = 2.
a(n) = A029707(n)-1. - R. J. Mathar, Feb 23 2009

Extensions

Edited and extended by R. J. Mathar, Feb 23 2009

A057467 GCD of n-th and (n+1)-st term in the sequence of first differences between primes, A001223.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 6, 2, 2, 2, 2, 2, 2, 12, 4, 2, 2, 2, 2, 2, 2, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 4, 4, 4, 4, 2, 6, 2, 2, 6, 2, 2, 6, 2, 2
Offset: 1

Views

Author

Labos Elemer, Dec 06 2000

Keywords

Crossrefs

Cf. A001223.

Programs

  • Mathematica
    Map[GCD @@ # &, Partition[#, 2, 1] &@ Differences@ Prime@ Range@ 100] (* Michael De Vlieger, Feb 18 2017 *)

Formula

a(n) = gcd( A001223(n+1), A001223(n) ).

A127596 Numbers k such that 1 + Sum_{i=1..k-1} A001223(i)*(-1)^i = 0.

Original entry on oeis.org

2, 4, 14, 22, 28, 233, 249, 261, 488, 497, 511, 515, 519, 526, 531, 534, 548, 562, 620, 633, 635, 2985, 3119, 3123, 3128, 3157, 4350, 4358, 4392, 4438, 4474, 4484, 4606, 4610, 4759, 5191, 12493, 1761067, 2785124, 2785152, 2785718, 2785729, 2867471
Offset: 1

Views

Author

Manuel Valdivia, Apr 03 2007

Keywords

Comments

Or, with prime(0) = 1, numbers k such that Sum_{i=0..k-1} (prime(i+1)-prime(i))*(-1)^i = Sum_{i=0..k-1} (A008578(i+1)-A008578(i))*(-1)^i = 0.
There are 313 terms < 10^7, 846 terms < 10^8, 1161 terms < 10^9.

Examples

			1 - A001223(1) = 1 - 1 = 0, hence 2 is a term.
1 - A001223(1) + A001223(2) - A001223(3) = 1 - 1 + 2 - 2 = 0, hence 4 is a term.

Links

Crossrefs

Cf. A001223 (differences between consecutive primes), A008578 (prime numbers at the beginning of the 20th century), A000101 (increasing gaps between primes, upper end), A002386 (increasing gaps between primes, lower end).
Cf. A282178 (prime(a(n))), A330545, A330547.

Programs

  • Mathematica
    S=0; Do[j=Prime[n+1]; i=Prime[n]; d[n]=j-i; S=S+(d[n]*(-1)^n); If[S+1==0, Print[Table[j|PrimePi[j]|S+1]]], {n,1,10^7,1}]
  • PARI
    {m=10^8; n=1; p=1; e=1; s=0; while(nKlaus Brockhaus, Apr 29 2007 */

Extensions

Edited by Klaus Brockhaus, Apr 29 2007

A052376 Primes followed by a [10,2,10] prime difference pattern of A001223.

Original entry on oeis.org

409, 1039, 2017, 2719, 3571, 4219, 4231, 4261, 4327, 6079, 6121, 6679, 6781, 8209, 11047, 11149, 11959, 12241, 15277, 19531, 19687, 21577, 21589, 26881, 27529, 28087, 28297, 29389, 30829, 30859, 31069, 32401, 42061, 45307, 47797, 48109
Offset: 1

Views

Author

Labos Elemer, Mar 22 2000

Keywords

Comments

Subsequence of lesser terms of 10-twins (A031928).
Start primes of quadruples consisting of two consecutive 10-twins of prime which are in minimal distance [minD = A052380(10/2) = 12].
First term of this sequence is 409 = A052381(5).

Examples

			p=1039 begins [1039,1049,1051,1061] prime quadruple with the appropriate difference pattern: [10,2,10]=[d,D-d,d], so d=10, D=12.

Links

Crossrefs

Cf. A031928, A053323, A052380, A052381, A052377, A052378.

Programs

  • Mathematica
    {p, q, r, s} = {2, 3, 5, 7}; lst = {}; While[p < 50000, If[ Differences[{p, q, r, s}] == {10, 2, 10}, AppendTo[lst, p]]; {p, q, r, s} = {q, r, s, NextPrime@ s}]; lst (* Robert G. Wilson v, Jul 15 2015 *)

Formula

a(n)=p, a prime which begins a [p, p+d, p+D, p+D+d]=[p, p+10, p+12, p+22] prime quadruple.
a(n) = A259025(n)-11. - Robert G. Wilson v, Jul 15 2015

A161913 Numbers k such that A001223(k) <> A000005(k).

Original entry on oeis.org

4, 9, 10, 11, 12, 15, 16, 19, 20, 21, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 39, 40, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78
Offset: 1

Views

Author

Omar E. Pol, Jul 04 2009

Keywords

Links

Crossrefs

Cf. A000005, A001223, A068526, A161911, A161912, A162188, A162189.

Programs

  • Maple
    p2:=2:for n from 1 to 78 do p1:=p2:p2:=ithprime(n+1):if(p2-p1<>tau(n))then print(n);fi:od: # Nathaniel Johnston, Apr 15 2011

A347102 Totally additive with a(prime(k)) = A001223(k), where A001223 gives the distance from the k-th prime to the next larger prime.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 19 2021

Keywords

Examples

			For n = 12 = 2*2*3, the corresponding prime gaps are 1, 1 and 2, thus a(12) = 1+1+2 = 4.
For n = 42 = 2*3*7, the corresponding prime gaps are 1, 2 and 4, thus a(42) = 1+2+4 = 7.

Links

Crossrefs

Cf. A001223, A001414, A003961, A007814, A056239, A064989, A347101, A347123.

Programs

  • PARI
    A347102(n) = { my(f=factor(n), s=0); for(i=1, #f~, s += f[i, 2]*(nextprime(f[i, 1]+1)-f[i,1])); (s); };

Formula

a(n) = A001414(A003961(n)) - A001414(n).
a(n) = A007814(n) + 2*A056239(A064989(A347123(n))).
For all n >= 0, a(2^n) = n.

A082508 Differences between consecutive primes that are powers of 2 in order of their appearance. Differences that are not powers of 2 are deleted from A001223.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 8, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 2, 4, 2, 4, 2, 4, 2, 2, 4, 2, 4, 2, 4, 2, 4, 8, 4, 8, 4, 8, 2, 2, 4, 8, 4, 2, 4, 8, 4, 8, 4, 2, 2, 2, 4, 2, 2, 4, 2, 4, 8, 8, 8, 4, 8, 4, 8, 4, 2, 2, 4, 2, 4, 2, 4, 4, 2, 4, 4, 8, 8, 4, 4, 8, 4, 2, 2, 2, 2, 4, 2, 4, 8, 2, 8, 8, 4, 2
Offset: 1

Views

Author

Labos Elemer, Apr 28 2003

Keywords

Links

Crossrefs

Cf. A001223, A061771.

Programs

  • Mathematica
    Do[s=Log[2, Prime[n+1]-Prime[n]]; If[IntegerQ[s], Print[Prime[n+1]]], {n, 1, 1000}]
  • PARI
    lista(pmax) = {my(p1 = 2, gap); forprime(p2 = 3, pmax, gap = p2 - p1; if(gap >> valuation(gap, 2) == 1, print1(gap, ", ")); p1 = p2);} \\ Amiram Eldar, Jun 06 2024

Formula

a(n) = A001223(A061771(n)). - Amiram Eldar, Jun 06 2024

A083550 Product of 2 consecutive prime differences of two successive terms of A001223.

Original entry on oeis.org

2, 4, 8, 8, 8, 8, 8, 24, 12, 12, 24, 8, 8, 24, 36, 12, 12, 24, 8, 12, 24, 24, 48, 32, 8, 8, 8, 8, 56, 56, 24, 12, 20, 20, 12, 36, 24, 24, 36, 12, 20, 20, 8, 8, 24, 144, 48, 8, 8, 24, 12, 20, 60, 36, 36, 12, 12, 24, 8, 20, 140, 56, 8, 8, 56, 84, 60, 20, 8, 24, 48, 48, 36, 24, 24, 48
Offset: 1

Views

Author

Labos Elemer, May 22 2003

Keywords

Links

Crossrefs

Cf. A001223, A083538-A083555, A057467.

Programs

  • Mathematica
    f[x_] := Prime[x+1]-Prime[x]
Table[f[w+1]*f[w], {w, 1, 128}]

Formula

a(n) = A001223(n)*A001223(n+1) = (prime(n+1)-prime(n))*(prime(n+2)-prime(n+1)).
Previous Showing 21-30 of 981 results. Next

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