A065516 -id:A065516 - cp's OEIS frontend

A114415 Records in 5-almost prime gaps ordered by merit.

16, 24, 28, 42, 56, 70

Offset: 1

Jonathan Vos Post, Nov 25 2005

Next term, if it exists, is associated with indices above 100000 in A114405 and A014614. - R. J. Mathar, May 10 2007

			Records defined in terms of A114405 and A014614:
  n  A114405(n)  A114405(n)/log_10(A014614(n))
  =  ==========  =============================
  1      16      16/log_10(32)  = 10.6301699
  2      24      24/log_10(48)  = 14.2751673
  3      8       8/log_10(72)   = 4.30725248
  4      28      28/log_10(80)  = 14.7129144
  5      4       4/log_10(108)  = 1.96712564
  6      8       8/log_10(112)  = 3.90392819
  7      42      42/log_10(120) = 20.2002592
  8      6       6/log_10(168)  = 2.69625443
  ...
  22     56      56/log_10(312) = 22.4524976

Cf. A014614, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

A014614 := proc(nmax) local a,i; a := [] ; i := 1 ; while nops(a) < nmax do if numtheory[bigomega](i) = 5 then a := [op(a),i] ; fi ; i := i+1 ; end: a ; end: A114405 := proc(a014614) local a,i; a := [] ; for i from 2 to nops(a014614) do a := [op(a), op(i,a014614)-op(i-1,a014614)] ; od ; a ; end: a014614 := A014614(100000) : a114405 := A114405(a014614) : Digits := 30 : rec := -1 : for i from 1 to nops(a114405) do if evalf(a114405[i]/log(a014614[i])) > rec then printf("%d, ",a114405[i]) ; rec := evalf(a114405[i]/log(a014614[i])) ; fi ; od ; # R. J. Mathar, May 10 2007

a(n) = records in A114405(n)/log_10(A014614(n)) = records in (A014614(n+1) - A014614(n))/log_10(A014614(n)).

a(6) from R. J. Mathar, May 10 2007

A131749 Triangle of successive absolute differences of semiprimes.

Original entry on oeis.org

4, 2, 6, 1, 3, 9, 1, 2, 1, 10, 0, 1, 3, 4, 14, 1, 1, 0, 3, 1, 15, 0, 1, 2, 2, 5, 6, 21, 1, 1, 0, 2, 0, 5, 1, 22, 1, 0, 1, 1, 3, 3, 2, 3, 25, 1, 0, 0, 1, 0, 3, 0, 2, 1, 26, 1, 0, 0, 0, 1, 1, 4, 4, 6, 7, 33, 1, 0, 0, 0, 0, 1, 0, 4, 0, 6, 1, 34, 0, 1, 1, 1, 1, 1, 2, 2, 6, 6, 0, 1, 35

Offset: 1

Jonathan Vos Post, Oct 23 2007

Semiprime analog of A036262. The conjecture analogous to Gilbreath's conjecture is that the leading term (after the second row) is always 0 or 1. First diagonal is semiprimes (A001358). Second diagonal is first differences of semiprimes (A065516).

			Table begins:
4  6  9 10 14 15 21 22 25 26 33 34 35 38 39 46 49 51 55 57 58 62 65 69 74 77 82 85
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  2  3  3  5  5  2  2  6  6  0  2  2  6  4  1  2  2  1  3  1  1  1  2  2  2
1  1  0  2  0  3  0  4  0  6  2  0  4  2  3  1  0  1  2  2  0  0  1  0  0
0  1  2  2  3  3  4  4  6  4  2  4  2  1  2  1  1  1  0  2  0  1  1  0
1  1  0  1  0  1  0  2  2  2  2  2  1  1  1  0  0  1  2  2  1  0  1
0  1  1  1  1  1  2  0  0  0  0  1  0  0  1  0  1  1  0  1  1  1
1  0  0  0  0  1  2  0  0  0  1  1  0  1  1  1  0  1  1  0  0
1  0  0  0  1  1  2  0  0  1  0  1  1  0  0  1  1  0  1  0
1  0  0  1  0  1  2  0  1  1  1  0  1  0  1  0  1  1  1
1  0  1  1  1  1  2  1  0  0  1  1  1  1  1  1  0  0
1  1  0  0  0  1  1  1  0  1  0  0  0  0  0  1  0
0  1  0  0  1  0  0  1  1  1  0  0  0  0  1  1
1  1  0  1  1  0  1  0  0  1  0  0  0  1  0
0  1  1  0  1  1  1  0  1  1  0  0  1  1
1  0  1  1  0  0  1  1  0  1  0  1  0
1  1  0  1  0  1  0  1  1  1  1  1
0  1  1  1  1  1  1  0  0  0  0
1  0  0  0  0  0  1  0  0  0
1  0  0  0  0  1  1  0  0
1  0  0  0  1  0  1  0
1  0  0  1  1  1  1
1  0  1  0  0  0
1  1  1  0  0
0  0  1  0
0  1  1
1  0
1
etc.

Robert G. Wilson v, Table of n, a(n) for n = 1..10011
Index entries for sequences related to Gilbreath conjecture and transform

Cf. A001358, A036262, A065516.

SemiPrimePi[n_] := Sum[ PrimePi[n/Prime[i]] - i + 1, {i, PrimePi[ Sqrt[n]]}]; SemiPrime[n_] := Block[{e = Floor[Log[2, n] + 1], a, b}, a = 2^e; Do[b = 2^p; While[SemiPrimePi@a < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; t[0, n_] := SemiPrime[n]; t[r_, c_] := Abs[t[r - 1, c] - t[r - 1, c + 1]]; Table[t[r - c, c], {r, 13}, {c, r}] // Flatten
(* to construct the table as shown *) mx = 13; Table[t[r, c], {r, 0, mx - 1}, {c, mx - r}] // TableForm (* Robert G. Wilson v, Jun 13 2018 *)

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

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

A114417 Records in 7-almost prime gaps, ordered by merit.

Original entry on oeis.org

64, 96, 112, 168, 210, 280

Offset: 1

Jonathan Vos Post, Nov 25 2005

			Records defined in terms of A114407 and A046308:
n A114407(n) A114407(n)/log(A046308(n))
1 64 64/log 128 = 30.371914
2 96 96/log 192 = 42.0443868
3 32 32/log 288 = 13.0113433
4 112 112/log 320 = 44.7079021
5 16 16/log 432 = 6.07099172
6 32 32/log 448 = 12.0696509
7 168 168/log 480 = 62.6575474
8 24 24/log 648 = 8.53614076

Cf. A046308, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

a(n) = Records in A114417(n)/log(A046308(n)) = Records in (A046308(n+1) - A046308(n))/log(A046308(n)).

a(5)-a(6) from Donovan Johnson, Feb 17 2010

A114418 Records in 8-almost prime gaps ordered by merit.

Original entry on oeis.org

128, 192, 224, 336, 420, 560

Offset: 1

Jonathan Vos Post, Dec 03 2005

			Records defined in terms of A114408 and A046310:
n A114418(n) A114418(n)/log(A046310(n)).
1 128 128/log 256 = 53.1508495
2 192 192/log 384 = 74.2938824
3 64 64/log 576 = 23.1848568
4 224 224/log 640 = 79.8238182
5 32 32/log 864 = 10.8972758
6 64 64/log 896 = 21.6779549
7 336 336/log 960 = 112.665809
8 48 48/log 1296 = 15.4211665
22 420 420/log 2496 = 123.629603

Cf. A046310, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

a(n) = records in A114418(n)/log(A046310(n)) = records in (A046310(n+1) - A046310(n))/log(A046310(n)).

Offset corrected and a(6) from Donovan Johnson, Feb 17 2010

A123386 Largest difference between successive semiprimes up to 10^n inclusive.

Original entry on oeis.org

3, 7, 14, 24, 38, 47, 74, 74, 95, 112, 146, 163, 174

Offset: 1

Alexander Adamchuk, Nov 09 2006

There are 4 semiprimes up to 10^1 {4, 6, 9, 10}. The differences between successive semiprimes are {2, 3, 1}. Thus a(1) = Max[ {2, 3, 1} ] = 3.

Cf. A001358, A065516, A036352, A038460, A053303.

A001358(prev)={ local(a=prev+1) ; while(bigomega(a)!=2, a++ ; ) ; return(a) ; }
A123386(n)={ local(sp1=4,sp2=6,a=2) ; while(sp2<=10^n, a=max(a,sp2-sp1) ; sp1=sp2 ; sp2=A001358(sp1) ; ) ; return(a) ; }
{ for(n=1,13, print(A123386(n)) ; ) ; } \\ 2 more terms from R. J. Mathar, Jan 17 2008

2 more term from R. J. Mathar, Jan 17 2008
a(8)-a(9) from Donovan Johnson, Sep 05 2008
a(10)-a(11) from Donovan Johnson, Apr 14 2010
a(12)-a(13) from Donovan Johnson, Sep 20 2012

A135406 Sum of squares of gaps between consecutive semiprimes.

Original entry on oeis.org

4, 13, 14, 30, 31, 67, 68, 77, 78, 127, 128, 129, 138, 139, 188, 197, 201, 217, 221, 222, 238, 247, 263, 288, 297, 322, 331, 332, 333, 349, 353, 354, 355, 476, 501, 517, 526, 527, 531, 532, 533, 569, 585, 586, 635, 636, 637, 641, 642, 723, 732, 733, 737, 762

Offset: 1

Jonathan Vos Post, Dec 09 2007

This is to semiprimes A001358 as A074741 is to primes A000040. What is the semiprime analog of D. R. Heath-Brown's conjecture: Sum_{prime(n)<=N} (prime(n)-prime(n-1))^2 ~ 2*N*log(N) and Marek Wolf's conjecture: Sum_{prime(n)A000720(n).

			a(10) = (6-4)^2 + (9-6)^2 + (10-9)^2 + (14-10)^2 + (15-14)^2 + (21-15)^2 + (22-21)^2 + (25-22)^2 + (26-25)^2 + (33-26)^2 = (2^2) + (3^2) + (1^2) + (4^2) + (1^2) + (6^2) + (1^2) + (3^2) + (1^2) + (7^2) = 127.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A001358, A065516, A074741.

A001358 := proc(n) option remember ; local a ; if n = 1 then 4; else for a from A001358(n-1)+1 do if numtheory[bigomega](a) = 2 then RETURN(a) ; fi ; od: fi ; end: A065516 := proc(n) A001358(n+1)-A001358(n) ; end: A135406 := proc(n) add( (A065516(k))^2,k=1..n) ; end: seq(A135406(n),n=1..80) ; # R. J. Mathar, Jan 07 2008

Accumulate[Differences[Select[Range[200],PrimeOmega[#]==2&]]^2] (* Harvey P. Dale, Mar 05 2015 *)

a(n) = SUM[k=1..n] A065516(k)^2 = SUM[k=1..n] (A001358(n+1) - A001358(n))^2.

More terms from R. J. Mathar, Jan 07 2008

A137462 a(n) + a(n-1) = n-th semiprime.

Original entry on oeis.org

1, 3, 3, 6, 4, 10, 5, 16, 6, 19, 7, 26, 8, 27, 11, 28, 18, 31, 20, 35, 22, 36, 26, 39, 30, 44, 33, 49, 36, 50, 37, 54, 39, 55, 40, 66, 45, 70, 48, 71, 50, 72, 51, 78, 55, 79, 62, 80, 63, 82, 64, 91, 67, 92, 69, 97, 72, 105, 73, 110, 75, 112, 82, 119, 83, 120, 85, 121, 88, 125

Offset: 0

Jonathan Vos Post, Apr 19 2008

This is to A001358 as A036467 is to A000040.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A001358, A036467, A065516.

A001358 := proc(n) option remember ; if n =1 then 4; else for a from A001358(n-1)+1 do if numtheory[bigomega](a) = 2 then RETURN(a) ; fi ; od: fi ; end: A137462 := proc(n) option remember; if n =0 then 1; else A001358(n)-A137462(n-1) ; fi ; end: seq(A137462(n),n=0..100) ; # R. J. Mathar, Apr 23 2008

Module[{nn=300,sp,k=1},sp=Select[Range[nn],PrimeOmega[#]==2&];Join[{1}, Table[k=sp[[n]]-k,{n,Length[sp]}]]] (* Harvey P. Dale, Apr 30 2015 *)

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

More terms from R. J. Mathar, Apr 23 2008

A200927 Difference between (least semiprime >= n) and (largest semiprime <= n).

Original entry on oeis.org

0, 2, 0, 3, 3, 0, 0, 4, 4, 4, 0, 0, 6, 6, 6, 6, 6, 0, 0, 3, 3, 0, 0, 7, 7, 7, 7, 7, 7, 0, 0, 0, 3, 3, 0, 0, 7, 7, 7, 7, 7, 7, 0, 3, 3, 0, 2, 0, 4, 4, 4, 0, 2, 0, 0, 4, 4, 4, 0, 3, 3, 0, 4, 4, 4, 0, 5, 5, 5, 5, 0, 3, 3, 0, 5, 5, 5, 5, 0, 3, 3, 0, 0, 0, 4, 4, 4

Offset: 4

Arkadiusz Wesolowski, Nov 24 2011

nonn

a(n) = 0 if and only if n is semiprime.

Arkadiusz Wesolowski, Table of n, a(n) for n = 4..10000

Cf. A001358, A065516.

A106325 := proc(n)
    for a from n do
        if numtheory[bigomega](a) = 2 then
            return a;
        end if;
    end do:
end proc;
prevSpr := proc(n)
    for a from n by -1 do
        if numtheory[bigomega](a) = 2 then
            return a;
        end if;
    end do:
end proc;
A200927 := proc(n)
    A106325(n)-prevSpr(n) ;
end proc:
seq(A200927(n),n=4..80) ; # R. J. Mathar, Nov 26 2011

Table[a = b = 0; While[! PrimeOmega[n - a] == 2, a++]; While[! PrimeOmega[n + b] == 2, b++]; a + b, {n, 4, 100}]

cp's OEIS Frontend

A114415 Records in 5-almost prime gaps ordered by merit.

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A131749 Triangle of successive absolute differences of semiprimes.

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A264046 Numbers k such that k and k+6 are consecutive semiprimes.

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PARI

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

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A114417 Records in 7-almost prime gaps, ordered by merit.

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A114418 Records in 8-almost prime gaps ordered by merit.

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A123386 Largest difference between successive semiprimes up to 10^n inclusive.

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PARI

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A135406 Sum of squares of gaps between consecutive semiprimes.

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