A031924 -id:A031924 - cp's OEIS frontend

A076976 Product of the smallest prime divisors of composite numbers between successive primes.

1, 2, 2, 12, 2, 12, 2, 12, 120, 2, 120, 12, 2, 12, 168, 120, 2, 120, 12, 2, 168, 12, 120, 1680, 12, 2, 12, 2, 12, 2217600, 12, 168, 2, 15840, 2, 120, 168, 12, 312, 120, 2, 15840, 2, 12, 2, 221760, 262080, 12, 2, 12, 120, 2, 18720, 264, 168, 120, 2, 120, 12, 2, 34272

Offset: 1

Amarnath Murthy, Oct 23 2002

nonn

From Bernard Schott, Apr 09 2020: (Start)
a(n) = 2 iff prime(n) is in A001359 (prime gap=2).
a(n) = 12 iff prime(n) is in A029710 (prime gap=4).
a(n) = 24 * p with p prime >= 5 iff prime(n) is in A031924 (prime gap=6).
a(n) = 2^m * q with q odd >= 3 iff prime(n+1) - prime(n) = 2*m where m = A007814(a(n)). (End)

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

Cf. A029707 (a(n)=2), A029709 (a(n)=12), A076977.

Cf. A001359, A029710, A031924.

p:= 2:
for i from 1 to 100 do
  q:= p; p:= nextprime(p);
  A[i]:= mul(min(numtheory:-factorset(i)),i=q+1..p-1);
od:
seq(A[i],i=1..100); # Robert Israel, Mar 30 2020

pspd[{p1_,p2_}]:=Times@@(FactorInteger[#][[1,1]]&/@Range[p1+1,p2-1]); pspd/@Partition[ Prime[Range[70]],2,1] (* Harvey P. Dale, Jan 12 2024 *)

a(n) = {my(p=1, pn=prime(n)); forcomposite(c=pn, nextprime(pn+1)-1, p *= vecmin(factor(c)[,1]);); p;} \\ Michel Marcus, Mar 31 2020

More terms from Sascha Kurz, Jan 22 2003

A080841 Number of pairs (p,q) of (not necessarily consecutive) primes with q-p = 6 and q < 10^n.

Original entry on oeis.org

0, 15, 74, 411, 2447, 16386, 117207, 879908, 6849047, 54818296, 448725003, 3741217498

Offset: 1

Jason Earls, Mar 28 2003

nonn

Note that one has to be careful to distinguish between pairs of consecutive primes (p,q) with q-p = 6 (A031924), and pairs of primes (p,q) with q-p = 6 (A023201). Here we consider the latter, whereas A093738 considers the former. - N. J. A. Sloane, Mar 07 2021

A. Granville, G. Martin, Prime number races, Amer. Math. Monthly vol 113, no 1 (2006) p 1.
Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021].

Cf. A023201, A046117, A007508, A080840, A006512, A046132, A046117.

A204672 Primes followed by a gap of 120.

Original entry on oeis.org

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))

A287050 Square array read by antidiagonals upwards: M(n,k) is the initial occurrence of first prime p1 of consecutive primes p1, p2, where p2 - p1 = 2*k, and p1, p2 span a multiple of 10^n, n>=1, k>=1.

Original entry on oeis.org

29, 599, 7, 2999, 97, 47, 179999, 1999, 1097, 89, 23999999, 69997, 21997, 1193, 139, 23999999, 199999, 369997, 23993, 691, 199, 29999999, 19999999, 3199997, 149993, 10993, 199, 113, 17399999999, 19999999, 6999997, 1199999, 139999, 997, 293, 1831

Offset: 1

Hartmut F. W. Hoft, May 18 2017

The unit digits of the numbers in the matrix representation M(n,k) are 9's for column 1, 7's or 9's for column 2, 7's for column 3, 3's or 9's for column 4, and 1's, 3's, 7's or 9's for column 5.
The following matrix terms appear as first terms in sequence
A060229(1) = M(1,1)
A288021(1) = M(1,2)
A288022(1) = M(1,3)
A288024(1) = M(1,4)
A031928(1) = M(1,5)
A158277(1) = M(2,1)
A160440(1) = M(2,2)
A160370(1) = M(2,3)
A287049(1) = M(2,4)
A160500(1) = M(2,5)
A158861(1) = M(3,1).

			The matrix representation of the sequence with row n indicating the spanned power of 10 and column k indicating the difference of 2*k between the first pair of consecutive primes spanning a multiple of 10^n:
--------------------------------------------------------------------------
n\k   1             2             3             4            5
--------------------------------------------------------------------------
1 |   29            7             47            89           139
2 |   599           97            1097          1193         691
3 |   2999          1999          21997         23993        10993
4 |   179999        69997         369997        149993       139999
5 |   23999999      199999        3199997       1199999      1999993
6 |   23999999      19999999      6999997       38999993     1999993
7 |   29999999      19999999      159999997     659999999    379999999
8 |   17399999999   7699999999    9399999997    8999999993   499999993
9 |   92999999999   135999999997  85999999997   8999999993   28999999999
10|   569999999999  519999999997  369999999997  29999999993  819999999997
...
Every column in the matrix is nondecreasing.
For the first and fourth columns, ceiling(M[n,1]/10^n) and ceiling(M[n,4]/10^n) are divisible by 3, for all n>=1 (see A158277 and A287049).

Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049.

Cf. A060229, A288021, A288022, A288024.

M(n,k) = min( p_i : p_(i+1) - p_i = 2*k, p_i and p_(i+1) consecutive primes and p_i < m*10^n < p_(i+1) for some integer m) where p_j is the j-th prime, n>=1 and k>=1.

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

A124590 Primes p such that q-p <= 6, where q is the next prime after p.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 97, 101, 103, 107, 109, 127, 131, 137, 149, 151, 157, 163, 167, 173, 179, 191, 193, 197, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 277, 281, 307, 311, 313, 331, 347, 349, 353, 367

Offset: 1

N. J. A. Sloane, Dec 19 2006

Goldston, Graham, Pintz, & Yilidirm give a conditional proof that this sequence is infinite; see their Theorem 4. - Charles R Greathouse IV, Jul 31 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yilidirm, Small gaps between primes or almost primes (2005)
K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc. 44 (2007), pp. 1-18.
Index entries for primes, gaps between

Cf. A083371, A124589, A031924.

v=List([2]);p=3;forprime(q=5,1e3,if(q-p<=6,listput(v,p));p=q);Vec(v) \\ Charles R Greathouse IV, Jul 31 2013

list(lim)=my(v=List(),p=2); forprime(q=3,nextprime(lim\1+1), if(q-p<7, listput(v,p)); p=q); Vec(v) \\ Charles R Greathouse IV, Jan 31 2017

A000040 MINUS A083371. - R. J. Mathar, Jun 15 2008
A124589 UNION A031924. - R. J. Mathar, Jan 23 2022
a(n) >> n log^2 n. - Charles R Greathouse IV, Jan 31 2017

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

A210477 Product of adjacent primes with a gap of 6.

Original entry on oeis.org

667, 1147, 2491, 3127, 4087, 5767, 7387, 17947, 23707, 25591, 28891, 30967, 55687, 64507, 67591, 70747, 75067, 111547, 126727, 136891, 141367, 148987, 190087, 198907, 256027, 295927, 313591, 320347, 329467, 348091, 355207, 364807, 372091, 422491, 430327, 462391, 532891

Offset: 1

R. J. Mathar, Jan 23 2013

nonn

Subsequence of A111192.

Sum_{n>=1} 1/a(n) > 0.00405067912.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A210477 := proc(n)
    A031924(n)*(A031924(n)+6) ;
end proc:
seq(A210477(n),n=1..100) ; # R. J. Mathar, Jan 23 2013

Times@@@Select[Partition[Prime[Range[200]],2,1],#[[2]]-#[[1]]==6&] (* Harvey P. Dale, Aug 22 2019 *)

a(n) = A031924(n)*(A031924(n)+6).

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.

A258578 Primes p such that difference between p and next prime after p is multiple of 6.

Original entry on oeis.org

23, 31, 47, 53, 61, 73, 83, 131, 151, 157, 167, 173, 199, 211, 233, 251, 257, 263, 271, 331, 353, 367, 373, 383, 433, 443, 467, 503, 509, 523, 541, 557, 563, 571, 587, 593, 601, 607, 619, 647, 653, 661, 677, 727, 733, 751, 797, 941, 947, 971, 977, 991, 997

Offset: 1

Zak Seidov, Jun 04 2015

nonn

A031924 is subsequence: first 12 terms are the same.

			a(1)=23 because next prime after 23 is 29=23+6,
a(13)=199 because next prime after 199 is 211=199+12,
a(30)=523 because next prime after 523 is 541=523+18,
a(90)=1669 because next term after 1669 is 1693=1669+24,
a(199)=4297 because next prime after 4297 is 4327=4297+30.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A031924, A124582, A130796.

Select[Partition[Prime[Range[200]],2,1],Mod[#[[2]]-#[[1]],6]==0&][[All,1]] (* Harvey P. Dale, Jun 20 2019 *)

lista(nn) = forprime(p=2, nn, if (!((nextprime(p+1) - p) % 6), print1(p, ", "));); \\ Michel Marcus, Jun 04 2015

v=List();p=2; forprime(q=3,1e4,if((q-p)%6==0,listput(v,p));p=q); v \\ Charles R Greathouse IV, Jun 04 2015

cp's OEIS Frontend

A076976 Product of the smallest prime divisors of composite numbers between successive primes.

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A080841 Number of pairs (p,q) of (not necessarily consecutive) primes with q-p = 6 and q < 10^n.

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A204672 Primes followed by a gap of 120.

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A287050 Square array read by antidiagonals upwards: M(n,k) is the initial occurrence of first prime p1 of consecutive primes p1, p2, where p2 - p1 = 2*k, and p1, p2 span a multiple of 10^n, n>=1, k>=1.

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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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A124590 Primes p such that q-p <= 6, where q is the next prime after p.

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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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A210477 Product of adjacent primes with a gap of 6.

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