A136196 -id:A136196 - cp's OEIS frontend

A264043 Numbers n such that n and n+3 are consecutive semiprimes.

6, 22, 35, 46, 62, 74, 82, 115, 155, 166, 206, 259, 262, 295, 323, 355, 358, 362, 395, 466, 478, 482, 502, 511, 559, 562, 583, 586, 611, 623, 626, 671, 703, 718, 731, 734, 746, 755, 763, 799, 835, 838, 862, 866, 886, 895, 914, 923, 955, 979, 982

Offset: 1

Zak Seidov, Nov 02 2015

nonn

			6 and 9 are 2nd and 3rd semiprimes, or 6=A001358(2) and 9=A001358(3);
22=A001358(8) and 25=A001358(9).

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

Cf. A001358, A065516, A136196.

Flatten[Position[Partition[Table[If[PrimeOmega[n]==2,1,0],{n,1000}], 4,1],?(#=={1,0,0,1}&)]] (* _Harvey P. Dale, Feb 08 2016 *)

for(n=1, 1e3, if(bigomega(n) == 2 && bigomega(n+3) == 2 && bigomega(n+1) !=2 && bigomega(n+2) !=2, print1(n", "))) \\ Altug Alkan, Nov 02 2015

is(n)=if(n%4==2, isprime(n/2) && bigomega(n+3)==2 && bigomega(n+1)!=2, n%4==3 && isprime((n+3)/2) && bigomega(n)==2 && bigomega(n+2)!=2) \\ Charles R Greathouse IV, Nov 02 2015

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

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

Zak Seidov, Nov 02 2015

nonn

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

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

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

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

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

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 *)

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

Zak Seidov, Nov 02 2015

nonn

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

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

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

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

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 *)

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

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

Zak Seidov, Nov 02 2015

nonn

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

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

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

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

Select[Partition[Select[Range[3000],PrimeOmega[#]==2&],2,1],#[[2]]-#[[1]]==6&][[;;,1]] (* Harvey P. Dale, Dec 24 2023 *)

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

A133609 Numbers k such that k, k+2 and k+4 are consecutive semiprimes.

Original entry on oeis.org

183, 287, 319, 411, 413, 469, 515, 533, 579, 667, 685, 789, 813, 1055, 1077, 1133, 1145, 1165, 1203, 1253, 1313, 1347, 1383, 1385, 1387, 1389, 1561, 1685, 1687, 1793, 1795, 1817, 1839, 1849, 1919, 1937, 1957, 1959, 2045, 2047, 2155, 2227, 2315, 2317

Offset: 1

Zak Seidov, Dec 28 2007

nonn

Terms k in A136196 such that k+2 are also in A136196.

All terms are odd, so it is a subsequence of A161945. - Michel Marcus, Oct 15 2013

			183, 185 and 187 are 59th, 60th and 61st semiprimes,
287, 289 and 291 are 89th, 90th and 91st semiprimes,
319, 321 and 323 are 101st, 102nd and 103rd semiprimes.

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

Cf. A136196.

Select[Range[2317],AllTrue[{#,#+2,#+4},PrimeOmega[#]==2&]&&AllTrue[{#+1,#+3},PrimeOmega[#]!=2&]&] (* James C. McMahon, Mar 29 2025 *)

isok(n) = (bigomega(n) == 2) && (bigomega(n+1) != 2) && (bigomega(n+2) == 2) && (bigomega(n+3) != 2) && (bigomega(n+4) == 2); \\ Michel Marcus, Oct 15 2013

A133597 Array of semiprimes, read by antidiagonals, where row k is the first of pairs of consecutive semiprimes j and j+k.

Original entry on oeis.org

9, 4, 14, 6, 49, 21, 10, 22, 55, 25, 69, 51, 35, 91, 33, 15, 77, 58, 46, 119, 34, 26, 123, 106, 65, 62, 143, 38, 169, 39, 365, 161, 87, 74, 159, 57, 146, 437, 134, 371, 178, 111, 82, 183, 85, 237, 226, 458, 187, 505, 221, 129, 115, 185, 86

Offset: 1

Jonathan Vos Post, Dec 27 2007

Every semiprime occurs in this table exactly once. Note that similar tables exist for k-almost primes (integers with exactly k prime factors, with multiplicity), this being the k=2 slice of a 3-dimensional array.

			The array begins:
==================================================================
n=......1....2.....3....4....5....6....7....8....9...10
==================================================================
k=1.|...9...14....21...25...33...34...38...57...85...86....A070552
k=2.|...4...49....55...91..119..143..159..183..185..203....A136196
k=3.|...6...22....35...46...62...74...82..115..155..166....A264043
k=4.|. 10...51....58...65...87..111..129..209..249..274....A264044
k=5.|..69...77...106..161..178..221..254..309..314..329....A264045
k=6.|..15..123...365..371..505..545..573..591..649..707....A264046
k=7.|..26...39...134..187..194..267..519..566..655..771....
k=8.|.169..437...458..614..723..737..905..965.1047.1059....
k=9.|.146..226...278..346.1018.1177.1273.1546.1594.1865....
k=10|.237..427..1027.1101.1661.2723.2747.3173.3295.3669....A217030
==================================================================

Cf. A001358, A070552, A136196.

v = Select[Range[5000], PrimeOmega[#]==2 &]; L[k_] := L[k] = v[[Select[Range[Length[v]-1], v[[#+1]] - v[[#]] == k &]]]; Flatten@ Table[ Table[L[k-j+1][[j]], {j, k}], {k, 10}] (* Giovanni Resta, Jun 20 2016 *)

Corrected and edited by Giovanni Resta, Jun 20 2016

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A264043 Numbers n such that n and n+3 are consecutive semiprimes.

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

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

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

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A133609 Numbers k such that k, k+2 and k+4 are consecutive semiprimes.

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A133597 Array of semiprimes, read by antidiagonals, where row k is the first of pairs of consecutive semiprimes j and j+k.

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