A085809 -id:A085809 - cp's OEIS frontend

A065516 Differences between products of 2 primes.

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

A215231 Increasing gaps between semiprimes.

Original entry on oeis.org

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

A349995 Record gaps between odd squarefree semiprimes (A046388).

Original entry on oeis.org

6, 12, 16, 20, 22, 24, 26, 28, 32, 36, 38, 40, 44, 50, 52, 60, 64, 70, 74, 84, 90, 92, 100, 102, 116, 118, 120, 132, 136, 138, 140, 142, 146, 152, 154, 156, 164, 170, 184, 186, 210

Offset: 1

Hugo Pfoertner, Dec 25 2021

			  n  A350098(n)  A350099(n)  a(n)
  1      15          21        6
  2      21          33       12
  3      95         111       16
  4     267         287       20
  5    2369        2391       22

Lucas A. Brown, Python program.

Records in A341828.
Cf. A350098 lower ends of the record gaps, A350099 upper ends of the record gaps.
Cf. A000101, A002386, A005250, A085809, A114057.

a(35)-a(41) from Lucas A. Brown, Feb 29 2024

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

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

A350098 a(n) is the lower end of a record gap A349995(n) between consecutive odd squarefree semiprimes (A046388).

Original entry on oeis.org

15, 21, 95, 267, 2369, 6559, 8817, 13705, 15261, 21583, 35981, 66921, 113009, 340891, 783757, 872219, 3058853, 3586843, 5835191, 12345473, 108994623, 248706917, 268749691, 679956119, 709239621, 3648864859, 3790337723, 4171420481, 33955869693, 34279038379, 34840796369

Offset: 1

Hugo Pfoertner, Dec 26 2021

nonn

			See A349995.

Lucas A. Brown, Table of n, a(n) for n = 1..41 (first 34 terms from Hugo Pfoertner)
Lucas A. Brown, Python program.

Cf. A046388, A085809, A341828, A349995, A350099.

Starting at a(3)=95 the terms coincide with the known terms of A114057.

a(n) = A350099(n) - A349995(n).

A350099 a(n) is the upper end of a record gap A349995(n) between consecutive odd squarefree semiprimes (A046388).

Original entry on oeis.org

21, 33, 111, 287, 2391, 6583, 8843, 13733, 15293, 21619, 36019, 66961, 113053, 340941, 783809, 872279, 3058917, 3586913, 5835265, 12345557, 108994713, 248707009, 268749791, 679956221, 709239737, 3648864977, 3790337843, 4171420613, 33955869829, 34279038517, 34840796509

Offset: 1

Hugo Pfoertner, Dec 26 2021

nonn

			See A349995.

Lucas A. Brown, Table of n, a(n) for n = 1..41 (first 34 terms from Hugo Pfoertner)
Lucas A. Brown, Python program.

Cf. A046388, A085809, A114057, A341828, A349995, A350098.

a(n) = A350098(n) + A349995(n).

A114057 Start of record gap in odd semiprimes A046315.

Original entry on oeis.org

9, 25, 39, 95, 267, 2369, 6559, 8817, 13705, 15261, 21583, 35981, 66921, 113009, 340891, 783757, 872219, 3058853, 3586843, 5835191, 12345473, 108994623, 248706917, 268749691, 679956119, 709239621, 3648864859, 3790337723, 4171420481, 33955869693, 34279038379

Offset: 1

Jonathan Vos Post, Feb 02 2006

nonn

3 of the first 5 values of record gaps in odd semiprimes are also record merits = (A046315(k+1)-A046315(k))/log_10(A046315(k)), namely: (15 - 9) / log_10(9) = 6.28770982; (111 - 95) / log_10(95) = 8.09010923; (287 - 267) / log_10(267) = 8.24228608. It is easy to prove that there are gaps of arbitrary length in even semiprimes (A100484); can we prove that there are gaps of arbitrary length in odd semiprimes (A046315) and in semiprimes (A001358)?

The record gaps have lengths 6, 8, 10, 16, 20, 22, 24, 26, 28, 32, 36, 38, 40, 44, 50, 52, 60, 64, 70, 74. - T. D. Noe, Feb 03 2006

			a(1) = A046315(2)-A046315(1) = 15 - 9 = 6.
a(2) = A046315(5)-A046315(4) = 33 - 25 = 8.
a(3) = A046315(8)-A046315(7) = 49 - 39 = 10.
a(4) = A046315(20)-A046315(19) = 111 - 95 = 16.
a(5) = A046315(55)-A046315(54) = 287 - 267 = 20.

Cf. A001358, A046315, A065516, A085809, A100484, A114412, A114021.
Starting at a(4)=95 the known terms of this sequence coincide with A350098.
Cf. A341828, A349995, A350099.

f[n_] := Block[{k = n + 2}, While[ Plus @@ Last /@ FactorInteger@k != 2, k += 2]; k]; lst = {}; d = 0; a = b = 9; Do[{a, b} = {b, f[a]}; If[b - a > d, d = b - a; AppendTo[lst, a]], {n, 10^8}]; lst (* Robert G. Wilson v, Feb 03 2006 *)

{a(n)} = {A046315(k) such that A046315(k+1)-A046315(k) is a record}.

More terms from Robert G. Wilson v and T. D. Noe, Feb 03 2006
a(23)-a(28) from Donovan Johnson, Mar 14 2010
a(29)-a(31) from Donovan Johnson, Oct 20 2012

A114058 Start of record gap in even semiprimes (A100484).

Original entry on oeis.org

4, 6, 14, 46, 178, 226, 1046, 1774, 2258, 2654, 19102, 31366, 39218, 62794, 311842, 721306, 740522, 984226, 2699066, 2714402, 4021466, 9304706, 34103414, 41662646, 94653386, 244329494, 379391318, 383825566, 774192266

Offset: 1

Jonathan Vos Post, Feb 02 2006

5 of the first 6 values of record gaps in even semiprimes are also record merits = (A100484(k+1)-A100484(k))/log_10(A100484(k)), namely: (6 - 4) / log_10(4) = 3.32192809; (10 - 6) / log_10(6) = 5.14038884; (22 - 14) / log_10(14) = 6.98002296; (58 - 46) / log_10(46) = 7.21692586; (254 - 226) / log_10(226) = 11.8940995. It is easy to prove that there are gaps of arbitrary length in even semiprimes (A100484), as 2*(n!+2), 2*(n!+3), 2*(n!+4), ..., 2*(n!+n) gives (n-1) consecutive even nonsemiprimes. Can we prove that there are gaps of arbitrary length in odd semiprimes (A046315) and in semiprimes (A001358)?

For every n, a(n) = 2*A002386(n). - John W. Nicholson, Jul 26 2012

			gap[a(1)] = A100484(2)-A100484(1) = 6 - 4 = 2.
gap[a(2)] = A100484(3)-A100484(2) = 10 - 6 = 4.
gap[a(3)] = A100484(5)-A100484(4) = 22 - 14 = 8.
gap[a(4)] = A100484(10)-A100484(9) = 58 - 46 = 12.
gap[a(5)] = A100484(25)-A100484(24) = 194 - 178 = 16.
gap[a(6)] = A100484(31)-A100484(30) = 254 - 226 = 28.

John W. Nicholson, Table of n, a(n) for n = 1..75

Cf. A001358, A046315, A065516, A085809, A100484, A114412, A114021. Maximal gap small prime A002386.

f[n_] := Block[{k = n + 2}, While[ Plus @@ Last /@ FactorInteger@k != 2, k += 2]; k]; lst = {}; d = 0; a = b = 4; Do[{a, b} = {b, f[a]}; If[b - a > d, d = b - a; AppendTo[lst, a]], {n, 10^8}]; lst (* Robert G. Wilson v *)

a(n) = A100484(k) such that A100484(k+1)-A100484(k) is a record.

a(7)-a(25) from Robert G. Wilson v, Feb 03 2006

a(26)-a(31) from Donovan Johnson, Mar 14 2010

cp's OEIS Frontend

A065516 Differences between products of 2 primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A215231 Increasing gaps between semiprimes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Haskell

Mathematica

Extensions

A349995 Record gaps between odd squarefree semiprimes (A046388).

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Perl

Formula

Extensions

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

Views

Author

Keywords

Crossrefs

Programs

Haskell

PARI

Formula

Extensions

A350098 a(n) is the lower end of a record gap A349995(n) between consecutive odd squarefree semiprimes (A046388).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A350099 a(n) is the upper end of a record gap A349995(n) between consecutive odd squarefree semiprimes (A046388).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A114057 Start of record gap in odd semiprimes A046315.

Views

Author

Keywords

Comments

Examples

Crossrefs