A065516 -id:A065516 - cp's OEIS frontend

A124319 Semiprime(3almostprime(n))-3almostprime(semiprime(n)). Commutator[A001358, A014612] at n.

2, 6, 7, 12, 16, 17, -11, 24, 23, 20, -1, 10, 48, 40, 39, 26, 14, 4, -1, 51, 60, 48, 48, 43, 31, 39, 22, 15, 37, 32, 39, 60, 90, 82, 68, 63, 64, 58, 66, 51, 53, 48, 28, 34, 42, 24, 28, 39, 87, 96, 106, 124, 124, 135, 131, 131, 88, 91, 72, 96, 103, 83, 83, 81, 91

Offset: 1

Jonathan Vos Post, Oct 26 2006

			a(1) = semiprime(3almostprime(1)) - 3almostprime(semiprime(1)) = 22 - 20 = 2.
a(2) = semiprime(3almostprime(2)) - 3almostprime(semiprime(2)) = 34 - 28 = 6.
a(3) = semiprime(3almostprime(3)) - 3almostprime(semiprime(3)) = 51 - 44 = 7.
a(4) = semiprime(3almostprime(4)) - 3almostprime(semiprime(4)) = 57 - 45 = 12.
a(7) = semiprime(3almostprime(7)) - 3almostprime(semiprime(7)) = 87 - 98 = -11, which is the first negative value in the commutators we have seen in these related set of sequences, exposing an incorrect assumption.

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, A124318.

p[k_] := p[k] = Select[Range[1000], PrimeOmega[#] == k &]; p[2][[ Take[p[3], 70]]] - p[3][[Take[p[2], 70]]] (* Giovanni Resta, Jun 13 2016 *)

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A124319(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)))
    def A001358(n):
        m, k = n, g(n)+n
        while m != k:
            m, k = k, g(k)+n
        return m
    m, k = n, f(n)+n
    while m != k:
        m, k = k, f(k)+n
    r, k = (p:=A001358(n)), f(p)+p
    while r != k:
        r, k = k, f(k)+p
    return A001358(m)-r # Chai Wah Wu, Aug 17 2024

a(18) corrected and a(22)-a(65) from Giovanni Resta, Jun 13 2016

A184728 a(n) = largest k such that A001358(n+1) = A001358(n) + (A001358(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 8, 6, 13, 9, 20, 19, 24, 19, 32, 33, 32, 37, 32, 43, 47, 47, 53, 56, 54, 59, 61, 64, 71, 72, 79, 84, 85, 83, 89, 92, 93, 84, 101, 107, 112, 117, 117, 120, 121, 117, 125, 132, 127, 140, 141, 141, 144, 137, 152, 157, 157

Offset: 1

Rémi Eismann, Jan 20 2011

a(n) = A001358(n) - A065516(n) if A001358(n) - A065516(n) > A065516(n), 0 otherwise.

A001358(n): semiprimes; A065516(n): first difference of semiprimes.

			For n = 1 we have A001358(n) = 4, A001358(n+1) = 6; there is no k such that 6 - 4 = 2 = (4 mod k), hence a(1) = 0.
For n = 3 we have A001358(n) = 9, A001358(n+1) = 10; 8 is the largest k such that 10 - 9 = 1 = (9 mod k), hence a(3) = 8; a(3) = A001358(3) - A065516(3) = 8.
For n = 20 we have A001358(n) = 57, A001358(n+1) = 58; 56 is the largest k such that 58 - 57 = 1 = (57 mod k), hence a(20) = 56; a(20) = A001358(20) - A065516(20) = 56.

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

Cf. A001358, A065516, A130533, A184729, A117078, A117563, A001223, A118534.

A184729 a(n) = A184728(n)/A130533(n) unless A130533(n) = 0 in which case a(n) = 0.

Original entry on oeis.org

0, 0, 4, 1, 1, 1, 10, 1, 12, 1, 16, 11, 8, 1, 4, 1, 1, 1, 1, 28, 9, 1, 1, 8, 1, 12, 1, 42, 17, 1, 1, 46, 31, 7, 1, 1, 28, 39, 39, 60, 11, 13, 25, 66, 1, 70, 47, 47, 72, 1, 38, 1, 1, 26, 1, 7, 88, 1, 1, 61, 20, 17, 100, 67, 67, 102, 29, 41, 106

Offset: 1

Rémi Eismann, Jan 20 2011

nonn

a(n) is the "level" of semiprimes.
The decomposition of semiprimes into weight * level + gap is A001358(n) = A130533(n) * a(n) + A065516(n) if a(n) > 0.
A184728(n) = A001358(n) - A065516(n) if A001358(n) - A065516(n) > A065516(n), 0 otherwise.

			For n = 1 we have A130533(1) = 0, hence a(1) = 0.
For n = 3 we have A184728(3)/A130533(3)= 8 / 2 = 4; hence a(3) = 4.
For n = 20 we have A184728(20)/A130533(20)= 56 / 2 = 28; hence a(20) = 28.

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

Cf. A001358, A065516, A130533, A184728, A117078, A117563, A001223, A118534.

A217357 Semiprimes p such that next semiprime after p is p+30.

Original entry on oeis.org

32777, 88649, 91799, 113107, 165697, 273257, 310103, 322211, 326137, 460963, 466063, 468877, 480443, 483223, 506509, 509131, 553349, 564347, 565493, 587611, 616771, 623257, 624959, 625619, 739177, 766799, 777163, 826657, 832357, 834123, 845177, 860873, 916163

Offset: 1

Zak Seidov, Oct 01 2012

nonn

Smallest difference between two consecutive terms occurs first at a(329) = 5861197 because a(330) = 5861227 and 5861227 - 5861197 = 30. Same difference for a(1212) = 16179703, a(1630) = 20611897 and a(1641) = 20703923.- Zak Seidov, Feb 14 2017

			32777 =A001358(8112)  = 73*449, 32807 = A001358(8113) = 3*619,
88649 =A001358(20880)  = 11*8059, 88679 = A001358(20881) = 71*1249.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A001358, A065516, A217030, A217335.

IsSemiprime:=func; [n: n in [4..1000000] | IsSemiprime(n) and IsSemiprime(n+30) and forall{n+i: i in [1..29] | not IsSemiprime(n+i)}]; // Bruno Berselli, Oct 01 2012

Select[Partition[Select[Range[10^6],PrimeOmega[#]==2&],2,1],#[[2]]-#[[1]] == 30&][[All,1]] (* Harvey P. Dale, May 06 2022 *)

A264045 Numbers n such that n and n+5 are consecutive semiprimes.

Original entry on oeis.org

69, 77, 106, 161, 178, 221, 254, 309, 314, 329, 341, 386, 398, 417, 422, 473, 554, 662, 674, 689, 758, 794, 934, 974, 998, 1094, 1121, 1149, 1169, 1214, 1294, 1502, 1517, 1522, 1541, 1569, 1673

Offset: 1

Zak Seidov, Nov 02 2015

nonn

Note that a(1)=69=A131109(k=5).

			a(1)=69=A001358(24) and a(1)+k=74=A001358(25).

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

Cf. A001358, A065516, A123375, A131109, A136196, A264043, A264044.

Flatten[Position[Partition[Table[If[PrimeOmega[n]==2,1,0],{n,2000}],6,1], ?(#=={1,0,0,0,0,1}&)]] (* _Harvey P. Dale, Dec 16 2015 *)

is(n)=if(n%4==1, isprime((n+5)/2) && bigomega(n)==2, n%4==2 && isprime(n/2) && bigomega(n+5)==2) && bigomega(n+1)!=2 && bigomega(n+2)!=2 && bigomega(n+3)!=2 && bigomega(n+4)!=2 \\ Charles R Greathouse IV, Nov 02 2015

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

A108197 Number of composite numbers between two successive semiprimes.

Original entry on oeis.org

0, 1, 0, 1, 0, 3, 0, 1, 0, 4, 0, 0, 1, 0, 4, 1, 1, 2, 1, 0, 1, 2, 2, 2, 2, 3, 1, 0, 0, 2, 1, 0, 0, 7, 2, 2, 2, 0, 1, 0, 0, 4, 2, 0, 4, 0, 0, 1, 0, 6, 1, 0, 1, 3, 1, 6, 0, 2, 1, 1, 4, 4, 0, 0, 1, 0, 2, 2, 0, 0, 1, 0, 0, 1, 3, 5, 1, 7, 1, 2, 0, 3, 2, 1, 1, 4, 2, 6, 1, 1, 2, 2, 0, 1, 0, 0, 1, 2, 2, 3, 1, 1, 2, 0, 1

Offset: 1

Giovanni Teofilatto, Jun 15 2005

This is to A046933 as semiprimes A001358 are to primes A000040. This is to composites A002808 as A088700 is to primes. a(A070552(i)) = 0. - Jonathan Vos Post, Oct 10 2007

a(n) = 0 if A001358(n) is in A070552. - Jonathan Vos Post, Mar 11 2007

			a(1) = 0 because between 2*2 and 2*3 there is 5 and it is not composite.
a(2) = 1 because between 2*3 and 3*3 there is 8 = 2*2*2;
a(6) = 3 because between 3*5 and 3*7 there are three composite numbers: {16, 18, 20}.
a(10) = 4 because between 2*13 and 3*11 there are four composite numbers: {27, 28, 30, 32}.
a(15) = 4 because there are four composites {40,42,44,45} between semiprime(15)=39 and semiprime(16)=46.

Semiprime analog of A046933.

Cf. A001358, A002808, A046933, A065855, A070552, A088700.

with(numtheory): sp:=proc(n) if bigomega(n)=2 then n else fi end: SP:=[seq(sp(n),n=1..450)]: for j from 1 to nops(SP)-1 do ct:=0: for i from SP[j]+1 to SP[j+1]-1 do if isprime(i)=false then ct:=ct+1 else ct:=ct fi: od: a[j]:=ct: od:seq(a[j],j=1..nops(SP)-1); # Emeric Deutsch, Mar 30 2007
A001358 := proc(nmin) local a,n ; a :=[] ; n := 1 ; while nops(a) < nmin do if numtheory[bigomega](n) = 2 then a := [op(a),n] ; fi ; n := n+1 ; od: RETURN(a) ; end: A000720 := proc(n) numtheory[pi](n) ; end: A065855 := proc(n) n-A000720(n)-1 ; end: A108197 := proc(nmin) local a,n,a001358 ; a001358 := A001358(nmin+1) ; a := [] ; for n from 1 to nops(a001358)-1 do a := [op(a), A065855(op(n+1,a001358))-A065855(op(n,a001358))-1 ] ; od; RETURN(a) ; end: A108197(100) ; # R. J. Mathar, Oct 23 2007

terms = 105;
cc = Select[Range[4 terms], CompositeQ] /. c_ /; PrimeOmega[c] == 2 -> 0;
SequenceReplace[cc, {0, c___ /; FreeQ[{c}, 0]} :> Length[{c}]][[;; terms]] (* Jean-François Alcover, Mar 31 2020 *)

a(n) = A065855(A001358(n+1)) - A065855(A001358(n)) - 1. - R. J. Mathar, Oct 23 2007

a(n)=A065516(n)-1-A088700(n). - R. J. Mathar, Jul 31 2008

Corrected and extended by Ray Chandler, Jul 07 2005
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jul 13 2007
Further edited by N. J. A. Sloane at the suggestion of R. J. Mathar, Jul 01 2008

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).

cp's OEIS Frontend

A124319 Semiprime(3almostprime(n))-3almostprime(semiprime(n)). Commutator[A001358, A014612] at n.

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A184728 a(n) = largest k such that A001358(n+1) = A001358(n) + (A001358(n) mod k), or 0 if no such k exists.

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A184729 a(n) = A184728(n)/A130533(n) unless A130533(n) = 0 in which case a(n) = 0.

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A217357 Semiprimes p such that next semiprime after p is p+30.

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A264045 Numbers n such that n and n+5 are consecutive semiprimes.

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A108197 Number of composite numbers between two successive semiprimes.

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Maple

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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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