A239673 -id:A239673 - cp's OEIS frontend

A215231 Increasing gaps between semiprimes.

2, 3, 4, 6, 7, 11, 14, 19, 20, 24, 25, 28, 30, 32, 38, 47, 54, 55, 70, 74, 76, 82, 85, 87, 88, 95, 98, 107, 110, 112, 120, 123, 126, 146, 163, 166, 171, 174

Offset: 1

T. D. Noe, Aug 07 2012

See A215232 and A217851 for the semiprimes that begin and end the gaps.
Records in A065516. - R. J. Mathar, Aug 09 2012
How long can these gaps be? In the Cramér model, with x = A215232(n), they are of length log(x)^2/log(log(x))(1 + o(1)) with probability 1. - Charles R Greathouse IV, Sep 07 2012
a(n) = A065516(A085809(n)). - Reinhard Zumkeller, Mar 23 2014

			4 is here because the difference between 10 and 14 is 4, and there is no smaller semiprimes with this property.

Cf. A001358 (semiprimes), A131109, A215232, A217851.
Cf. A005250 (increasing gaps between primes).
Cf. A239673 (increasing gaps between sphenic numbers).

a215231 n = a215231_list !! (n-1)
(a215231_list, a085809_list) = unzip $ (2, 1) : f 1 2 a065516_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

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][[1]]

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

a(32)-a(38) from Donovan Johnson, Sep 20 2012

A239656 First differences of sphenic numbers, cf. A007304.

Original entry on oeis.org

12, 24, 4, 8, 24, 3, 5, 4, 16, 8, 16, 11, 5, 4, 8, 4, 4, 5, 27, 8, 1, 7, 8, 9, 3, 8, 7, 9, 3, 1, 4, 20, 8, 4, 23, 9, 3, 9, 4, 4, 11, 14, 3, 4, 4, 8, 8, 3, 1, 4, 1, 3, 4, 13, 10, 5, 4, 9, 11, 4, 8, 12, 12, 4, 21, 6, 13, 8, 8, 5, 3, 4, 4, 3, 1, 5, 3, 9, 11, 4

Offset: 1

Reinhard Zumkeller, Mar 23 2014

nonn

a(n) = A007304(n+1) - A007304(n);

see A239673 and A239674 for record values and where they occur: A239673(n) = a(A239674(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A065516.

a239656 n = a239656_list !! (n-1)
a239656_list = zipWith (-) (tail a007304_list) a007304_list

A007304 := proc(n)
    option remember;
    if n = 1 then
        30;
    else
        for a from procname(n-1)+1 do
            if numtheory[bigomega](a) =3 and nops(numtheory[factorset](a)) = 3 then
                return a;
            end if;
        end do:
    end if;
end proc:
A239656 := proc(n)
    A007304(n+1)-A007304(n) ;
end proc:

With[{upto=1000},Differences[Sort[Select[Times@@@Subsets[Prime[ Range[ Ceiling[upto/6]]],{3}],#<=upto&]]]] (* Harvey P. Dale, Jan 08 2015 *)

A239674 Where records occur in A239656 (the first differences of sphenic numbers).

Original entry on oeis.org

1, 2, 19, 498, 2114, 8351, 8381, 59704, 233890, 291963, 1119181, 1507131, 1839746, 9768399, 40844982, 94852115, 138032741, 443653568, 453853664, 2491818901

Offset: 1

Reinhard Zumkeller, Mar 23 2014

Cf. A007304, A085809, A239656, A239673.

a239674 n = a239674_list !! (n-1)
-- See A239673 for definition of A239674_list.

list(lim)=my(v=List(), t); forprime(p=2, (lim)^(1/3), forprime(q=p+1, sqrt(lim\p), t=p*q; forprime(r=q+1, lim\t, listput(v, t*r)))); vecsort(Vec(v)) ; \\ A007304
chk(lim) = my(v=list(lim), dv = vector(#v-1, k, v[k+1] - v[k]), r=0); for (i=1, #dv, if (dv[i] > r, r=dv[i]; print1(i, ", "));); \\ Michel Marcus, Sep 20 2023

A239656(a(n)) = A239673(n).

a(12)-a(14) from Michel Marcus, Sep 20 2023

a(15)-a(20) from Amiram Eldar, May 19 2024

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A215231 Increasing gaps between semiprimes.

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A239656 First differences of sphenic numbers, cf. A007304.

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A239674 Where records occur in A239656 (the first differences of sphenic numbers).

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