A126721 -id:A126721 - cp's OEIS frontend

A134120 Primes p such that q-p = 42, where q is the next prime after p.

16141, 34981, 38501, 39251, 45439, 52391, 58789, 61169, 68821, 75437, 78737, 78989, 81239, 81569, 82657, 84871, 88547, 96601, 97729, 104417, 105277, 108301, 116047, 116747, 117991, 119447, 120431, 121081, 122401, 125821, 126781, 126989

Offset: 1

Rick L. Shepherd, Oct 08 2007

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

Cf. A126721, A134121.

Select[Partition[Prime[Range[12000]],2,1],#[[2]]-#[[1]]==42&][[All,1]] (* Harvey P. Dale, Aug 24 2019 *)

A134118 Primes p such that q - p = 38, where q is the next prime after p.

Original entry on oeis.org

30593, 38393, 39461, 45779, 49559, 51071, 56999, 58271, 60821, 63863, 68399, 72173, 72383, 75041, 78653, 79493, 85259, 90749, 95819, 104243, 105563, 109751, 110183, 111053, 111149, 111539, 116201, 119321, 125963, 126359, 129803, 134951

Offset: 1

Rick L. Shepherd, Oct 08 2007

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

Cf. A001223, A134117, A126721.

a:=proc(n) if nextprime(ithprime(n))-ithprime(n)=38 then ithprime(n) else end if end proc: seq(a(n),n=1..15000); # Emeric Deutsch, Oct 24 2007

Select[Prime[Range[5000]], NextPrime[#] == # + 38 &] (* Alonso del Arte, Apr 10 2016 *)
Select[Partition[Prime[Range[13000]],2,1],#[[2]]-#[[1]]==38&][[All,1]] (* Harvey P. Dale, Oct 03 2020 *)

lista(nn) = {p = 2; forprime (q = 3, nn, if (q-p == 38, print1(p, ", ")); p = q;);} \\ Michel Marcus, Apr 11 2016

A290450 Primes with property that the next prime has the same last digit.

Original entry on oeis.org

139, 181, 241, 283, 337, 409, 421, 547, 577, 631, 691, 709, 787, 811, 829, 887, 919, 1021, 1039, 1051, 1153, 1171, 1249, 1399, 1471, 1627, 1637, 1699, 1723, 1801, 1879, 2017, 2029, 2053, 2089, 2143, 2521, 2647, 2719, 2731, 2767, 2887, 2917, 3001, 3089, 3109, 3361, 3413, 3517, 3547, 3571

Offset: 1

Alonso del Arte, Aug 06 2017

Starts off the same as A031928, primes p such that the next prime is p + 10. First term that differs is 887, since 897 = 3 * 13 * 23 and the next prime is 907.
As the primes get larger and more sparsely distributed, the difference between successive primes is less likely to be less than 10.
One might expect that a prime is 1/4 as likely to be followed by a prime with the same least significant digit in base 10 (since the possibilities are 1, 3, 7, 9).
One might also expect this sequence to consist of a quarter of the primes. And yet pi(a(50)) = pi(3547) = 497; the 200th prime is 1223.

			139 is in the sequence because the immediately following prime is 149, which also ends in 9.
But 149 is not in the sequence because the next prime after that one is 151, which ends in 1, not 9.

Marius A. Burtea, Table of n, a(n) for n = 1..5005
Evelyn Lamb, "Peculiar Pattern Found in 'Random' Prime Numbers", Nature, March 14, 2016, republished by Scientific American.
Robert J. Lemke Oliver and Kannan Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv preprint arXiv:1603.03720 [math.NT], submitted on March 11, 2016.

Cf. A031928 (subset), A050434 (with 2 digits).

f:=func; a:=[]; for p in PrimesUpTo(4000) do if f(p,1) or f(p,3) or f(p,7) or f(p,9) then Append(~a,p); end if; end for; a; // Marius A. Burtea, Oct 16 2019

Select[Partition[Prime[Range[1000]], 2, 1], Mod[#[[1]], 10] == Mod[#[[2]], 10] &][[All, 1]] (* Harvey P. Dale, Aug 21 2017 *)
Module[{nn=1000,prs,p},prs=Prime[Range[nn]];p=Divisible[#,10]&/@ Differences[prs];Pick[Most[prs],p]] (* Harvey P. Dale, Aug 22 2017 *)

A031928 UNION A031938 UNION A124596 UNION A126721 UNION ... - R. J. Mathar, Jan 23 2022

A271981 Primes p such that p + 40 is also prime.

Original entry on oeis.org

3, 7, 13, 19, 31, 43, 61, 67, 73, 97, 109, 127, 139, 151, 157, 193, 199, 211, 223, 229, 241, 271, 277, 307, 313, 349, 379, 409, 421, 439, 463, 523, 547, 577, 601, 607, 613, 619, 643, 661, 733, 757, 769, 787, 823, 907, 937, 991, 1009, 1021, 1051, 1063, 1069

Offset: 1

Karl V. Keller, Jr., Apr 17 2016

nonn

A126721 is a subsequence of this sequence.

			3 is a term since 3 + 40 = 43 is also prime.
7 is a term since 7 + 40 = 47 is also prime.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000

Cf. A000040, A126721.

q:= n-> andmap(isprime, [n, n+40]):
select(q, [$2..2000])[];  # Alois P. Heinz, Jul 21 2022

Select[Prime@ Range@ 180, PrimeQ[# + 40] &] (* Michael De Vlieger, Apr 18 2016 *)

lista(nn) = forprime(p=2, nn, if (isprime(p+40), print1(p, ", "))); \\ Michel Marcus, Apr 19 2016

from sympy import isprime
for i in range(3, 2001,2):
     if isprime(i) and isprime(i+40): print (i,end=', ')

A174350 Square array: row n >= 1 lists the primes p for which the next prime is p+2n; read by antidiagonals.

Original entry on oeis.org

3, 5, 7, 11, 13, 23, 17, 19, 31, 89, 29, 37, 47, 359, 139, 41, 43, 53, 389, 181, 199, 59, 67, 61, 401, 241, 211, 113, 71, 79, 73, 449, 283, 467, 293, 1831, 101, 97, 83, 479, 337, 509, 317, 1933, 523, 107, 103, 131, 491, 409, 619, 773, 2113, 1069, 887

Offset: 1

Clark Kimberling, Mar 16 2010

Every odd prime p = prime(i), i > 1, occurs in this array, in row (prime(i+1) - prime(i))/2. Polignac's conjecture states that each row contains an infinite number of indices. In case this does not hold, we can use the convention to continue finite rows with 0's, to ensure the sequence is well defined. - M. F. Hasler, Oct 19 2018

A permutation of the odd primes (A065091). - Robert G. Wilson v, Sep 13 2022

			Upper left hand corner of the array:
     3     5    11    17    29    41    59    71   101 ...
     7    13    19    37    43    67    79    97   103 ...
    23    31    47    53    61    73    83   131   151 ...
    89   359   389   401   449   479   491   683   701 ...
   139   181   241   283   337   409   421   547   577 ...
   199   211   467   509   619   661   797   997  1201 ...
   113   293   317   773   839   863   953  1409  1583 ...
  1831  1933  2113  2221  2251  2593  2803  3121  3373 ...
   523  1069  1259  1381  1759  1913  2161  2503  2861 ...
  (...)
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: (lesser of) twin primes A001359.
Row 2: p(4) = 7, p(6) = 13, p(8) = 19,... these being the primes for which the next prime is 4 greater: (lesser of) cousin primes A029710.

Robert G. Wilson v, Falling antidiagonals 1..115, flattened (first 50 from T. D. Noe).
Fred B. Holt and Helgi Rudd, On Polignac's Conjecture, arxiv:1402.1970 [math.NT], 2014.

Cf. A000040, A001223, A065091, A174349.
Rows 1, 2, 3, ...: A001359, A029710, A031924, A031926, A031928 (row 5), A031930, A031932, A031934, A031936, A031938 (row 10), A061779, A098974, A124594, A124595, A124596 (row 15), A126784, A134116, A134117, A134118, A126721 (row 20), A134120, A134121, A134122, A134123, A134124 (row 25), A204665, A204666, A204667, A204668, A126771 (row 30), A204669, A204670.
Rows 35, 40, 45, 50, ...: A204792, A126722, A204764, A050434 (row 50), A204801, A204672, A204802, A204803, A126724 (row 75), A184984, A204805, A204673, A204806, A204807 (row 100); A224472 (row 150).
Column 1: A000230.
Column 2: A046789.

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}]; Table[t2[[b, a - b + 1]], {a, rows}, {b, a}] (* T. D. Noe, Feb 11 2014 *)
t[r_, 0] = 2; t[r_, c_] := Block[{p = NextPrime@ t[r, c - 1], q}, q = NextPrime@ p; While[ p + 2r != q, p = q; q = NextPrime@ q]; p]; Table[ t[r - c + 1, c], {r, 10}, {c, r, 1, -1}] (* Robert G. Wilson v, Nov 06 2020 *)

A174350_row(g, N=50, i=0, p=prime(i+1), L=[])={g*=2; forprime(q=1+p, , i++; if(p+g==p=q, L=concat(L, q-g); N--||return(L)))} \\ Returns the first N terms of row g. - M. F. Hasler, Oct 19 2018

a(n) = A000040(A174349(n)). - Michel Marcus, Mar 30 2016

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

A320718 Indices of primes followed by a gap (distance to next larger prime) of 40.

Original entry on oeis.org

2191, 2344, 2524, 2788, 3562, 4058, 4677, 5030, 5349, 6076, 6145, 6256, 6320, 6442, 6454, 6902, 7232, 7488, 8119, 8152, 8245, 8366, 8553, 8567, 8591, 8746, 9260, 9361, 10536, 10735, 11095, 11407, 11534, 11781, 12227, 12312, 12663, 12815, 12940, 13015, 13333, 13676, 13873, 14065, 14123

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A126721.

Index entries for primes, gaps between.

Equals A000720 o A126721.
Row 20 of A174349.
Subsequence of A107730 (prime(n+1) ends in same digit as prime(n)).
Indices of 40'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).

A(N=100,g=40,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

a(n) = A000720(A126721(n)).

A320718 = { i > 0 | prime(i+1) = prime(i) + 40 } = A001223^(-1)({40}).

A271323 Numbers n such that n - 41, n - 1, n + 1, n + 41 are consecutive primes.

Original entry on oeis.org

383220, 1269642, 1528938, 2590770, 3014700, 3158298, 3697362, 3946338, 4017312, 4045050, 4545642, 4711740, 4851618, 4871568, 5141178, 5194602, 5925042, 5972958, 5990820, 6075030, 6179862, 6212202, 6350760, 6442938, 6549312, 6910638, 6912132

Offset: 1

Karl V. Keller, Jr., May 15 2016

nonn

This sequence is a subsequence of A014574 (average of twin prime pairs) and A256753.
The terms ending in 0 belong to A249674 (divisible by 30).
The terms ending in 2 (resp. 8) are congruent to 12 (resp. 18) mod 30.
The numbers n - 40 and n + 1 belong to A126721 (p such that p + 40 is the next prime) and A271981 (p and p + 40 are primes).
The numbers n - 40 and n - 1 belong to A271982 (p and p + 42 are primes).

			383220 is the average of the four consecutive primes 383179, 383219, 383221, 383261.
1269642 is the average of the four consecutive primes 1269601, 1269641, 1269643, 1269683.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A014574, A077800 (twin primes), A249674, A256753.

Mean/@Select[Partition[Prime[Range[472000]],4,1],Differences[#] == {40,2,40}&] (* Harvey P. Dale, Oct 16 2021 *)

from sympy import isprime,prevprime,nextprime
for i in range(0,12000001,6):
  if isprime(i-1) and isprime(i+1) and prevprime(i-1) == i-41 and nextprime(i+1) == i+41: print (i,end=', ')

cp's OEIS Frontend

A134120 Primes p such that q-p = 42, where q is the next prime after p.

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A134118 Primes p such that q - p = 38, where q is the next prime after p.

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A290450 Primes with property that the next prime has the same last digit.

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A271981 Primes p such that p + 40 is also prime.

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A174350 Square array: row n >= 1 lists the primes p for which the next prime is p+2n; read by antidiagonals.

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A320718 Indices of primes followed by a gap (distance to next larger prime) of 40.

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A271323 Numbers n such that n - 41, n - 1, n + 1, n + 41 are consecutive primes.

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