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 1-5 of 5 results.

A080376 Numbers where A080374 increases.

Original entry on oeis.org

2, 4, 9, 24, 30, 34, 99, 189, 217, 282, 367, 738, 3302, 3427, 3644, 3793, 4612, 7970, 8688, 14357, 23283, 34202, 49414, 85633, 85787, 103520, 224659, 273413, 415069, 474029, 685903, 2386432, 2398788, 2959782, 4875380, 6169832, 9330121, 12768473, 13879771, 17681799
Offset: 1

Views

Author

Labos Elemer, Feb 27 2003

Keywords

Comments

Numbers where a consecutive prime-difference (prime(a(n)+1)-prime(a(n))) arises with a new prime-power factor.

Examples

			From _Michael De Vlieger_, May 12 2017: (Start)
Values of A080374 starting at a(n).
    n     a(n)   A080374(a(n))
    1       2    1
    2       4    2
    3       9    4
    4      24    12
    5      30    24
    6      34    168
    7      99    840
    8     189    2520
    9     217    27720
   10     282    471240
   11     367    942480
   12     738    12252240
   13    3302    24504480
   14    3427    465585120
   15    3644    2327925600
   16    3793    72165693600
   17    4612    216497080800
   18    7970    6278415343200
   19    8688    144403552893600
   20   14357    288807105787200
   21   23283    12418705548849600
   22   34202    509166927502833600
   23   49414    18839176317604843200
   24   85633    131874234223233902400
   25   85787    6989334413831396827200
...
(End)

Links

Crossrefs

Cf. A001223, A080374.

Programs

  • Mathematica
    s=1; Do[s1=s; s=LCM[s, Prime[n+1]-Prime[n]]; If[Greater[s, s1], Print[n]], {n, 1, 100000}]
(* Second program: *)
Most[Accumulate@ #2 + 1] & @@ Transpose@ Map[{First@ #, Length@ #} &, Split@ FoldList[LCM @@ {#1, #2} &, Differences@ Array[Prime, 10^4]]] (* Michael De Vlieger, May 12 2017 *)
  • PARI
    lista(pmax) = {my(k = 1, p1 = 2, lcmmax = 1, lcm1 = 1, d); forprime(p2 = 3, pmax, d = p2 - p1;  lcm1 = lcm(lcm1, d); if(lcm1 > lcmmax, lcmmax = lcm1; print1(k, ", ")); p1 = p2; k++);} \\ Amiram Eldar, Jun 09 2024

Extensions

Edited by N. J. A. Sloane, May 13 2017 at the suggestion of Michael De Vlieger.
More terms from Amiram Eldar, Jun 09 2024

A080375 Distinct values of A080374, where A080374(n) is the lcm of the first n consecutive prime differences.

Original entry on oeis.org

1, 2, 4, 12, 24, 168, 840, 2520, 27720, 471240, 942480, 12252240, 24504480, 465585120, 2327925600, 72165693600, 216497080800, 6278415343200, 144403552893600, 288807105787200, 12418705548849600, 509166927502833600, 18839176317604843200, 131874234223233902400, 6989334413831396827200, 328498717450075650878400
Offset: 1

Views

Author

Labos Elemer, Feb 27 2003

Keywords

Links

Crossrefs

Cf. A001223, A080374, A080376.

Programs

  • Mathematica
    s=1; Do[s1=s; s=LCM[s, Prime[n+1]-Prime[n]]; If[Greater[s, s1], Print[s]], {n, 1, 100000}]
Module[{nn=100000,dprs},dprs=Differences[Prime[Range[nn]]];Table[LCM@@ Take[ dprs,n],{n,nn-1}]]//Union (* Harvey P. Dale, Nov 06 2021 *)

Extensions

More terms from Harvey P. Dale, Nov 06 2021

A080377 Prime gaps where A080374 increases.

Original entry on oeis.org

2, 4, 6, 8, 14, 10, 18, 22, 34, 16, 26, 32, 38, 50, 62, 54, 58, 46, 64, 86, 82, 74, 98, 106, 94, 118, 122, 128, 146, 134, 142, 162, 178, 158, 166, 202, 194, 206, 214, 218, 242, 250, 226, 254, 274, 262, 256, 278, 326, 302, 298, 314, 382, 346, 358, 338, 394, 334, 386, 362, 398, 446, 454, 486
Offset: 1

Views

Author

Labos Elemer, Feb 27 2003

Keywords

Comments

a(n+1) is the smallest prime gap (A001223) that has a prime factor not present in previous gaps or was present but at a lower power.

Examples

			18 is the 7th term: in the first 6 terms, {2, 4, 6, 8, 14, 10}, 3 does not occur with power 2 unlike in 18 = 2 * 3^2.
22 is the 8th term: in the first 7 terms 11 is not a prime factor unlike 22.
Several even numbers do not arise in this sequence, e.g., 12, 20, 36, 48, etc..

Crossrefs

Cf. A001223, A080374, A080375, A080376.

Programs

  • Mathematica
    s=1; Do[s1=s; s=LCM[s, d=Prime[n+1]-Prime[n]]; If[Greater[s, s1], Print[d]], {n, 1, 10000000}]

Formula

a(n) = prime(1+A080376(n)) - prime(A080376(n)).

Extensions

More terms from Amiram Eldar, Feb 09 2025

A081411 Partial product of prime gaps: a(n) = a(n-1)*(prime(n+1) - prime(n)).

Original entry on oeis.org

1, 2, 4, 16, 32, 128, 256, 1024, 6144, 12288, 73728, 294912, 589824, 2359296, 14155776, 84934656, 169869312, 1019215872, 4076863488, 8153726976, 48922361856, 195689447424, 1174136684544, 9393093476352, 37572373905408, 75144747810816, 300578991243264, 601157982486528
Offset: 1

Views

Author

Labos Elemer, Apr 01 2003

Keywords

Comments

Original name was: Generated by recursion: a(n)=(Mod[Prime[n+1],Prime[n]]*n[n-1]; a[0]=1; Product of the first n consecutive prime-differences.

Links

Crossrefs

Cf. A001223, A080374, A080375, A080376, A099002.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = a[n - 1] * (Prime[n + 1] - Prime[n]); Array[a, 30] (* Amiram Eldar, Nov 19 2020 *)
  • PARI
    diff(v)=vector(#v-1,i,v[i+1]-v[i])
pprod(v)=my(t=1); vector(#v,i,t*=v[i])
pprod(diff(primes(50))) \\ Charles R Greathouse IV, Mar 27 2014

Formula

Sum_{n>=1} 1/a(n) = A099002. - Amiram Eldar, Nov 19 2020

Extensions

New name from Charles R Greathouse IV, Mar 27 2014
More terms from Amiram Eldar, Nov 19 2020

A083273 a(n) is the quotient of lcm of first n consecutive prime differences and A001223(n), the n-th difference between consecutive primes.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 6, 2, 3, 6, 3, 2, 2, 6, 2, 3, 6, 2, 3, 2, 3, 6, 12, 6, 12, 6, 12, 42, 28, 84, 84, 420, 140, 140, 210, 140, 140, 420, 84, 420, 210, 420, 70, 70, 210, 420, 210, 140, 420, 84, 140, 140, 140, 420, 140, 210, 420, 84, 60, 210, 420, 210, 60, 140, 84, 420, 210
Offset: 1

Views

Author

Labos Elemer, May 15 2003

Keywords

Links

Crossrefs

Cf. A001223, A080374, A080375, A080376.

Programs

  • Mathematica
    ld[x_] := Apply[LCM, Table[Prime[w+1]-Prime[w], {w, 1, x}]]; Table[ld[j]/(Prime[j+1]-Prime[j]), {j, 1, 100}]
seq[len_] := Module[{d = Differences[Prime[Range[len+1]]]}, FoldList[LCM, d]/d]; seq[70] (* Amiram Eldar, Feb 08 2025 *)

Formula

a(n) = A080374(n)/A001223(n).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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