A131109 -id:A131109 - cp's OEIS frontend

A123375 Least k such that the difference between consecutive semiprimes A065516(k) equals n, or 0 if no such k exists.

3, 1, 2, 4, 24, 6, 10, 56, 50, 78, 34, 320, 249, 186, 463, 762, 598, 1238, 422, 760, 3760, 3585, 9214, 1765, 4112, 13447, 6675, 4585, 68498, 8112, 10083, 8650, 86203, 49433, 35085, 20641, 458421, 8861, 366314, 157857, 169147, 487115, 277440, 563951, 511757, 920602, 75150

Offset: 1

Alexander Adamchuk, Nov 09 2006

a(n) equals least k such that A065516(k) = n, or 0 if no such k exists. Conjecture: a(n)>0 exists for all n.

			A065516(n) begins {2, 3, 1, 4, 1, 6, 1, 3, 1, 7, 1, 1, 3, 1, 7, 3, 2, 4, 2, 1, 4, 3, 4, 5, ...}.
Thus a(1) = 3, a(2) = 1, a(3) = 2, a(4) = 4, a(5) = 24.

David A. Corneth, Table of n, a(n) for n = 1..82
Eric Weisstein's World of Mathematics, Semiprime

Cf. A001358 (semiprimes), A065516 (differences between semiprimes), A131109.

nextSprime := proc(n) local res ; res := n+1 ; while numtheory[bigomega](res) <> 2 do res := res+1 ; od ; RETURN(res) ; end ; nmax := 500 ; kmax := 500000 ; a := array(1..nmax) ; for i from 1 to nmax do a[i] := 0 ; od : sp1 := 4 : sp2 := nextSprime(sp1) : n := sp2-sp1 : a[n] := 1 : for k from 2 to kmax do sp1 := sp2 ; sp2 := nextSprime(sp1) ; n := sp2-sp1 ; if a[n] = 0 then a[n] := k ; fi ; od : for i from 1 to nmax do if a[i] = 0 then break ; else printf("%d,",a[i]) ; fi ; od : # R. J. Mathar, Jan 13 2007

Table[k=6;While[FreeQ[b=Differences[Select[Range@k++,PrimeOmega[#]==2&]],n]];
Length@b,{n,11}] (* Giorgos Kalogeropoulos, Apr 02 2021 *)

Corrected and extended by R. J. Mathar, Jan 13 2007

More terms from David A. Corneth, Apr 02 2021

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

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

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

Original entry on oeis.org

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.

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

A278351 Least number that is the start of a prime-semiprime gap of size n.

Original entry on oeis.org

2, 7, 26, 97, 341, 241, 6091, 3173, 2869, 2521, 16022, 26603, 114358, 41779, 74491, 39343, 463161, 104659, 248407, 517421, 923722, 506509, 1930823, 584213, 2560177, 4036967, 4570411, 4552363, 7879253, 4417813, 27841051, 5167587, 13683034, 9725107, 47735342, 25045771, 63305661

Offset: 1

Bobby Jacobs, Charles R Greathouse IV, Jonathan Vos Post, and Robert G. Wilson v, Nov 23 2016

nonn

A prime-semiprime gap of n is defined as the difference between p & q, p being either a prime, A000040, or a semiprime, A001358, and q being the next greater prime or semiprime, see examples.
The corresponding numbers at the end of the prime-semiprime gaps, i.e., a(n)+n, are in A278404.
In the first 52 terms, 19 are primes and the remaining 33 are semiprime. Of the end-of-gap terms a(n)+n, 20 are primes and 32 are not. There are only 6 pairs of p and q that are both primes, and 19 pairs that are both semiprime.

			a(1) = 2 since there is a gap of 1 between 2 and 3, both of which are primes.
a(2) = 7 since there is a gap of 2 between 7 and 9, the first is a prime and the second is a semiprime.
a(3) = 26 since there is a gap of 3 between 26, a semiprime, and 29, a prime.
a(6) = 241 because the first prime-semiprime gap of size 6 is between 241 and 247.

Dana Jacobsen, Table of n, a(n) for n = 1..106 (first 52 terms from Bobby Jacobs, Charles R Greathouse IV, Jonathan Vos Post and Robert G. Wilson v)

Cf. A000230, A037143, A131109, A275108, A275013, A275014, A278404.

nxtp[n_] := Block[{m = n + 1}, While[ PrimeOmega[m] > 2, m++]; m]; gp[_] = 0; p = 2; While[p < 1000000000, q = nxtp[p]; If[ gp[q - p] == 0, gp[q -p] = p; Print[{q -p, p}]]; p = q]; Array[gp, 40]

use ntheory ":all";
my($final,$p,$nextn,@gp) = (40,2,1);  # first 40 values in order
forfactored {
  if (scalar(@) <= 2) { my $q = $;
    if (!defined $gp[$q-$p]) {
      $gp[$q-$p] = $p;
      while ($nextn <= $final && defined $gp[$nextn]) {
        print "$nextn $gp[$nextn]\n";
        $nextn++;
      }
      lastfor if $nextn > $final;
    }
    $p = $q;
  }
} 3,10**14; # Dana Jacobsen, Sep 10 2018

A282407 Semiprimes p such that next semiprime after p is p + 40.

Original entry on oeis.org

741841, 1633213, 1889467, 1946677, 2210557, 2440203, 2655427, 2660857, 2729091, 2749273, 2774911, 3077323, 3724909, 3977473, 4021507, 4030891, 4323301, 4372337, 4408581, 4421713, 4608574, 4640419, 4836223, 5640861, 5691531, 6148599, 6166101, 6429853, 6786523

Offset: 1

Zak Seidov, Feb 14 2017

nonn

Note that a(1) = 741841 = A131109(40).

Smallest difference between two consecutive terms occurs first at a(22060) = 1141901643 because a(22061) = 1141901683 = 1141901643 + 40.

Zak Seidov, Table of n, a(n) for n = 1..23671

Cf. A001358, A131109, A217030, A217335, A217357.

is(p) = if(bigomega(p)==2 && bigomega(p+40)==2, for(i=p+1, p+39, if(bigomega(i)==2, return(0))); 1); \\ Jinyuan Wang, May 23 2021

A282424 Semiprimes p such that next semiprime after p is p + 50.

Original entry on oeis.org

1226777, 4732631, 5093729, 9210671, 12515461, 12917989, 16121409, 16183253, 16698881, 17263069, 19418903, 23003807, 24534161, 26519563, 26726659, 27625067, 29605299, 29772471, 32655031, 34027277, 34366179, 35340719, 37570683, 38117319, 38687461, 39038399, 39479381

Offset: 1

Zak Seidov, Feb 14 2017

nonn

All terms are odd because even semiprime 2*p plus 50 = 2*(p+25) is multiple of 4 and not semiprime.
Note that a(1) = 1226777 = A131109(50).
Smallest possible difference is 50 but among first 10000 terms
the least difference 100 is between a(325) = 228601303 and a(326) = 228601403.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A001358, A131109, A217030, A217335, A217357, A282407.

lista(nn) = my(r); for(k=1, nn, if(bigomega(k)==2, if(k-r==50, print1(r, ", ")); r=k)); \\ Jinyuan Wang, May 23 2021

cp's OEIS Frontend

A123375 Least k such that the difference between consecutive semiprimes A065516(k) equals n, or 0 if no such k exists.

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A215231 Increasing gaps between semiprimes.

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

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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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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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A278351 Least number that is the start of a prime-semiprime gap of size n.

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