A114422 -id:A114422 - cp's OEIS frontend

A114403 Triprime gaps. First differences of A014612.

4, 6, 2, 7, 1, 2, 12, 2, 1, 5, 2, 11, 3, 2, 2, 5, 1, 2, 14, 6, 1, 3, 3, 5, 4, 2, 1, 7, 1, 5, 8, 9, 1, 5, 1, 10, 1, 5, 1, 1, 2, 1, 7, 4, 2, 2, 5, 12, 5, 10, 8, 1, 5, 2, 4, 2, 1, 1, 9, 3, 3, 5, 2, 5, 2, 4, 3, 2, 1, 1, 4, 2, 18, 6, 2, 4, 3, 7, 1, 5, 5, 2, 9, 2, 1

Offset: 1

Jonathan Vos Post, Nov 25 2005

			a(1) = 4 = 12-8 where 8 is the first triprime and 12 is the second.
a(2) = 6 = 18-12
a(3) = 2 = 20-18
a(4) = 7 = 27-20

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

Cf. A014612, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

is3Alm := proc(n::integer) local ifa,ex,i ; ifa := op(2,ifactors(n)) ; ex := 0 ; for i from 1 to nops(ifa) do ex := ex+ op(2,op(i,ifa)) ; od : if ex = 3 then RETURN(true) ; else RETURN(false) ; fi ; end: A014612 := proc(n::integer) local resul,i; i :=1; resul := 8 ; while i < n do resul := resul + 1 ; if is3Alm(resul) then i := i+1 ; fi ; od ; RETURN(resul) ; end: A114403 := proc(n::integer) RETURN(A014612(n+1)-A014612(n)) ; end: for n from 1 to 160 do printf("%d,",A114403(n)) ; od: # R. J. Mathar, Apr 25 2006

Differences[Select[Range[425], PrimeOmega[#] == 3 &]] (* Jayanta Basu, Jul 01 2013 *)

a(n) = A014612(n+1) - A014612(n).

Corrected and extended by R. J. Mathar, Apr 25 2006

A114412 Records in semiprime gaps ordered by merit.

Original entry on oeis.org

2, 3, 4, 6, 11, 19, 24, 28, 30, 32, 38, 47, 54, 70, 74, 107, 110, 112, 120, 126, 146

Offset: 1

Jonathan Vos Post, Nov 25 2005

There is an associated index list n = 1, 2, 4, 6, 34, 422, 1765, 4585, 8112, 8650, 8861, 75150, ... and an associated semiprime list A001358(n) = 4, 6, 10, 15, 1418, 6559, 17965, 32777, 35103, 35981, 340894, ... - R. J. Mathar, Mar 15 2009

			Records defined in terms of A065516 and A001358:
.
  n  A065516(n)  A065516(n)/log_10(A001358(n))
  =  ==========  ==============================
  1       2      2 / log_10(4)  = 3.32192809...
  2       3      3 / log_10(6)  = 3.85529162...
  3       1      1 / log_10(9)  = 1.04795163...
  4       4      4 / log_10(10) = 4.00000000
  5       1      1 / log_10(14) = 0.87250286...
  6       6      6 / log_10(15) = 5.10164492...
  7       1      1 / log_10(21) = 0.75630419...
  8       3      3 / log_10(22) = 2.23476557...
  9       1      1 / log_10(25) = 0.71533827...

Eric Weisstein's World of Mathematics, Semiprime.
Eric Weisstein's World of Mathematics, Prime Gaps.

Cf. A001358, A065516, A111870, A111871, A113688-A113693, A114403-A114411, A114412-A114422.

sp = 4; m0 = 0;  l = {}; lim = 1000000;
For[i = 5, i <= lim, i++, If[PrimeOmega[i] == 2, m = (i - sp)/Log[sp]; If[m > m0, m0 = m; AppendTo[l, i - sp]]; sp = i] ]; l (* Robert Price, Oct 29 2018 *)

a(n) = records in A065516(n)/log_10(A001358(n)) = records in (A001358(n+1) - A001358(n))/log_10(A001358(n)).

Corrected and extended by Charles R Greathouse IV, Oct 05 2006

a(16)-a(21) from Donovan Johnson, Feb 17 2010

A114405 5-almost prime gaps. First differences of A014614.

Original entry on oeis.org

16, 24, 8, 28, 4, 8, 42, 6, 8, 4, 20, 8, 35, 9, 12, 6, 2, 8, 20, 4, 8, 56, 10, 14, 4, 9, 3, 12, 20, 10, 6, 8, 4, 28, 4, 20, 32, 15, 21, 4, 2, 18, 4, 14, 26, 4, 15, 5, 4, 4, 8, 4, 2, 26, 16, 6, 2, 8, 20, 48, 20, 34, 6, 3, 27, 2, 4, 20, 1, 7, 16, 8, 4, 4, 6, 30, 6, 6, 12, 6, 3, 11

Offset: 1

Jonathan Vos Post, Nov 25 2005

First occurrences of a(n)=1,2,3,.. are at n=69, 17, 27, 5, 48, 8, 70, 3, 14, 23, 82, 15, 150, 24, 38, 1, 172, 42, 258, 11, 39, 135, 102, 2, 779, 45, 65, 4, 518, 76, 263, 37, 211, 62, 13, 1009, 2463, 606, 254, 151, 3348, 7, 4513,... - R. J. Mathar, Oct 06 2007

			a(1) = 16 = 48-32 where 32 is the first 5-almost prime and 48 is the second.
a(2) = 24 = 72-48.
a(3) = 8 = 80-72.
a(4) = 28 = 108-80.
a(5) = 4 = 112-108.
a(6) = 8 = 120-112.
a(7) = 42 = 162-120.
a(8) = 6 = 168-162.
a(13) = 35 = 243-208.
a(22) = 56 = 368-312.

R. J. Mathar, Table of n, a(n) for n = 1..9999

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

Differences[Select[Range[2000],PrimeOmega[#]==5&]] (* Harvey P. Dale, Sep 28 2019 *)

a(n) = A014614(n+1) - A014614(n).

More terms from R. J. Mathar, Oct 06 2007

A114414 Records in 4-almost prime gaps ordered by merit.

Original entry on oeis.org

8, 12, 14, 21, 28

Offset: 1

Jonathan Vos Post, Nov 25 2005

Next term (if it exists) associated with A014613 > 1030000. - R. J. Mathar, Mar 13 2007

			Records defined in terms of A114404 and A014613:
  n  A114404(n)  A114404(n)/log_10(A014613(n))
  =  ==========  =============================
  1      8       8/log_10(16)   = 6.64385619
  2      12      12/log_10(24)  = 8.6943213
  3      4       4/log_10(36)   = 2.57019442
  4      14      14/log_10(40)  = 8.73874891
  5      2       2/log_10(54)   = 1.15447195
  6      4       4/log_10(56)   = 2.2880834
  7      21      21/log_10(60)  = 11.810019
  ...
  13     22      22/log_10(104) = 10.9071078
  ...
  21     28      28/log_10(156) = 12.7671725

Cf. A014613, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

Digits := 16 : A114414 := proc() local n,a014613,a114414,rec ; a014613 := 16 ; a114414 := 8 ; rec := a114414/log(a014613) ; print(a114414) ; n := 17 ; while true do while numtheory[bigomega](n) <> 4 do n := n+1 ; od ; a114414 := n-a014613 ; if ( evalf(a114414/log(a014613)) > evalf(rec) ) then rec := a114414/log(a014613) ; print(a114414) ; fi ; a014613 := n ; n := n+1 : od ; end: A114414() ; # R. J. Mathar, Mar 13 2007

a(n) = records in A114404(n)/log_10(A014613(n)) = records in (A014613(n+1) - A014613(n))/log_10(A014613(n)).

A159965 Riordan array (1/sqrt(1-4x), (1-2x-(1-3x)c(x))/(x*sqrt(1-4x))), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 20, 21, 8, 1, 70, 84, 45, 11, 1, 252, 330, 220, 78, 14, 1, 924, 1287, 1001, 455, 120, 17, 1, 3432, 5005, 4368, 2380, 816, 171, 20, 1, 12870, 19448, 18564, 11628, 4845, 1330, 231, 23, 1, 48620, 75582, 77520, 54264, 26334, 8855, 2024, 300, 26, 1

Offset: 0

Paul Barry, Apr 28 2009

Product of A007318 and A114422. Product of A007318^2 and A116382. Row sums are A108080.
Diagonal sums are A108081.
Riordan array (1/sqrt(1 - 4*x), x*c(x)^3) obtained from A092392 by taking every third column starting from column 0; x*c(x)^3 is the o.g.f. for A000245. - Peter Bala, Nov 24 2015

			Triangle begins
1,
2, 1,
6, 5, 1,
20, 21, 8, 1,
70, 84, 45, 11, 1,
252, 330, 220, 78, 14, 1,
924, 1287, 1001, 455, 120, 17, 1,
3432, 5005, 4368, 2380, 816, 171, 20, 1

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

Cf. A000245, A007318, A092392, A108080, A108081, A114422, A116382.

/* As triangle */ [[Binomial(2*n+k, n+2*k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 27 2015

Number triangle T(n,k) = Sum_{j = 0..n} binomial(n+k,j-k)*binomialC(n,j).

T(n,k) = binomial(2*n + k, n + 2*k). - Peter Bala, Nov 24 2015

A114404 4-almost prime gaps. First differences of A014613.

Original entry on oeis.org

8, 12, 4, 14, 2, 4, 21, 3, 4, 2, 10, 4, 22, 6, 3, 1, 4, 10, 2, 4, 28, 5, 7, 2, 6, 6, 10, 5, 3, 4, 2, 14, 2, 10, 16, 18, 2, 1, 9, 2, 7, 13, 2, 10, 2, 2, 4, 2, 1, 13, 8, 3, 1, 4, 10, 24, 10, 17, 3, 15, 1, 2, 10, 4, 8, 4, 2, 2, 3, 15, 3, 3, 6, 3, 7, 4, 10, 4, 8, 6, 4, 2, 2, 8, 4, 1, 35, 1, 4, 7, 4, 8, 6

Offset: 1

Jonathan Vos Post, Nov 25 2005

			a(1) = 8 = 24-16 where 16 is the first 4-almost prime and 24 is the second.
a(2) = 12 = 36-24.
a(3) = 4 = 40-36.
a(4) = 14 = 54-40.
a(5) = 2 = 56-54.
a(6) = 4 = 60-56.
a(7) = 21 = 81-60.
a(13) = 22 = 126-104.
a(21) = 28 = 184-156.

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

Cf. A014613, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

A114404 := proc(nmax) local a,i,a014613 ; a := [] ; i := 1 ; a014613 := -1 ; while nops(a) < nmax do if numtheory[bigomega](i) = 4 then if a014613 > 0 then a := [op(a),i-a014613] ; fi ; a014613 := i ; fi ; i := i+1 ; end: a ; end: A114404(200) ; # R. J. Mathar, May 10 2007

Differences[Select[Range[800],Total[FactorInteger[#][[All,2]]]==4&]] (* Harvey P. Dale, Feb 14 2017 *)
Select[Range[1000],PrimeOmega[#]==4&]//Differences (* Harvey P. Dale, May 12 2018 *)

a(n) = A014613(n+1) - A014613(n).

Corrected and extended by R. J. Mathar, May 10 2007

A114406 6-almost prime gaps. First differences of A046306.

Original entry on oeis.org

32, 48, 16, 56, 8, 16, 84, 12, 16, 8, 40, 16, 70, 18, 24, 12, 4, 16, 40, 8, 16, 105, 7, 20, 28, 8, 18, 6, 24, 40, 20, 12, 16, 8, 56, 8, 40, 64, 30, 42, 8, 4, 27, 9, 8, 28, 52, 8, 30, 10, 8, 8, 16, 8, 4, 52, 32, 12, 4, 16, 40, 96, 40, 5, 63, 12, 6, 54, 4, 8, 40, 2, 14, 32, 16, 8, 8, 12, 45

Offset: 1

Jonathan Vos Post, Nov 25 2005

			a(1) = 32 = 96-64 where 64 is the first 6-almost prime and 96 is the second.
a(2) = 48 = 144-96.
a(3) = 16 = 160-144.
a(4) = 56 = 216-160.
a(5) = 8 = 224-216.
a(6) = 16 = 240-224.
a(7) = 84 = 324-240.
a(8) = 12 = 336-324.
a(22) = 105 = 729-624.

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

Cf. A046306, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

a(n) = A046306(n+1) - A046306(n).

More terms from R. J. Mathar, Aug 31 2007

A114407 7-almost prime gaps. First differences of A046308.

Original entry on oeis.org

64, 96, 32, 112, 16, 32, 168, 24, 32, 16, 80, 32, 140, 36, 48, 24, 8, 32, 80, 16, 32, 210, 14, 40, 56, 16, 36, 12, 48, 80, 40, 24, 32, 16, 112, 16, 80, 107, 21, 60, 84, 16, 8, 54, 18, 16, 56, 104, 16, 60, 20, 16, 16, 32, 16, 8, 104, 64, 24, 8, 32, 80, 192, 80, 10, 126, 24, 12

Offset: 1

Jonathan Vos Post, Nov 25 2005

			a(1) = 64 = 192-128 where 128 is the first 7-almost prime and 192 is the second.
a(2) = 96 = 288-192.
a(3) = 32 = 320-288.
a(4) = 112 = 432-320.
a(5) = 16 = 448-432.
a(6) = 32 = 480-448.
a(7) = 168 = 648-480.
a(8) = 24 = 672-648.

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

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

Differences[Select[Range[10000],PrimeOmega[#]==7&]] (* Harvey P. Dale, Oct 13 2019 *)

a(n) = A046308(n+1) - A046308(n).

Corrected and extended by R. J. Mathar, Aug 31 2007

A114408 8-almost prime gaps. First differences of A046310.

Original entry on oeis.org

128, 192, 64, 224, 32, 64, 336, 48, 64, 32, 160, 64, 280, 72, 96, 48, 16, 64, 160, 32, 64, 420, 28, 80, 112, 32, 72, 24, 96, 160, 80, 48, 64, 32, 224, 32, 160, 214, 42, 120, 168, 32, 16, 108, 36, 32, 112, 208, 32, 120, 40, 32, 32, 64, 32, 16, 208, 128, 48

Offset: 1

Jonathan Vos Post, Dec 03 2005

			a(1) = 128 = 384-256 = A046310(2) - A046310(1).
a(2) = 192 = 576-384.
a(3) = 64 = 640-576.
a(4) = 224 = 864-640.
a(5) = 32 = 896-864.
a(6) 64 = 960-896.
a(7) = 336 = 1296-960.
a(8) = 48 = 1344-1296.
a(22) = 420 = 2916-2496.

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

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

a(n) = A046310(n+1) - A046310(n).

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

Original entry on oeis.org

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

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A114403 Triprime gaps. First differences of A014612.

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A114412 Records in semiprime gaps ordered by merit.

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A114405 5-almost prime gaps. First differences of A014614.

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A114414 Records in 4-almost prime gaps ordered by merit.

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A159965 Riordan array (1/sqrt(1-4x), (1-2x-(1-3x)c(x))/(x*sqrt(1-4x))), c(x) the g.f. of A000108.

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A114404 4-almost prime gaps. First differences of A014613.

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A114406 6-almost prime gaps. First differences of A046306.

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A114407 7-almost prime gaps. First differences of A046308.