A065516 -id:A065516 - cp's OEIS frontend

A239656 First differences of sphenic numbers, cf. A007304.

12, 24, 4, 8, 24, 3, 5, 4, 16, 8, 16, 11, 5, 4, 8, 4, 4, 5, 27, 8, 1, 7, 8, 9, 3, 8, 7, 9, 3, 1, 4, 20, 8, 4, 23, 9, 3, 9, 4, 4, 11, 14, 3, 4, 4, 8, 8, 3, 1, 4, 1, 3, 4, 13, 10, 5, 4, 9, 11, 4, 8, 12, 12, 4, 21, 6, 13, 8, 8, 5, 3, 4, 4, 3, 1, 5, 3, 9, 11, 4

Offset: 1

Reinhard Zumkeller, Mar 23 2014

nonn

a(n) = A007304(n+1) - A007304(n);

see A239673 and A239674 for record values and where they occur: A239673(n) = a(A239674(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A065516.

a239656 n = a239656_list !! (n-1)
a239656_list = zipWith (-) (tail a007304_list) a007304_list

A007304 := proc(n)
    option remember;
    if n = 1 then
        30;
    else
        for a from procname(n-1)+1 do
            if numtheory[bigomega](a) =3 and nops(numtheory[factorset](a)) = 3 then
                return a;
            end if;
        end do:
    end if;
end proc:
A239656 := proc(n)
    A007304(n+1)-A007304(n) ;
end proc:

With[{upto=1000},Differences[Sort[Select[Times@@@Subsets[Prime[ Range[ Ceiling[upto/6]]],{3}],#<=upto&]]]] (* Harvey P. Dale, Jan 08 2015 *)

A264043 Numbers n such that n and n+3 are consecutive semiprimes.

Original entry on oeis.org

6, 22, 35, 46, 62, 74, 82, 115, 155, 166, 206, 259, 262, 295, 323, 355, 358, 362, 395, 466, 478, 482, 502, 511, 559, 562, 583, 586, 611, 623, 626, 671, 703, 718, 731, 734, 746, 755, 763, 799, 835, 838, 862, 866, 886, 895, 914, 923, 955, 979, 982

Offset: 1

Zak Seidov, Nov 02 2015

nonn

			6 and 9 are 2nd and 3rd semiprimes, or 6=A001358(2) and 9=A001358(3);
22=A001358(8) and 25=A001358(9).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001358, A065516, A136196.

Flatten[Position[Partition[Table[If[PrimeOmega[n]==2,1,0],{n,1000}], 4,1],?(#=={1,0,0,1}&)]] (* _Harvey P. Dale, Feb 08 2016 *)

for(n=1, 1e3, if(bigomega(n) == 2 && bigomega(n+3) == 2 && bigomega(n+1) !=2 && bigomega(n+2) !=2, print1(n", "))) \\ Altug Alkan, Nov 02 2015

is(n)=if(n%4==2, isprime(n/2) && bigomega(n+3)==2 && bigomega(n+1)!=2, n%4==3 && isprime((n+3)/2) && bigomega(n)==2 && bigomega(n+2)!=2) \\ Charles R Greathouse IV, Nov 02 2015

a(n) >> n log n. - Charles R Greathouse IV, Nov 02 2015

A114021 Number of semiprimes between n and n + sqrt(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 2, 2, 2, 2, 2, 1, 0, 0, 1, 2, 3, 3, 3, 3, 3, 2, 2, 2, 1, 0, 1, 1, 1, 2, 2, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 5, 5, 4, 4, 4, 4, 3, 3, 2, 1, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2

Offset: 0

Jonathan Vos Post, Jan 31 2006

It appears that for n > 37 it is always true that a(n) > 0. The exponent can be reduced further. Since 597 + 597^(0.4129) > 611, leaping the record semiprime gap between 597 and 611, it seems that for n > 597 it is always true that there is a semiprime between n and n^(0.4129). It seems that for n > 2705 it is always true that there is a semiprime between n and n^(0.3509). These conjectures are related to the various sequences about semiprime gaps and the merit of such gaps.

a(96) appears to be the last zero term. - T. D. Noe, Aug 12 2008

			a(0) = 0 because there are no semiprimes between 0 and 0+sqrt(0) = 0.
a(2) = 0 because there are no semiprimes between 2 and 2+sqrt(2) = 3.414...
a(3) = 1 as the semiprime 4 falls between 3 and 3 + sqrt(3) = 4.732...
a(5) = 1 as the semiprime 6 falls between 5 and 5 + sqrt(5) = 7.236...

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A001358, A060715, A065516, A077463, A085809, A097824, A114412, A115766.

SemiPrimeQ[n_] := TrueQ[Plus@@Last/@FactorInteger[n]==2]; Table[hi=n+Sqrt[n]; If[IntegerQ[hi], hi--, hi=Floor[hi]]; Length[Select[Range[n+1,hi], SemiPrimeQ]], {n,0,150}] (* T. D. Noe, Aug 12 2008 *)

use ntheory ":all"; print "$ ",semiprime_count($+1, $+sqrtint($)-($ && is_square($))),"\n" for 0..1000; # Dana Jacobsen, Mar 04 2019

a(n) = card{S such that S is an element of A001358 and n < S < n + n^(1/2)}.

Corrected and extended by T. D. Noe, Aug 12 2008

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)).

A217335 Semiprimes p such that next semiprime after p is p+20.

Original entry on oeis.org

2681, 3523, 6953, 8227, 16817, 26101, 28533, 28563, 28617, 29011, 34249, 37007, 42401, 49983, 50117, 55703, 60657, 65083, 66938, 71381, 71873, 73443, 76247, 92773, 92978, 101109, 101271, 109129, 111479, 112051, 113018, 113721, 115586, 116267, 119969, 124177

Offset: 1

Zak Seidov, Oct 01 2012

nonn

			2681 = A001358(760)  = 7*383, 2701 = A001358(761) = 37*73,
3523 = A001358(986)  = 13*271, 3543 = A001358(987) = 3*1181.

Zak Seidov, Table of n, a(n) for n = 1..2000

Cf. A001358, A065516, A217030.

IsSemiprime:=func; [n: n in [4..130000] | IsSemiprime(n) and IsSemiprime(n+20) and forall{n+i: i in [1..19] | not IsSemiprime(n+i)}]; // Bruno Berselli, Oct 01 2012

f = Flatten@Position[Differences@(s = Select[Range@100000, PrimeOmega@# == 2 &]), 20]; s[[f]] (* Hans Rudolf Widmer, Aug 19 2024 *)

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

A114057 Start of record gap in odd semiprimes A046315.

Original entry on oeis.org

9, 25, 39, 95, 267, 2369, 6559, 8817, 13705, 15261, 21583, 35981, 66921, 113009, 340891, 783757, 872219, 3058853, 3586843, 5835191, 12345473, 108994623, 248706917, 268749691, 679956119, 709239621, 3648864859, 3790337723, 4171420481, 33955869693, 34279038379

Offset: 1

Jonathan Vos Post, Feb 02 2006

nonn

3 of the first 5 values of record gaps in odd semiprimes are also record merits = (A046315(k+1)-A046315(k))/log_10(A046315(k)), namely: (15 - 9) / log_10(9) = 6.28770982; (111 - 95) / log_10(95) = 8.09010923; (287 - 267) / log_10(267) = 8.24228608. It is easy to prove that there are gaps of arbitrary length in even semiprimes (A100484); can we prove that there are gaps of arbitrary length in odd semiprimes (A046315) and in semiprimes (A001358)?

The record gaps have lengths 6, 8, 10, 16, 20, 22, 24, 26, 28, 32, 36, 38, 40, 44, 50, 52, 60, 64, 70, 74. - T. D. Noe, Feb 03 2006

			a(1) = A046315(2)-A046315(1) = 15 - 9 = 6.
a(2) = A046315(5)-A046315(4) = 33 - 25 = 8.
a(3) = A046315(8)-A046315(7) = 49 - 39 = 10.
a(4) = A046315(20)-A046315(19) = 111 - 95 = 16.
a(5) = A046315(55)-A046315(54) = 287 - 267 = 20.

Cf. A001358, A046315, A065516, A085809, A100484, A114412, A114021.
Starting at a(4)=95 the known terms of this sequence coincide with A350098.
Cf. A341828, A349995, A350099.

f[n_] := Block[{k = n + 2}, While[ Plus @@ Last /@ FactorInteger@k != 2, k += 2]; k]; lst = {}; d = 0; a = b = 9; Do[{a, b} = {b, f[a]}; If[b - a > d, d = b - a; AppendTo[lst, a]], {n, 10^8}]; lst (* Robert G. Wilson v, Feb 03 2006 *)

{a(n)} = {A046315(k) such that A046315(k+1)-A046315(k) is a record}.

More terms from Robert G. Wilson v and T. D. Noe, Feb 03 2006
a(23)-a(28) from Donovan Johnson, Mar 14 2010
a(29)-a(31) from Donovan Johnson, Oct 20 2012

A124317 Semiprimes indexed by 3-almost primes.

Original entry on oeis.org

22, 34, 51, 57, 82, 85, 87, 123, 133, 134, 146, 158, 201, 205, 209, 214, 221, 226, 237, 295, 305, 309, 321, 327, 341, 361, 365, 371, 394, 395, 413, 447, 478, 481, 497, 501, 529, 533, 543, 545, 551, 554, 559, 583, 597, 614, 623, 635, 689, 699, 734, 763, 766

Offset: 1

Jonathan Vos Post, Oct 26 2006

Note that a(10)-a(9) = a(30)-a(29) = 1, achieving the minimum possible, due to a combination of the appropriate semiprime gap (A065516) and 3-almost prime gap (A114403).

			a(1) = semiprime(3almostprime(1)) = semiprime(8 = 2^3) = 22 = 2 * 11.
a(2) = semiprime(3almostprime(2)) = semiprime(12 = 2^2 * 3) = 34 = 2 * 17.
a(3) = semiprime(3almostprime(3)) = semiprime(18 = 2 * 3^2) = 51 = 3 * 17.

Giovanni Resta, Table of n, a(n) for n = 1..5000

Cf. A124318 3-almost primes indexed by semiprimes. A124319 semiprime(3almostprime(n)) - 3almostprime(semiprime(n)). A124308 Primes indexed by 5-almost primes. A124309 5-almost primes indexed by primes. A124310 prime(5almostprime(n)) - 5almostprime(prime(n)). 4-almost primes indexed by primes = A124283. prime(4almostprime(n)) - 4almostprime(prime(n)) = A124284. Primes indexed by 3-almost primes = A124268. 3-almost primes indexed by primes = A124269. prime(3almostprime(n)) - 3almostprime(prime(n)) = A124270. See also A106349 Primes indexed by semiprimes. See also A106350 Semiprimes indexed by primes. See also A122824 Prime(semiprime(n)) - semiprime(prime(n)). Commutator [A000040, A001358] at n.

Cf. A000040, A001358, A014612, A065516, A114403, A122824, A124269, A124270, A124282-A124284, A124309, A124310, A124318, A124319.

p[k_] := Select[Range[1000], PrimeOmega[#] == k &]; p[2][[Take[p[3], 60]]] (* Giovanni Resta, Jun 13 2016 *)

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A124317(n):
    def f(x): return int(x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    def g(x): return int(x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)+n
    while m != k:
        m, k = k, f(k)+n
    r, k = m, g(m)+m
    while r != k:
        r, k = k, g(k)+m
    return r # Chai Wah Wu, Aug 17 2024

a(n) = semiprime(3almostprime(n)) = A001358(A014612(n)).

Data corrected by Giovanni Resta, Jun 13 2016

A124318 3-almost primes indexed by semiprimes.

Original entry on oeis.org

20, 28, 44, 45, 66, 68, 98, 99, 110, 114, 147, 148, 153, 165, 170, 188, 207, 222, 238, 244, 245, 261, 273, 284, 310, 322, 343, 356, 357, 363, 374, 387, 388, 399, 429, 438, 465, 475, 477, 494, 498, 506, 531, 549, 555, 590, 595, 596, 602, 603, 628, 639, 642

Offset: 1

Jonathan Vos Post, Oct 26 2006

			a(1) = 3almostprime(semiprime(1)) = 3almostprime(4 = 2^2) = 20 = 2^2 * 5.
a(2) = 3almostprime(semiprime(2)) = 3almostprime(6 = 2 * 3) = 28 = 2^2 * 7.
a(3) = 3almostprime(semiprime(3)) = 3almostprime(9 = 3^2) = 44 = 2^2 * 11.
a(4) = 3almostprime(semiprime(4)) = 3almostprime(10 = 2 * 5) = 45 = 3^2 * 5.

Cf. A124317 Semiprimes indexed by 3-almost primes. A124318 3-almost primes indexed by semiprimes. A124319 semiprime(3almostprime(n)) - 3almostprime(semiprime(n)). A124308 Primes indexed by 5-almost primes. A124309 5-almost primes indexed by primes. A124310 prime(5almostprime(n)) - 5almostprime(prime(n)). 4-almost primes indexed by primes = A124283. prime(4almostprime(n)) - 4almostprime(prime(n)) = A124284. Primes indexed by 3-almost primes = A124268. 3-almost primes indexed by primes = A124269. prime(3almostprime(n)) - 3almostprime(prime(n)) = A124270. See also A106349 Primes indexed by semiprimes. See also A106350 Semiprimes indexed by primes. See also A122824 Prime(semiprime(n)) - semiprime(prime(n)). Commutator [A000040, A001358] at n.

Cf. A000040, A001358, A014612, A065516, A114403, A122824, A124269, A124270, A124282-A124284, A124309, A124310, A124317, A124319.

p[k_] := Select[Range[1000], PrimeOmega[#] == k &]; p[3][[Take[p[2], 60]]] (* Giovanni Resta, Jun 13 2016 *)

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A124318(n):
    def g(x): return int(x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    def f(x): return int(x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)+n
    while m != k:
        m, k = k, f(k)+n
    r, k = m, g(m)+m
    while r != k:
        r, k = k, g(k)+m
    return r # Chai Wah Wu, Aug 17 2024

a(n) = 3almostprime(semiprime(n)) = A014612(A001358(n)).

a(22)-a(53) from Giovanni Resta, Jun 13 2016

cp's OEIS Frontend

A239656 First differences of sphenic numbers, cf. A007304.

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A264043 Numbers n such that n and n+3 are consecutive semiprimes.

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A114021 Number of semiprimes between n and n + sqrt(n).

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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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A217335 Semiprimes p such that next semiprime after p is p+20.

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

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