A114403 -id:A114403 - cp's OEIS frontend

A184753 a(n) = A184752(n)/A130650(n) unless A130650(n) = 0 in which case a(n) = 0.

0, 0, 4, 1, 13, 2, 1, 10, 1, 5, 16, 1, 15, 16, 22, 5, 37, 2, 4, 2, 1, 24, 11, 10, 2, 28, 23, 11, 41, 20, 2, 3, 73, 13, 76, 12, 1, 20, 13, 85, 34, 1, 21, 2, 46, 62, 5, 3, 2, 2, 2, 1, 2, 78, 39, 80, 81, 122, 3, 63, 51, 32, 88, 1, 1, 1, 69, 70

Offset: 1

Rémi Eismann, Jan 21 2011

nonn

a(n) is the "level" of 3-almost primes.
The decomposition of 3-almost primes into weight * level + gap is A014612(n) = A130650(n) * a(n) + A114403(n) if a(n) > 0.
a(n) = A014612(n) - A114403(n) if A014612(n) - A114403(n) > A114403(n), 0 otherwise.

			For n = 1 we have A130650(1) = 0, hence a(1) = 0.
For n = 3 we have A184752(3)/A130650(3)= 16 / 4 = 4; hence a(3) = 4.
For n = 21 we have A184752(21)/A130650(21)= 97 / 97 = 28; hence a(21) = 1.

Rémi Eismann, Table of n, a(n) for n = 1..10000

Cf. A014612, A114403, A130650, A184752, A117078, A117563, A001223, A118534.

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

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

A121906 Excess of n-th 3-almost prime A014612 over previous prime.

Original entry on oeis.org

1, 1, 1, 1, 4, 5, 1, 1, 1, 2, 3, 5, 2, 5, 1, 3, 2, 3, 5, 3, 1, 2, 1, 2, 1, 1, 3, 4, 11, 12, 3, 1, 8, 9, 2, 3, 1, 2, 3, 4, 5, 1, 2, 1, 5, 7, 9, 2, 8, 1, 11, 1, 2, 3, 5, 1, 3, 4, 5, 4, 1, 4, 3, 5, 2, 4, 2, 1, 1, 2, 3, 7, 9, 3, 3, 1, 5, 8, 1, 2

Offset: 1

Jonathan Vos Post, Sep 01 2006

nonn

a(n) = 1 iff n-th 3-almost prime is of form prime + 1 (not yet in OEIS). See also: A109067 3-almost primes of the form semiprime + 1.

			a(1) = 8 - 7 = 1; a(2) = 12 - 11 = 1; a(3) = 18 - 17 = 1;
a(4) = 20 - 19 = 1; a(5) = 27 - 23 = 4; a(6) = 28 - 23 = 5;
a(7) = 30 - 29 = 1.

Cf. A000040, A014612, A114403.

A014612 := proc(n) option remember; local a; if n = 1 then 8; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 3 then return a; end if; end do: end if; end proc:
A121906 := proc(n) local t; t := A014612(n) ; t-prevprime(t) ; end proc:
seq(A121906(n),n=1..80) ; # R. J. Mathar, Dec 22 2010

lim=335;p3=Select[Range[lim], PrimeOmega[#] == 3 &] ;l3=Length[p3];Table[p3[[n]]-NextPrime[p3[[n]],-1],{n,l3}] (* James C. McMahon, Oct 24 2024 *)

a(n) = Min{A014612(n) - p where p is in A000040 and 1

A014612(n)}.

a(13) corrected by R. J. Mathar, Dec 22 2010

A110934 Difference between 3-almostprime(n) and 3-almostprime(n+2).

Original entry on oeis.org

10, 8, 9, 8, 3, 14, 14, 3, 6, 7, 13, 14, 5, 4, 7, 6, 3, 16, 20, 7, 4, 6, 8, 9, 6, 3, 8, 8, 6, 13, 17, 10, 6, 6, 11, 11, 6, 6, 2, 3, 3, 8, 11, 6, 4, 7, 17, 17, 15, 18, 9, 6, 7, 6, 6, 3, 2, 10, 12, 6, 8, 7, 7, 7, 6, 7, 5, 3, 2, 5, 6, 20, 24, 8, 6, 7, 10, 8, 6, 10, 7

Offset: 1

Jonathan Vos Post, Jan 21 2006

This is the 3-almost prime analog of what A113784 is for semiprimes and what A031131 is for primes. The minimum values in the sequence are 2 because we have, for example, the 3 consecutive 3-almost primes 170, 171, 172, so a(39) = A014612(41) - A014612(39) = 172 - 170 = 2. Equivalently, there are 2 consecutive 1 values of A114403 (3-almost prime gaps; first differences of A014612). This happens for elements of A113789 (numbers n such that n, n+1 and n+2 are 3-almost primes).

			a(1) = 10 because the difference between the first and third 3-almost primes is A014612(3) - A014612(1) = 18 - 8 = 10.
a(2) = A014612(4) - A014612(2) = 20 - 12 = 8.
a(3) = A014612(5) - A014612(3) = 27 - 18 = 9.

Cf. A014612, A031131, A067813, A113784, A113789, A114403.

a(n) = A014612(n+2) - A014612(n).

a(28) corrected by R. J. Mathar, Dec 22 2010

cp's OEIS Frontend

A184753 a(n) = A184752(n)/A130650(n) unless A130650(n) = 0 in which case a(n) = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

Extensions

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

Views

Author

Keywords

Examples

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Examples

Crossrefs

Formula

Extensions

A121906 Excess of n-th 3-almost prime A014612 over previous prime.

Views

Author

Keywords