A097824 -id:A097824 - cp's OEIS frontend

A096003 Smallest semiprime (A001358) which is at the end of an arithmetic progression of n semiprimes.

4, 6, 14, 46, 58, 221, 445, 497, 1211, 1561, 4195, 4393, 6347, 10717, 14233, 28213, 31451, 72965, 119029, 121603, 124177, 611261, 632171, 2003171, 2012771, 7466993, 7619243, 7771493, 7923743, 68029097, 142247113, 168901739

Offset: 1

N. J. A. Sloane, Jul 28 2004

nonn

a(33) <= 1320196303 (gap of 39029760). a(34) <= 1359226063 (gap of 39029760). a(35) <= 1398255823 (gap of 39029760). - Donovan Johnson, Jun 03 2012

			a(3)=14 since (4,9,14) [or (6,10,14)] is an arithmetic progression of 3 semiprimes ending in 14 and 14 is the smallest semiprime with this property.

Cf. A001358 (semiprimes), A005115 (analog for primes).

For the associated gaps see A097824.

More terms from T. D. Noe and Ray Chandler, Jul 29 2004
a(18)-a(20) from Ray Chandler, Aug 01 2004
More terms from Hugo Pfoertner, Aug 27 2004
a(26)-a(30) from Hugo Pfoertner, Sep 07 2004
a(31)-a(32) from Donovan Johnson, Jun 03 2012

A114021 Number of semiprimes between n and n + sqrt(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 2, 2, 2, 2, 2, 1, 0, 0, 1, 2, 3, 3, 3, 3, 3, 2, 2, 2, 1, 0, 1, 1, 1, 2, 2, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 5, 5, 4, 4, 4, 4, 3, 3, 2, 1, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2

Offset: 0

Jonathan Vos Post, Jan 31 2006

It appears that for n > 37 it is always true that a(n) > 0. The exponent can be reduced further. Since 597 + 597^(0.4129) > 611, leaping the record semiprime gap between 597 and 611, it seems that for n > 597 it is always true that there is a semiprime between n and n^(0.4129). It seems that for n > 2705 it is always true that there is a semiprime between n and n^(0.3509). These conjectures are related to the various sequences about semiprime gaps and the merit of such gaps.

a(96) appears to be the last zero term. - T. D. Noe, Aug 12 2008

			a(0) = 0 because there are no semiprimes between 0 and 0+sqrt(0) = 0.
a(2) = 0 because there are no semiprimes between 2 and 2+sqrt(2) = 3.414...
a(3) = 1 as the semiprime 4 falls between 3 and 3 + sqrt(3) = 4.732...
a(5) = 1 as the semiprime 6 falls between 5 and 5 + sqrt(5) = 7.236...

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A001358, A060715, A065516, A077463, A085809, A097824, A114412, A115766.

SemiPrimeQ[n_] := TrueQ[Plus@@Last/@FactorInteger[n]==2]; Table[hi=n+Sqrt[n]; If[IntegerQ[hi], hi--, hi=Floor[hi]]; Length[Select[Range[n+1,hi], SemiPrimeQ]], {n,0,150}] (* T. D. Noe, Aug 12 2008 *)

use ntheory ":all"; print "$ ",semiprime_count($+1, $+sqrtint($)-($ && is_square($))),"\n" for 0..1000; # Dana Jacobsen, Mar 04 2019

a(n) = card{S such that S is an element of A001358 and n < S < n + n^(1/2)}.

Corrected and extended by T. D. Noe, Aug 12 2008

A226833 Triangle whose n-th row has the smallest n semiprimes in an arithmetic progression.

Original entry on oeis.org

4, 4, 6, 6, 10, 14, 10, 22, 34, 46, 10, 22, 34, 46, 58, 201, 205, 209, 213, 217, 221, 133, 185, 237, 289, 341, 393, 445, 133, 185, 237, 289, 341, 393, 445, 497, 635, 707, 779, 851, 923, 995, 1067, 1139, 1211, 697, 793, 889, 985, 1081, 1177, 1273, 1369, 1465, 1561

Offset: 1

T. D. Noe, Jun 28 2013

The largest term of each row is as small as possible. Although Nowicki reports on the 28th row of this triangle, those terms are too large. Sequence A096003 reports the largest terms.

			Triangle:
4,
4,   6,
6,   10,  14,
10,  22,  34,  46,
10,  22,  34,  46,  58,
201, 205, 209, 213, 217,  221,
133, 185, 237, 289, 341,  393,  445,
133, 185, 237, 289, 341,  393,  445,  497,
635, 707, 779, 851, 923,  995,  1067, 1139, 1211,
697, 793, 889, 985, 1081, 1177, 1273, 1369, 1465, 1561

T. D. Noe, Rows n = 1..32 of triangle, flattened
Andrzej Nowicki, Second numbers in arithmetic progressions, arxiv 1306.6424

Cf. A226834 (first term), A096003 (last term), A097824 (gaps).

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; p2 = Select[Range[2000], SemiPrimeQ]; nn = Length[p2]; t = {}; n = 0; last = 1; While[n++; found = False; last = n; While[k = last - 1; While[d = p2[[last]] - p2[[k]]; nums = Table[p2[[last]] - i*d, {i, 0, n - 1}]; int = Intersection[nums, Take[p2, last]]; nums[[-1]] > 0 && Length[int] < n, k--]; nums[[-1]] <= 0 && last < nn, last++]; If[last < nn, AppendTo[t, Reverse[nums]]]; last < nn]; t

A112888 Least semiprime of a cluster of just n semiprimes.

Original entry on oeis.org

9, 33, 91, 299, 213, 1383, 3091, 8129

Offset: 1

Robert G. Wilson v, Nov 30 2005

Clusters are sets composed of odd numbers.
If we include even numbers then the sequence would start 4,9,33 and terminates because in any group of four consecutive numbers greater than 4, 4 is a divisor to at least one member leaving a quotient greater than 1.
Any set of 9 consecutive odd numbers contain a multiple of 9, which not semiprime (unless it is equal to 9). Hence there are no 9 consecutive odd semiprimes.

			a(8)=8129 because 8129=11*739, 8131=47*173, 8133=3*2711, 8135=5*1627, 8137=79*103, 8139=3*2713, 8141=7*1163, 8143=17*479.

Cf. A001358, A097824, A082919.

spQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; f[n_] := Block[{k = 1}, While[ s[[k]] + 2n != s[[k + n]] || s[[k]] + 2n + 2 == s[[k + n + 1]], k++ ]; s[[k]]]; s = {}; Do[ If[ spQ[n], AppendTo[s, n]], {n, 9, 7*10^6, 2}]; Table[ f[n], {n, 0, 7}]
Join[{9},Module[{osps=Select[Range[9,10001,2],PrimeOmega[#]==2&]}, #[[2]]& /@ Table[ SelectFirst[Partition[osps,n+2,1],Union[ Differences[ Rest[ Most[#]]]]=={2}&&Last[#]-#[[-2]]!=2&&#[[2]]-#[[1]]!=2&],{n,2,8}]]] (* Harvey P. Dale, Jun 01 2016 *)

fini, full from Max Alekseyev, Feb 03 2010

A226834 Smallest semiprime (A001358) which is at the beginning of an arithmetic progression of n semiprimes whose largest term is as small as possible.

Original entry on oeis.org

4, 4, 6, 10, 10, 201, 133, 133, 635, 697, 2215, 2215, 4979, 2995, 13561, 22903, 1691, 5951, 72697, 72697, 72697, 172151, 172151, 1782371, 1782371, 3660743, 3660743, 3660743, 3660743, 13298267, 2235913, 41963249

Offset: 1

T. D. Noe, Jun 28 2013

Smallest number in row A226833(n).

Cf. A096003 (largest semiprime in row), A097824 (gaps).

cp's OEIS Frontend

A096003 Smallest semiprime (A001358) which is at the end of an arithmetic progression of n semiprimes.

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A114021 Number of semiprimes between n and n + sqrt(n).

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A226833 Triangle whose n-th row has the smallest n semiprimes in an arithmetic progression.

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A112888 Least semiprime of a cluster of just n semiprimes.

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A226834 Smallest semiprime (A001358) which is at the beginning of an arithmetic progression of n semiprimes whose largest term is as small as possible.

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