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 25 results. Next

A116496 Numbers k such that difference between k-th prime and next prime is 100.

Original entry on oeis.org

33608, 66762, 100978, 124508, 125049, 172619, 202315, 233905, 256422, 286306, 306691, 320569, 326694, 334412, 362134, 374275, 382591, 395155, 414640, 428335, 440270, 467181, 493060, 511698, 518536, 555912, 561795, 567479, 590434, 592581
Offset: 1

Views

Author

Zak Seidov, Feb 18 2006

Keywords

Examples

			p(33609) - p(33608) = 396833 - 396733 = 100.

Links

Crossrefs

Cf. A050434, A116493, A116495, A116497.

Programs

  • Mathematica
    PrimePi/@Transpose[Select[Partition[Prime[Range[600000]],2,1],#[[2]]-#[[1]] == 100&]][[1]] (* Harvey P. Dale, Sep 22 2013 *)
  • PARI
    lista(nn) = {my(i=0, q=2); forprime(p=3, nn, i++; if(p-q==100, print1(i, ", ")); q=p); } \\ Jinyuan Wang, Jan 29 2020

Formula

A050434(n) = prime(a(n)). - R. J. Mathar, Apr 30 2024

A116495 Numbers k such that difference between k-th prime and next prime is 210.

Original entry on oeis.org

1319945, 5240989, 9223725, 9359659, 12606992, 13798935, 16163072, 17559961, 18805047, 18973547, 20278475, 22091310, 22358431, 24253281, 25377875, 26813808, 28784698, 29308594, 29819740, 29867343, 30573917, 30838655
Offset: 1

Views

Author

Zak Seidov, Feb 18 2006

Keywords

Examples

			p(1319946) - p(1319945) = 20831533 - 20831323 = 210.

Links

Crossrefs

Cf. A116493, A116496, A116497.

Programs

  • PARI
    lista(nn) = {my(i=0, q=2); forprime(p=3, nn, i++; if(p-q==210, print1(i, ", ")); q=p); } \\ Jinyuan Wang, Jan 29 2020

A116497 Indices n such that the difference between the n-th prime and the next larger prime is 200.

Original entry on oeis.org

20226285, 23169912, 25441017, 26172843, 27841352, 29657240, 30714253, 31998495, 33419255, 36637876, 37995065, 38182448, 38758272, 39019864, 40022755, 40296600, 40339473, 40985817, 41293073, 42116899, 45474429
Offset: 1

Views

Author

Zak Seidov, Feb 18 2006

Keywords

Examples

			p(20226286) - p(20226285) = 378044179 - 378043979 = 200.

Links

Crossrefs

Cf. A116493, A116495, A116496.
The actual primes are given in A204807. - M. F. Hasler, Jan 19 2012
See also A052187 and references therein.

Programs

  • PARI
    lista(nn) = {my(i=0, q=2); forprime(p=3, nn, i++; if(p-q==200, print1(i, ", ")); q=p); } \\ Jinyuan Wang, Jan 29 2020

Formula

A204807(n) = prime(a(n)). - M. F. Hasler, Jan 19 2012

A174349 Square array: row n gives the indices i for which prime(i+1) = prime(i) + 2n; read by falling antidiagonals.

Original entry on oeis.org

2, 3, 4, 5, 6, 9, 7, 8, 11, 24, 10, 12, 15, 72, 34, 13, 14, 16, 77, 42, 46, 17, 19, 18, 79, 53, 47, 30, 20, 22, 21, 87, 61, 91, 62, 282, 26, 25, 23, 92, 68, 97, 66, 295, 99, 28, 27, 32, 94, 80, 114, 137, 319, 180, 154, 33, 29, 36, 124, 82, 121, 146, 331, 205, 259, 189
Offset: 1

Views

Author

Clark Kimberling, Mar 16 2010

Keywords

Comments

It is conjectured that every positive integer except 1 occurs in the array.
From M. F. Hasler, Oct 19 2018: (Start)
The above conjecture is obviously true: the integer i appears in row (prime(i+1) - prime(i))/2.
Polignac's Conjecture states that all rows are of infinite length.
To ensure the sequence is well-defined in case the conjecture would not hold, we can use the convention that finite rows are continued by 0's. (End)

Examples

			Corner of the array:
   2    3    5    7    10    13 ...
   4    6    8   12    14    17 ...
   9   11   15   16    18    21 ...
  24   72   77   79    87    92 ...
  34   42   53   61    68    80 ...
  46   47   91   97   114   121 ...
  (...)
Row 1: p(2) = 3, p(3) = 5, p(5) = 11, p(7) = 17, ..., these being the primes for which the next prime is 2 greater, cf. A029707.
Row 2: p(4) = 7, p(6) = 13, p(8) = 19, ..., these being the primes for which the next prime is 4 greater, cf. A029709.

Links

Crossrefs

Cf. A000040, A000720, A001223, A174350.
Rows 1, 2, 3, ... are A029707, A029709, A320701, ..., A320720; A116493 (row 35), A116496 (row 50), A116497 (row 100), A116495 (row 105).
Column 1 is A038664.

Programs

  • Mathematica
    rows = 10; t2 = {}; Do[t = {}; p = Prime[2]; While[Length[t] < rows - off + 1, nextP = NextPrime[p]; If[nextP - p == 2*off, AppendTo[t, p]]; p = nextP]; AppendTo[t2, t], {off, rows}]; t3 = Table[t2[[b, a - b + 1]], {a, rows}, {b, a}]; PrimePi /@ t3 (* T. D. Noe, Feb 11 2014 *)

Formula

a(n) = A000720(A174350(n)). - Michel Marcus, Mar 30 2016

Extensions

Name corrected and other edits by M. F. Hasler, Oct 19 2018

A320701 Indices of primes followed by a gap (distance to next larger prime) of 6.

Original entry on oeis.org

9, 11, 15, 16, 18, 21, 23, 32, 36, 37, 39, 40, 51, 54, 55, 56, 58, 67, 71, 73, 74, 76, 84, 86, 96, 100, 102, 103, 105, 107, 108, 110, 111, 118, 119, 123, 129, 130, 133, 160, 161, 164, 165, 167, 170, 174, 179, 184, 185, 187, 188, 194, 195, 199, 200, 202, 208, 210, 216, 218, 219, 227, 231
Offset: 1

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes given in A031924.
Subsequence of indices of sexy primes A023201.

Links

Crossrefs

Equals A000720 o A031924.
Row 3 of A174349.
Indices of 6's in A001223.
Cf. A029707, A029709, A320702, A320703, ..., A320720 (analog for gaps 2, 4, 8, 10, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

Programs

  • Mathematica
    Position[Differences[Prime[Range[250]]],6]//Flatten (* Harvey P. Dale, Oct 13 2022 *)
  • PARI
    A(N=100,g=6,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

Formula

a(n) = A000720(A031924(n)).
A320701 = { i > 0 | prime(i+1) = prime(i) + 6 } = A001223^(-1)({6}).

A320702 Indices of primes followed by a gap (distance to next larger prime) of 8.

Original entry on oeis.org

24, 72, 77, 79, 87, 92, 94, 124, 126, 128, 132, 135, 156, 158, 166, 186, 192, 196, 220, 228, 241, 246, 248, 270, 281, 299, 304, 325, 330, 334, 338, 364, 370, 379, 386, 393, 400, 413, 417, 421, 432, 436, 454, 456, 482, 488, 507, 517, 519, 538, 589, 594, 620, 640, 661, 676, 689, 691, 712, 736, 750, 759
Offset: 1

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes given in A031926.

Links

Crossrefs

Equals A000720 o A031926.
Row 4 of A174349.
Indices of 8's in A001223.
Cf. A029707, A029709, A320701, A320703, ..., A320720 (analog for gaps 2, 4, 6, 10, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

Programs

  • Magma
    [n: n in [1..800] | NthPrime(n+1) - NthPrime(n) eq 8]; // Vincenzo Librandi, Mar 21 2019
  • Maple
    p:= 2: Res:= NULL: count:= 0:
for n from 1 while count < 100 do
  q:= nextprime(p);
  if q-p = 8 then count:= count+1; Res:= Res, n; fi;
  p:= q;
od:
Res; # Robert Israel, Oct 19 2018
  • Mathematica
    Select[Range[800], Prime[#] + 8 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 21 2019 *)
  • PARI
    A_vec(N=100,g=8,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)}

Formula

a(n) = A000720(A031926(n)) = A174349(4,n).
A320702 = { i > 0 | prime(i+1) = prime(i) + 8 } = A001223^(-1)({8}).

A320720 Indices of primes followed by a gap (distance to next larger prime) of 44.

Original entry on oeis.org

1831, 3861, 4009, 7499, 8937, 10328, 10427, 11725, 12904, 12926, 13011, 13051, 16596, 16915, 18280, 20055, 20160, 20352, 20619, 21458, 21465, 21550, 21659, 23752, 23934, 24107, 24384, 24445, 24651, 24871, 24933, 24992, 25027, 26089, 26166, 26483, 26923, 27038, 27048, 28898, 29343
Offset: 1

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes listed in A134121.

Links

Crossrefs

Cf. A029707, A029709 (analog for gaps 2 and 4), A320701, A320702, ... A320719 (analog for gaps 6, 8, 10, ..., 42), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).
Equals A000720 o A134121.
Indices of 44's in A001223.
Row 22 of A174349.

Programs

  • PARI
    A(N=100,g=44,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

Formula

a(n) = A000720(A134121(n)).

A107730 Numbers n such that prime(n+1) has the same last digit as prime(n).

Original entry on oeis.org

34, 42, 53, 61, 68, 80, 82, 101, 106, 115, 125, 127, 138, 141, 145, 154, 157, 172, 175, 177, 191, 193, 204, 222, 233, 258, 259, 266, 269, 279, 289, 306, 308, 310, 316, 324, 369, 383, 397, 399, 403, 418, 422, 431, 442, 443, 474, 480, 491, 497, 500, 502, 518
Offset: 1

Views

Author

Jonathan Vos Post, Jun 12 2007

Keywords

Examples

			a(1) = 34 because prime(34) = 139, prime(35) = 149, both end with the digit 9.
a(2) = 42 because prime(42) = 181, prime(43) = 191, both end with the digit 1.
a(4) = 61 because prime(61) = 283, prime(62) = 293, both end with the digit 3.
a(5) = 68 because prime(68) = 337, prime(69) = 347, both end with the digit 7.

Links

Crossrefs

Cf. A000040, A129750.
Union of rows r == 0 (mod 5) of A174349. Indices of multiples of 10 (A008592) in A001223.

Programs

  • GAP
    P:=List(Filtered([1..4000],IsPrime),n->Reversed(ListOfDigits(n)));;
a:=Filtered([1..Length(P)-1],i->P[i+1][1]=P[i][1]); # Muniru A Asiru, Oct 31 2018
  • Maple
    isA107730 := proc(n) local ldign, ldign2 ; ldign := convert(ithprime(n),base,10) ; ldign2 := convert(ithprime(n+1),base,10) ; if op(1,ldign) = op(1,ldign2) then true ; else false ; fi ; end: for n from 1 to 600 do if isA107730(n) then printf("%d, ",n) ; fi ; od ; # R. J. Mathar, Jun 15 2007
  • Mathematica
    Select[Range[200],IntegerDigits[Prime[ # ]][[ -1]]==IntegerDigits[Prime[ #+1]][[ -1]]&] (* Stefan Steinerberger, Jun 14 2007 *)
Flatten[Position[Partition[Prime[Range[600]],2,1],?(Mod[#[[1]],10] == Mod[#[[2]],10]&),{1},Heads->False]] (* _Harvey P. Dale, Aug 20 2015 *)
  • PARI
    isok(n) = (prime(n) % 10) == prime(n+1) % 10; \\ Michel Marcus, Feb 16 2017
  • PARI
    is_A107730(n)=!((nextprime(1+n=prime(n))-n)%10) \\ This (...) is twice as fast as prime(n+1)-prime(n), and prime(n) becomes very slow for n > 41538, even with primelimit = 10^7. - M. F. Hasler, Oct 24 2018

Formula

Numbers n such that A000040(n)==A000040(n+1) mod 10, or A000040(n+1) - A000040(n) = 10*k for some integer k, or n such that A129750(n) = 0. [Corrected and edited by M. F. Hasler, Oct 24 2018]
A107730 = A001223^(-1)(A008592) = { i > 0 | A001223(i) == 0 (mod 10)} = U_{k>0} {A174349(5k,j); j >= 1}. - M. F. Hasler, Oct 24 2018
Union of A320703, A320708, A320713, A320718, ... A116493,..., A116496 ... etc. - R. J. Mathar, Apr 30 2024

Extensions

More terms from Stefan Steinerberger and R. J. Mathar, Jun 14 2007

A320703 Indices of primes followed by a gap (distance to next larger prime) of 10.

Original entry on oeis.org

34, 42, 53, 61, 68, 80, 82, 101, 106, 115, 125, 127, 138, 141, 145, 157, 172, 175, 177, 191, 193, 204, 222, 233, 258, 266, 269, 279, 289, 306, 308, 310, 316, 324, 369, 383, 397, 399, 403, 418, 422, 431, 443, 474, 491, 497, 500, 502, 518, 525, 531, 535, 575
Offset: 1

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes given in A031928.

Links

Crossrefs

Equals A000720 o A031928.
Row 5 of A174349.
Indices of 10's in A001223.
Subsequence of A107730: prime(n+1) ends in same digit as prime(n).
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

Programs

Formula

a(n) = A000720(A031928(n)).
A320703 = { i > 0 | prime(i+1) = prime(i) + 10 }.

A320708 Indices of primes followed by a gap (distance to next larger prime) of 20.

Original entry on oeis.org

154, 259, 442, 480, 548, 753, 777, 783, 876, 971, 1035, 1066, 1095, 1106, 1147, 1254, 1277, 1302, 1337, 1345, 1355, 1381, 1396, 1400, 1423, 1438, 1562, 1592, 1613, 1662, 1669, 1808, 1955, 2016, 2043, 2081, 2116, 2129, 2147, 2226, 2302, 2307, 2387, 2517, 2547, 2563, 2694, 2724, 2745, 2755, 2766
Offset: 1

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes listed in A031938.

Links

Crossrefs

Equals A000720 o A031938.
Row 10 of A174349.
Subsequence of A107730 (prime(n+1) ends in same digit as prime(n)).
Indices of 20's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

Programs

  • Magma
    [n: n in [1..3000] | NthPrime(n+1) - NthPrime(n) eq 20]; // Vincenzo Librandi, Mar 22 2019
  • Mathematica
    Select[Range[3000], Prime[#] + 20 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 22 2019 *)
  • PARI
    A(N=100,g=20,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

Formula

a(n) = A000720(A031938(n)).
A320708 = { i > 0 | prime(i+1) = prime(i) + 20 } = A001223^(-1)({20}).
Showing 1-10 of 25 results. Next

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