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.

A131109 a(n) is the smallest semiprime such that difference between a(n) and next semiprime, b(n), is n.

Original entry on oeis.org

9, 4, 6, 10, 69, 15, 26, 169, 146, 237, 95, 1082, 818, 597, 1603, 2705, 2078, 4511, 1418, 2681, 14545, 13863, 37551, 6559, 16053, 55805, 26707, 17965, 308918, 32777, 41222, 35103, 393565, 219509, 153263, 87627, 2263057, 35981, 1789339, 741841, 797542
Offset: 1

Views

Author

Zak Seidov, Sep 24 2007

Keywords

Comments

This is the semiprime analogous to A000230. - Robert G. Wilson v, Jun 13 2013

Examples

			n, b(n)-a(n): 1=10-9, 2=6-4, 3=9-6, 4=14-10, 5=74-69, 6=21-15, 7=33-26, 8=177-169, 9=155-146, 10=247-237, 11=106-95, 12=1094-1082, 13=831-818, 14=611-597, 15=1618-1603, 16=2721-2705, 17=2095-2078, 18=4529-4511, 19=1437-1418, 20=2701-2681, 21=14566-14545, 22=13885-13863, 23=37574-37551, 24=6583-6559, 25=16078-16053, 26=55831-55805, 27=26734-26707, 28=17993-17965, 29=308947-308918, 30=32807-32777, 31=41253-41222, 32=35135-35103, 33=393598-393565, 34=219543-219509, 35=153298-153263, 36=87663-87627, 37=2263094-2263057, 38=36019-35981.

Links

Crossrefs

Cf. A065516, A133478.

Programs

  • Mathematica
    SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; NextSemiPrime[n_] := Module[{m = n + 1}, While[! SemiPrimeQ[m], m++]; m]; nn = 30; t = Table[0, {nn}]; found = 0; sp0 = 4; While[found < nn, sp1 = NextSemiPrime[sp0]; d = sp1 - sp0; If[d <= nn && t[[d]] == 0, t[[d]] = sp0; found++]; sp0 = sp1]; t (* T. D. Noe, Oct 02 2012 *)

Formula

a(n) = A001358(A123375(n)). - T. D. Noe, Sep 28 2007

Extensions

Corrected and extended by T. D. Noe and R. J. Mathar, Sep 28 2007

A264044 Numbers n such that n and n+4 are consecutive semiprimes.

Original entry on oeis.org

10, 51, 58, 65, 87, 111, 129, 209, 249, 274, 291, 305, 335, 377, 382, 403, 407, 447, 454, 485, 489, 493, 497, 529, 538, 629, 681, 699, 713, 749, 767, 781, 785, 803, 831, 889, 901, 917, 939, 951, 961, 985, 989, 1007, 1037, 1073, 1115, 1191, 1207
Offset: 1

Views

Author

Zak Seidov, Nov 02 2015

Keywords

Comments

Note that a(1)=10=A131109(k=4).
Subsequence of A175648: a(1)=10=A175648(2), a(2)=51=A175648(7), a(3)=58=A175648(8), etc. - Zak Seidov, Dec 20 2017

Examples

			10=A001358(4) and 14=A001358(5).

Links

Crossrefs

Cf. A001358, A065516, A123375, A131109, A136196, A175648, A264043.

Programs

  • Maple
    B:= select(numtheory:-bigomega=2, [$1..2000]):
B[select(t ->B[t+1]-B[t]=4, [$1..nops(B)-1])]; # Robert Israel, Dec 21 2017
  • Mathematica
    Select[Partition[Select[Range[1250], PrimeOmega@ # == 2 &], 2, 1], Differences@ # == {4} &][[All, 1]] (* Michael De Vlieger, Dec 20 2017 *)
SequencePosition[Table[If[PrimeOmega[n]==2,1,0],{n,1300}],{1,0,0,0,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 19 2020 *)
  • PARI
    is(n)=bigomega(n)==2 && bigomega(n+4)==2 && bigomega(n+1)!=2 && bigomega(n+2)!=2 && bigomega(n+3)!=2 \\ Charles R Greathouse IV, Nov 02 2015

A264045 Numbers n such that n and n+5 are consecutive semiprimes.

Original entry on oeis.org

69, 77, 106, 161, 178, 221, 254, 309, 314, 329, 341, 386, 398, 417, 422, 473, 554, 662, 674, 689, 758, 794, 934, 974, 998, 1094, 1121, 1149, 1169, 1214, 1294, 1502, 1517, 1522, 1541, 1569, 1673
Offset: 1

Views

Author

Zak Seidov, Nov 02 2015

Keywords

Comments

Note that a(1)=69=A131109(k=5).

Examples

			a(1)=69=A001358(24) and a(1)+k=74=A001358(25).

Links

Crossrefs

Cf. A001358, A065516, A123375, A131109, A136196, A264043, A264044.

Programs

  • Mathematica
    Flatten[Position[Partition[Table[If[PrimeOmega[n]==2,1,0],{n,2000}],6,1], ?(#=={1,0,0,0,0,1}&)]] (* _Harvey P. Dale, Dec 16 2015 *)
  • PARI
    is(n)=if(n%4==1, isprime((n+5)/2) && bigomega(n)==2, n%4==2 && isprime(n/2) && bigomega(n+5)==2) && bigomega(n+1)!=2 && bigomega(n+2)!=2 && bigomega(n+3)!=2 && bigomega(n+4)!=2 \\ Charles R Greathouse IV, Nov 02 2015

Formula

a(n) >> n log n. - Charles R Greathouse IV, Nov 02 2015

A264046 Numbers k such that k and k+6 are consecutive semiprimes.

Original entry on oeis.org

15, 123, 365, 371, 505, 545, 573, 591, 649, 707, 807, 843, 943, 1067, 1159, 1247, 1357, 1405, 1529, 1555, 1633, 1739, 1745, 1829, 1843, 1897, 1909, 1985, 2149, 2159, 2209, 2285, 2329, 2353, 2363, 2407, 2413, 2463, 2501, 2643, 2773, 2779
Offset: 1

Views

Author

Zak Seidov, Nov 02 2015

Keywords

Comments

Note that a(1) = 15 = A131109(k=6).

Examples

			15 = A001358(6) and 21 = A001358(7).

Links

Crossrefs

Cf. A001358, A065516, A123375, A131109, A136196, A264043, A264044, A264045.

Programs

  • Mathematica
    Select[Partition[Select[Range[3000],PrimeOmega[#]==2&],2,1],#[[2]]-#[[1]]==6&][[;;,1]] (* Harvey P. Dale, Dec 24 2023 *)
  • PARI
    is(n)=if(bigomega(n)!=2 || bigomega(n+6)!=2, return(0)); for(i=1,5,if(bigomega(n+i)==2, return(0))); 1 \\ Charles R Greathouse IV, Nov 02 2015

A131939 Least k such that the difference between consecutive 3-almost primes A014612(k) equals n, or 0 if no such k exists.

Original entry on oeis.org

5, 3, 13, 1, 10, 2, 4, 31, 32, 36, 12, 7, 136, 19, 302, 486, 1094, 73, 1366, 6763, 1092, 2006, 8924, 4785, 18345, 18487, 42798, 16571, 11095, 57831, 60912, 4528, 24846, 41304, 232350, 233678, 123279, 1779265, 740729, 177385, 1015228, 1772286
Offset: 1

Views

Author

Jonathan Vos Post, Oct 05 2007

Keywords

Comments

Index of smallest 3-almost prime gap equal to n. This is to 3-almost primes A014612 as A123375 is to semiprimes A001358. a(25) = 0 or a(25) > 10000. Conjecture: a(n) > 0 for all n.

Examples

			a(1) = 5 because A014612(6)-A014612(5) = 28-27 = 1.
a(2) = 3 because A014612(4)-A014612(3) = 20-18 = 2.
a(3) = 13 because 66-63 = 3.
a(4) = 1 because 12-8 = 4.
a(5) = 10 because 50-45 = 5.
a(6) = 2 because 18-12 = 6.
a(7) = 4 because 27-20 = 7.
a(8) = 31 because 138-130 = 8.
a(9) = 32 because 147-138 = 9
a(10) = 36 because 164-154 = 10.

Crossrefs

Cf. A014612, A114403, A123375.

Programs

  • Mathematica
    p3=Select[Range[2*10^6],PrimeOmega[#]==3&];s={};Do[d=0;Until[p3[[d+1]]-p3[[d]]==n,d++];AppendTo[s,d],{n,37}];s (* James C. McMahon, Mar 02 2025 *)

Formula

a(n) = MIN{k such that A114403(k) = n, or 0 if no such k exists}.
a(n) = MIN{k such that A014612(k+1) - A014612(k) = n, or 0 if no such k exists}.

Extensions

More terms from R. J. Mathar, Oct 07 2007
Showing 1-5 of 5 results.

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