A111870 -id:A111870 - cp's OEIS frontend

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

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)).

A241540 Indices of primes p in A182514, i.e., a(n) = primepi(p) = A000720(A182514(n)).

Original entry on oeis.org

1, 2, 4, 30, 217, 49749629143526

Offset: 1

M. F. Hasler, Apr 25 2014

a(6) = A214935(33) = A000720(A205827(33)).

Cf. A053302, A053976, A073861, A111870, A111943, A182514.

See also A214935(n)=A000720(A205827(n)), A005669(n) = A000720(A002386(n)).

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

A130789 The primes prime(n) sorted according to increasing prime(n)/prime(n+1).

Original entry on oeis.org

3, 7, 2, 5, 13, 23, 19, 31, 11, 47, 113, 17, 53, 37, 61, 43, 89, 73, 83, 139, 29, 199, 67, 211, 181, 79, 41, 293, 131, 317, 241, 97, 151, 103, 157, 109, 167, 283, 173, 523, 59, 127, 337, 71, 233, 467, 1327, 163, 409, 251, 421, 509, 257, 263, 887, 359, 271, 193

Offset: 1

R. J. Mathar, Jul 15 2007

nonn

Or: primes sorted according to decreasing ratio A001223(n)/A000040(n). All values are conjectural, derived from the finite list up to prime(200000): large prime gaps at higher indices may still insert numbers above prime(200000) at low positions of the sequence.

Using a table of prime gaps, it is easy to determine that the sequence is correct for all primes < 10^18. - T. D. Noe, Jul 17 2007

			3/5 < 7/11 < 2/3 < 5/7 < 13/17 < 23/29 < 19/23 < 31/37 < 11/13 < ...
Numerators of this chain of inequalities define the sequence.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Gaps
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]

Cf. A107664, A111870.

With[{nn=60},Take[Transpose[SortBy[Partition[Prime[Range[20*nn]],2,1], #[[1]]/ #[[2]]&]][[1]],nn]] (* Harvey P. Dale, Dec 03 2014 *)

A166597 Let p = largest prime <= n, with p(0)=p(1)=0, and let q = smallest prime > n; then a(n) = q-p.

Original entry on oeis.org

2, 2, 1, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 2, 2, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 2, 2, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 2, 2, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 4, 4, 4, 4, 2, 2, 4, 4

Offset: 0

Daniel Forgues, Oct 17 2009

nonn

Note the large prime gap of 72 between 31397 and 31469. This is the prime gap with the largest merit (cf. A111870), 72/log(31397)=6.95352 for primes less than 100000. Also 72/(log(31397))^2=0.67154 (cf. conjectures of Cramer-Granville, Shanks and Wolf) is largest for primes less than 100000. - Daniel Forgues, Oct 23 2009

			a(0) = 2 since the least prime greater than 0 is 2 (gap of 2 from 0 to 2).
a(9) = 4 since the least prime greater than 9 is 11 (gap of 4 from 7 to 11).
a(11) = 2 since the least prime greater than 11 is 13 (gap of 2 from 11 to 13).

Daniel Forgues, Table of n, a(n) for n = 0..100000
Eric Weisstein's World of Mathematics, Prime Gaps.
Eric Weisstein's World of Mathematics, Cramer-Granville Conjecture.
Eric Weisstein's World of Mathematics, Shanks Conjecture.

Cf. A151800, A166594.
Cf. A111870. - Daniel Forgues, Oct 23 2009
See A327441 for the classic G(n) version. - N. J. A. Sloane, Sep 11 2019
Cf. A001223, A000720, A007917, A007918, A151799.

2,2,seq(nextprime(n)-prevprime(n+1), n=2..100); # Ridouane Oudra, Dec 28 2024

f[n_]:=Module[{a=If[PrimeQ[n],n,NextPrime[n,-1]]}, NextPrime[n]-a]; Join[{2,2},Array[f,120,2]] (* Harvey P. Dale, May 17 2011 *)

a(n) = nextprime(n+1) - precprime(n); \\ Michel Marcus, Mar 02 2023

From Ridouane Oudra, Dec 28 2024: (Start)
a(n) = A001223(A000720(n)), for n>1.
a(n) = A151800(n) - A007917(n), for n>1.
a(n) = A007918(n+1) - A151799(n+1), for n>1. (End)

Definition rephrased by N. J. A. Sloane, Oct 25 2009

cp's OEIS Frontend

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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A241540 Indices of primes p in A182514, i.e., a(n) = primepi(p) = A000720(A182514(n)).

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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.

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

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

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