A215231 -id:A215231 - cp's OEIS frontend

A215232 Least semiprime m such that the next semiprime is m + A215231(n).

4, 6, 10, 15, 26, 95, 597, 1418, 2681, 6559, 16053, 17965, 32777, 35103, 35981, 340894, 1069541, 1589662, 3586843, 5835191, 139139887, 251306317, 285074689, 327023206, 751411951, 981270902, 2655397631, 5238280946, 6498130361, 8512915573, 16328958619

Offset: 1

T. D. Noe, Aug 07 2012

The semiprime m + A215231(n) is in A217851.

Matomäki & Teräväinen prove that there is almost always (in the sense of natural density) a semiprime in (x, x + log(x)^2.1]. Under RH the exponent can be chosen as 2 + e for any e > 0. - Charles R Greathouse IV, Oct 03 2022

Donovan Johnson, Table of n, a(n) for n = 1..38 (terms < 10^13)
Kaisa Matomäki and Joni Teräväinen, Almost primes in almost all short intervals II, arXiv:2207.05038 [math.NT].

Cf. A001358 (semiprimes), A131109, A215231, A217851.

Cf. A002386 (increasing gaps between primes).

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]; t = {{0, 0}}; s1 = nextSemiprime[1]; While[s1 < 10^7, s2 = nextSemiprime[s1]; d = s2 - s1; If[d > t[[-1, 1]], AppendTo[t, {d, s1}]; Print[{d, s1}]]; s1 = s2]; t = Rest[t]; Transpose[t][[2]]

r=0;s=2;for(n=3,1e7,if(bigomega(n)==2,if(n-s>r,r=n-s;print1(s", "));s=n)) \\ Charles R Greathouse IV, Sep 07 2012

a(27)-a(31) from Donovan Johnson, Aug 07 2012

A217851 Least semiprime m such that the previous semiprime is m - A215231(n).

Original entry on oeis.org

6, 9, 14, 21, 33, 106, 611, 1437, 2701, 6583, 16078, 17993, 32807, 35135, 36019, 340941, 1069595, 1589717, 3586913, 5835265, 139139963, 251306399, 285074774, 327023293, 751412039, 981270997, 2655397729, 5238281053, 6498130471, 8512915685, 16328958739

Offset: 1

T. D. Noe, Nov 07 2012

These are the semiprimes that are the upper end of the gaps of length A215231. The lower end semiprimes are given in A215232.

Donovan Johnson, Table of n, a(n) for n = 1..38

Cf. A001358 (semiprimes), A131109, A215231, A215232.

A065516 Differences between products of 2 primes.

Original entry on oeis.org

2, 3, 1, 4, 1, 6, 1, 3, 1, 7, 1, 1, 3, 1, 7, 3, 2, 4, 2, 1, 4, 3, 4, 5, 3, 5, 3, 1, 1, 4, 2, 1, 1, 11, 5, 4, 3, 1, 2, 1, 1, 6, 4, 1, 7, 1, 1, 2, 1, 9, 3, 1, 2, 5, 3, 8, 1, 5, 2, 2, 7, 7, 1, 1, 2, 1, 3, 4, 1, 1, 2, 1, 1, 2, 5, 9, 2, 10, 2, 4, 1, 5, 3, 3, 2, 7, 4, 9, 2, 2, 4, 3, 1, 2, 1, 1, 2, 4, 5, 5, 2, 2, 3, 1, 2

Offset: 1

Lior Manor, Nov 27 2001

See A215231 and A085809 for record values and where they occur: A215231(n) = a(A085809(n)). - Reinhard Zumkeller, Mar 23 2014

			a(6) = A001358(7) - A001358(6) = 21 - 15 = 6.

T. D. Noe, Table of n, a(n) for n=1..10000

A166237 is the version for distinct primes.

Cf. A001358, A239656.

a065516 n = a065516_list !! (n-1)
a065516_list = zipWith (-) (tail a001358_list) a001358_list
-- Reinhard Zumkeller, Mar 23 2014

Differences[Select[Range[329], PrimeOmega[#] == 2 &]] (* Arkadiusz Wesolowski, Nov 24 2011 *)

{spg(m)=local(a,b); a=0; b=4; for(n=5,m,if(bigomega(n) == 2,a=n; print1(a-b","); b=a; ))}

a(n) = A001358(n+1) - A001358(n).

More terms from Jason Earls, Jul 24 2003

A085809 Indices of semiprimes where largest gap occurs. Or, positions of records in A065516.

Original entry on oeis.org

1, 2, 4, 6, 10, 34, 186, 422, 760, 1765, 4112, 4585, 8112, 8650, 8861, 75150, 223993, 327048, 712605, 1135940, 23958638, 42367759, 47848742, 54626559, 121984495, 157877985, 413509327, 798321315, 983679985, 1277946119, 2403158480

Offset: 1

Jason Earls, Jul 24 2003

nonn

A215231(n) = A065516(a(n)). - Reinhard Zumkeller, Mar 23 2014

Cf. A239674.

a085809 n = a085809_list !! (n-1)
-- See A215231 for definition of a085809_list.
-- Reinhard Zumkeller, Mar 23 2014

{sgr(m)=local(a,b,rec,c); c=0; a=0; b=4; rec=0; for(n=5,m,if(bigomega(n)==2,c++; a=n; if(a-b>rec,rec=a-b; print1(c","); b=a,b=a; )))}

a(19)-a(26) from Donovan Johnson, Jan 28 2009

a(27)-a(31) from Donovan Johnson, Apr 14 2010

A239673 Record values in A239656 (the first differences of sphenic numbers).

Original entry on oeis.org

12, 24, 27, 28, 33, 35, 43, 44, 46, 48, 50, 52, 60, 65, 70, 72, 79, 82, 92, 98

Offset: 1

Reinhard Zumkeller, Mar 23 2014

Cf. A007304, A005250, A215231, A239656, A239674.

a239673 n = a239673_list !! (n-1)
(a239673_list, a239674_list) = unzip $ (12, 1) : f 1 12 a239656_list where
   f i v (q:qs) | q > v = (q, i) : f (i + 1) q qs
                | otherwise = f (i + 1) v qs
-- Reinhard Zumkeller, Mar 23 2014

lista(kmax) = {my(k1 = 30, d, dm = 0); forcomposite(k2 = k1 + 1, kmax, if(factor(k2)[,2] == [1,1,1]~, d = k2 - k1; if(d > dm, dm = d; print1(d, ", ")); k1 = k2));} \\ Amiram Eldar, May 19 2024

a(n) = A239656(A239674(n)).

a(12)-a(20) from Amiram Eldar, May 19 2024

cp's OEIS Frontend

A215232 Least semiprime m such that the next semiprime is m + A215231(n).

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A217851 Least semiprime m such that the previous semiprime is m - A215231(n).

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A065516 Differences between products of 2 primes.

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A085809 Indices of semiprimes where largest gap occurs. Or, positions of records in A065516.

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A239673 Record values in A239656 (the first differences of sphenic numbers).

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