A029710 -id:A029710 - cp's OEIS frontend

A204672 Primes followed by a gap of 120.

1895359, 2898239, 6085441, 7160227, 7784039, 7803491, 7826899, 8367397, 8648557, 9452959, 10052071, 10863973, 11630503, 11962823, 12109697, 12230233, 12415681, 14411737, 14531899, 15014557, 15020737, 15611909, 16179041

Offset: 1

M. F. Hasler, Jan 18 2012

nonn

M. F. Hasler, Table of n, a(n) for n = 1..433 (all a(n) < 10^8).
Index entries for primes, gaps between

Cf. A058193 (first gap of 6n), A140791 (first gap of 10n).
Cf. A126771 (gap 60), A126724 (gap 150), A204673 (gap 180).
Cf. A001359, A029710, A031924-A031938, A061779, A098974, A124594-A124596, A126784, A134116-A134124, A204665-A204670.

N = 2*10^7; % to get all terms <= N
P = primes(N+120);
J = find(P(2:end) - P(1:end-1) == 120);
P(J)  % Robert Israel, Feb 28 2017

Transpose[Select[Partition[Prime[Range[1100000]],2,1],Last[#]-First[#] == 120&]] [[1]] (* Harvey P. Dale, Jul 11 2014 *)

g=120;c=o=0;forprime(p=1,default(primelimit),(-o+o=p)==g&write("c:/temp/b204672.txt",c++" "p-g))

A271211 Composite integers sandwiched between primes p, q with q-p = 4.

Original entry on oeis.org

8, 9, 10, 14, 15, 16, 20, 21, 22, 38, 39, 40, 44, 45, 46, 68, 69, 70, 80, 81, 82, 98, 99, 100, 104, 105, 106, 110, 111, 112, 128, 129, 130, 164, 165, 166, 194, 195, 196, 224, 225, 226, 230, 231, 232, 278, 279, 280, 308, 309, 310, 314, 315, 316, 350, 351, 352, 380

Offset: 1

Michel Marcus, Apr 02 2016

nonn

			The composite number 8 is sandwiched between primes 7 and 11, and 11-7=4, so 8 is a member of the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Laurent Coppey, Décompositions multiplicatives directes des entiers, Diagrammes, 65-66 (2011), p. 1-68, in French, see J4 p. 11.

Cf. A014574, A029710, A045881.

Range[#[[1]]+1,#[[2]]-1]&/@Select[Partition[Prime[Range[100]],2,1],#[[2]]- #[[1]] == 4&]//Flatten (* Harvey P. Dale, Oct 12 2019 *)

lista(nn) = {forcomposite(c=4, nn, if ((p=precprime(c)) && ((nextprime(c)-p)==4), print1(c, ", ")););}

A339920 Primes p such that p^2 - p*q + q^2 is prime, where q is the next prime after p.

Original entry on oeis.org

2, 3, 53, 151, 167, 263, 373, 443, 467, 509, 523, 571, 1063, 1103, 1117, 1217, 1493, 1553, 1901, 1973, 2161, 2207, 2281, 2399, 2713, 2837, 2963, 3259, 3347, 3511, 4073, 4297, 4373, 4463, 4523, 4673, 4691, 4877, 5147, 5237, 5303, 5419, 5471, 5851, 6211, 6311, 6367

Offset: 1

Michel Marcus, Dec 23 2020

nonn

From Bernard Schott, Dec 23 2020: (Start)
Except for a(2)=3, (3, 5) gives A339698(2) = 19, there is no other pair of twin primes (p, p+2) (p in A001359) that gives a prime number of the form p^2-p*q+q^2 = p^2+2p+4.
There are no consecutive cousin primes (p, p+4) (p in A029710) that gives a prime number of the form p^2-pq+q^2 = p^2+4p+16.
There are no consecutive primes with a gap of 8 (p, p+8) (p in A031926) that give a prime number of the form p^2-pq+q^2 = p^2+8p+64. (End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A339698.

q:= 2: count:= 0: R:= NULL:
while count < 100 do
  p:= q; q:= nextprime(q);
  if isprime(p^2-p*q+q^2) then
    count:= count+1; R:= R, p;
  fi
od:
R; # Robert Israel, Dec 24 2020

forprime(p=1, 1e4, my(q=nextprime(p+1)); if(ispseudoprime(p^2-p*q+q^2), print1(p, ", ")));

A353072 Numbers k such that nextprime(k)-k is a positive square.

Original entry on oeis.org

1, 2, 4, 6, 7, 10, 12, 13, 16, 18, 19, 22, 25, 28, 30, 33, 36, 37, 40, 42, 43, 46, 49, 52, 55, 58, 60, 63, 66, 67, 70, 72, 75, 78, 79, 82, 85, 88, 93, 96, 97, 100, 102, 103, 106, 108, 109, 112, 118, 123, 126, 127, 130, 133, 136, 138, 140, 145, 148, 150, 153

Offset: 1

Tanya Khovanova, Apr 21 2022

nonn

Numbers p-1, where p is prime is a subsequence (see A006093).

			The next prime after 7 is 11, and 11-7 = 4 a square, so 7 is in this sequence.
The next prime after 118 is 127, 127-118 = 9 is a square, so 118 is a term.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A006093, A013632, A029710.

Select[Range[2000], NextPrime[#] - # > 0 && IntegerQ[Sqrt[NextPrime[#] - #]] &]
npsQ[n_]:=With[{c=NextPrime[n]-n},c>0&&IntegerQ[Sqrt[c]]]; Select[Range[200],npsQ] (* Harvey P. Dale, May 05 2023 *)

upto(n) = {my(res = List(1), q = 2, u = nextprime(n + 1)); forprime(p = 3, u, forstep(i = sqrtint(p - q), 1, -1, listput(res, p-i^2) ); q = p ); res } \\ David A. Corneth, Apr 22 2022

isok(k) = issquare(nextprime(k+1)-k); \\ Michel Marcus, Apr 22 2022

A078869 Number of n-tuples with elements in {2,4,6} which can occur as the differences between n+1 consecutive primes > n+1. (Values of a(11), ..., a(18) are conjectured to be correct, but are only known to be upper bounds.)

Original entry on oeis.org

3, 7, 15, 26, 38, 48, 67, 92, 105, 108, 109, 118, 130, 128, 112, 80, 36, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Dec 19 2002

nonn

The ">n+1" rules out n-tuples like (2,2), which only occurs for the primes 3, 5, 7. All terms from a(19) on equal 0.

An n-tuple (a_1,a_2,...,a_n) is counted iff the partial sums 0, a_1, a_1+a_2, ..., a_1+...+a_n do not contain a complete residue system (mod p) for any prime p.

Eric Weisstein's World of Mathematics, k-Tuple Conjecture

The 26 4-tuples and 38 5-tuples are in A078868 and A078870. Cf. A001359, A008407, A029710, A031924, A022004-A022007, A078852, A078858, A078946-A078969, A020497.

test[tuple_] := Module[{r, sums, i, j}, r=Length[tuple]; sums=Prepend[tuple.Table[If[j>=i, 1, 0], {i, 1, r}, {j, 1, r}], 0]; For[i=1, Prime[i]<=r+1, i++, If[Length[Union[Mod[sums, Prime[i]]]]==Prime[i], Return[False]]]; True]; tuples[0]={{}}; tuples[n_] := tuples[n]=Select[Flatten[Outer[Append, tuples[n-1], {2, 4, 6}, 1], 1], test]; a[n_] := Length[tuples[n]]

Edited by Dean Hickerson, Dec 20 2002

A111981 Numbers n such that 2n-1 and 2n+3 are consecutive primes.

Original entry on oeis.org

4, 7, 10, 19, 22, 34, 40, 49, 52, 55, 64, 82, 97, 112, 115, 139, 154, 157, 175, 190, 199, 220, 229, 232, 244, 250, 307, 322, 337, 370, 379, 385, 412, 427, 430, 439, 442, 454, 469, 484, 505, 544, 547, 607, 640, 649, 652, 712, 715, 724, 742, 745, 775, 784, 790

Offset: 1

Ray Chandler, Aug 24 2005

nonn

Essentially the same as A088762.

a(n) = (A029708(n)-1)/2 = (A029710(n)+1)/2 = (A031505(n)-3)/2.

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

A204813 Primes followed by a gap of 256 = nextprime(p)-p.

Original entry on oeis.org

1872851947, 2362150363, 2394261637, 2880755131, 2891509333, 3353981623, 3512569873, 3727051753, 3847458487, 4008610423, 4486630573, 4541745583, 4755895531, 4837532347, 5227869607, 5389475977, 6201260587, 6229685347, 6952228483, 7325665111, 7414468513

Offset: 1

M. F. Hasler, Jan 19 2012

nonn

Washington Bomfim, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between.

Cf. A204812, A204670, A126784, A031934, A031926, A029710, A001359; A062529, A138198, A123995.

list_gaps(g=256,f,N=25,p=0)=for(c=1,N,while(g+p!=p=nextprime(p+1),);if(f,write(f".txt",c" ",p-g),print1(", "p-g)))

a(8)-a(21) from Washington Bomfim

A224472 Primes followed by a gap of 300.

Original entry on oeis.org

4758958741, 5612345261, 6169169561, 6306815239, 6646984159, 7335508261, 8645089003, 8806019249, 9047808247, 9148138313, 9466071347, 9907846261, 10055451683, 11063821453, 11475026363, 11603081459, 12292390637, 12750876857, 13833827471, 14636472007, 15876700949

Offset: 1

Zak Seidov, Apr 07 2013

nonn

The first twin gap equal to 300 occurs for p = 6537587646371. - Giovanni Resta, Apr 07 2013

Zak Seidov, Table of n, a(n) for n = 1..3718
Index entries for primes, gaps between

Cf. A058193 (first gap of 6n), A140791 (first gap of 10n), A126771 (gap 60), A126724 (gap 150), A204673 (gap 180), A204807 (gap 200), A000230, A001359, A204672, A029710, A031924-A031938, A061779, A098974, A124594-A124596, A126784, A134116-A134124, A204665-A204670.

A346249 Möbius transform of A337549.

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 3, 8, 10, 1, 1, 8, 3, 3, 7, 28, 1, 10, 3, 12, 15, 1, 5, 28, 16, 3, 62, 24, 1, 7, 5, 92, 13, 1, 21, 52, 3, 3, 21, 44, 1, 15, 3, 24, 56, 5, 5, 92, 58, 16, 19, 36, 5, 62, 15, 84, 27, 1, 1, 44, 5, 5, 108, 292, 27, 13, 3, 36, 37, 21, 1, 168, 5, 3, 68, 48, 39, 21, 3, 148, 346, 1, 5, 84, 21, 3, 31, 92, 7, 56

Offset: 1

Antti Karttunen, Jul 14 2021

nonn

Cf. A000010, A000040, A001223, A003961, A003972, A008683, A337549.

Sequences A001359, A029710, A031924 give the subsets of positions of 1's, 3's and 5's in this sequence.

A003972(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); eulerphi(factorback(f)); };
A337549(n) = (A003972(n) - n);
A346249(n) = sumdiv(n,d,moebius(n/d)*A337549(d));

a(n) = A008683(n/d) * A337549(d).

a(A000040(n)) = A001223(n)-1. - Antti Karttunen, Dec 06 2021

cp's OEIS Frontend

A204672 Primes followed by a gap of 120.

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A271211 Composite integers sandwiched between primes p, q with q-p = 4.

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A339920 Primes p such that p^2 - p*q + q^2 is prime, where q is the next prime after p.

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A353072 Numbers k such that nextprime(k)-k is a positive square.

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A078869 Number of n-tuples with elements in {2,4,6} which can occur as the differences between n+1 consecutive primes > n+1. (Values of a(11), ..., a(18) are conjectured to be correct, but are only known to be upper bounds.)

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A111981 Numbers n such that 2n-1 and 2n+3 are consecutive primes.

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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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A204813 Primes followed by a gap of 256 = nextprime(p)-p.

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