A056809 -id:A056809 - cp's OEIS frontend

A202319 Lesser of two semiprimes sandwiched each between semiprimes thus forming a twin semiprime-triple.

214, 143098, 194758, 206134, 273418, 684898, 807658, 1373938, 1391758, 1516534, 1591594, 1610998, 1774798, 1882978, 1891762, 2046454, 2051494, 2163418, 2163958, 2338054, 2359978, 2522518, 2913838, 3108202, 4221754, 4297318, 4334938, 4866118, 4988878, 5108794

Offset: 1

V.J. Pohjola, Dec 16 2011

nonn

Conjecture: For every natural number k there are infinitely many semiprime pairs sp and sp' both sandwiched between semiprimes such that sp' - sp = 4k.
Note: For the case k=1 the pair of two consecutive semiprime triples will be called twin semiprime-triple with an analogy to twin prime.
The number centered between the triples of a twin semiprime-triple must be divisible by 36. Let m be the middle of such a twin semiprime-triple. It is trivial that m is divisible by 4, and that it is congruent to 0, 4, or 5 (mod 9). If it were congruent to 4, then m-1 and m+2 would both be divisible by 3, hence equal to 3 times a prime. But then those two primes would differ by 1, impossible except for primes 2, 3, which can be checked separately. A similar argument eliminates the case m == 5 (mod 9), so m must be divisible by 9. Conjecture by the author, proved by Franklin T. Adams-Watters, Dec 18 2011.
Members of this sequence must be twice the lesser of a twin prime. - Franklin T. Adams-Watters, Dec 18 2011
A number is in the sequence if and only if it has the form 6k-2, with 2k+-1 being twin primes, 3k+-1 twin primes, and 6k+-1 semiprimes. - Peter Munn, Oct 28 2017
By arguments similar to the above proof that m = a(n)+2 is divisible by 36, it can be shown that (a(n)+2)/36 == {-1, 0, 1} (mod 5) == {-1, 0, 1} (mod 7) and that a(n) == {214, 502, 538, 718, 754, 1042, 1258} (mod 1260). - Jon E. Schoenfield, Feb 26 2022

			The first twin semiprime-triple is {{213,214,215},{217,218,219}} whereby a(1)=214.
The fifteenth semiprime-triple is {{1891761,1891762,1891763},{1891765,1891766,1891767}} whereby a(15)=1891762.
The separating numbers 216 and 1891764 are divisible by 36.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A086005, A056809, A001359 Analogs: a(n) and A001359, a(n)+2 and A014574, a(n)+4 and A006512.

(* Run first to define sp *) nn = 10^8; p = Prime[Range[PrimePi[nn/2]]]; lim = Floor[Sqrt[nn]]; sp = {}; k = 0; While[k++; p[[k]] <= lim, sp = Join[sp, p[[k]]*Take[p, {k, PrimePi[nn/p[[k]]]}]]]; sp = Sort[sp]; lsp = Length[sp]; b = {}; Do[If[sp[[n]] == sp[[n - 1]] + 1 && sp[[n]] == sp[[n + 1]] - 1, AppendTo[b, sp[[n - 1]]]], {n, 2, lsp-1}]; lb = Length[b]; triples = {}; Do[If[b[[i]] == b[[i - 1]] + 4, AppendTo[triples, b[[i - 1]]]], {i, 2, lb}]; triples+1 (* V.J.Pohjola, Dec 18 2011 *)
SequencePosition[PrimeOmega[Range[5200000]],{2,2,2,,2,2,2}][[All,1]]+1 (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jul 06 2017 *)

A217222 Initial terms of sets of 8 consecutive semiprimes with gap 2.

Original entry on oeis.org

8129, 237449, 401429, 452639, 604487, 858179, 1471727, 1999937, 2376893, 2714987, 3111977, 3302039, 3869237, 4622087, 7813559, 9795449, 10587899, 10630739, 11389349, 14186387, 14924153, 15142547, 15757337, 18017687, 18271829, 19732979, 22715057, 25402907

Offset: 1

Zak Seidov, Sep 28 2012

nonn

All terms == 11 (mod 18).
Also all terms of sets of 8 consecutive semiprimes are odd, e.g., {8129, 8131, 8133, 8135, 8137, 8139, 8141, 8143} is the smallest set of 8 consecutive semiprimes.
Note that in all cases "9th term" (in this case 8143+2=8145) is divisible by 9 and hence is not semiprime.
Also note that all seven "intermediate" even integers (in this case {8130, 8132, 8134, 8136, 8138, 8140, 8142}) have at least three prime factors counting with multiplicity. Up to n = 40*10^9 there are 5570 terms of this sequence.

Zak Seidov, Table of n, a(n) for n = 1..5570

Subsequence of A082919.

Cf. A056809, A070552, A092125-A092129, A092207-A092209.

Transpose[Select[Partition[Select[Range[26*10^6],PrimeOmega[#] == 2&],8,1], Union[ Differences[#]]=={2}&]][[1]] (* Harvey P. Dale, Sep 02 2015 *)

A113784 Difference between semiprime(n) and semiprime(n+2).

Original entry on oeis.org

5, 4, 5, 5, 7, 7, 4, 4, 8, 8, 2, 4, 4, 8, 10, 5, 6, 6, 3, 5, 7, 7, 9, 8, 8, 8, 4, 2, 5, 6, 3, 2, 12, 16, 9, 7, 4, 3, 3, 2, 7, 10, 5, 8, 8, 2, 3, 3, 10, 12, 4, 3, 7, 8, 11, 9, 6, 7, 4, 9, 14, 8, 2, 3, 3, 4, 7, 5, 2, 3, 3, 2, 3, 7, 14, 11, 12, 12, 6, 5, 6, 8, 6, 5, 9, 11, 13, 11, 4, 6, 7, 4, 3, 3, 2, 3, 6, 9

Offset: 1

Jonathan Vos Post, Jan 20 2006

Semiprime analog of A031131 "Difference between n-th prime and (n+2)nd prime."

			a(1) = 5 because 3rd semiprime - first semiprime = 9 - 4 = 5.
a(2) = 4 because semiprime(4) - semiprime(2) = 10 - 6 = 4.
a(3) = 5 because semiprime(5) - semiprime(3) = 14 - 9 = 5.
a(4) = 5 because semiprime(6) - semiprime(4) = 15 - 10 = 5.

Cf. A001358, A056809.

t = Select[ Range@320, Plus @@ Last /@ FactorInteger@# == 2 &]; Drop[t, 2] - Drop[t, -2] (* Robert G. Wilson v *)

a(n) = A001358(n+2) - A001358(n).

More terms from Robert G. Wilson v, Jan 21 2006

A115394 List of triples of semiprimes: each three numbers are consecutive semiprimes.

Original entry on oeis.org

33, 34, 35, 85, 86, 87, 93, 94, 95, 121, 122, 123, 141, 142, 143, 201, 202, 203, 213, 214, 215, 217, 218, 219, 301, 302, 303, 393, 394, 395, 445, 446, 447, 633, 634, 635, 697, 698, 699, 841, 842, 843, 921, 922, 923, 1041, 1042, 1043, 1137, 1138, 1139, 1261

Offset: 1

Zak Seidov, Mar 08 2006

nonn

Cf. A056809, A086005, A086006.

Flatten[Select[Partition[Union[Times@@@Tuples[Prime[Range[150]],{2}]],3,1],#[[3]]-#[[2]]==#[[2]]-#[[1]]==1&]]  (* Harvey P. Dale, Jan 22 2011 *)

a(3n-2) = A056809(n), a(3n-1) = A086005(n), a(3n) = A086005(n) + 1 = 2*A086006(n) + 1, n=1,2,...

A115402 Difference between 3-almostprime(n) and 3-almostprime(n+3).

Original entry on oeis.org

12, 15, 10, 10, 15, 16, 15, 8, 8, 18, 16, 16, 7, 9, 8, 8, 17, 22, 21, 10, 7, 11, 12, 11, 7, 10, 9, 13, 14, 22, 18, 15, 7, 16, 12, 16, 7, 7, 4, 4, 10, 12, 13, 8, 9, 19, 22, 27, 23, 19, 14, 8, 11, 8

Offset: 1

Jonathan Vos Post, Jan 22 2006

			a(1) = A014612(1+3) - A014612(1) = 20 - 8 = 12.
a(2) = A014612(2+3) - A014612(2) = 27 - 12 = 15.
a(3) = A014612(3+3) - A014612(3) = 28 - 18 = 10.
a(39) = A014612(39+3) - A014612(39) = 174 - 170 = 4.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A014612, A056809, A067813, A113789, A115403.

Last[#]-First[#]&/@Partition[Select[Range[300],PrimeOmega[#]==3&],4,1] (* Harvey P. Dale, Nov 09 2012 *)

a(n) = A014612(n+3) - A014612(n).

A178034 a(n) = binomial(n*Omega(n),Omega(n)) / n.

Original entry on oeis.org

1, 1, 1, 7, 1, 11, 1, 253, 17, 19, 1, 595, 1, 27, 29, 39711, 1, 1378, 1, 1711, 41, 43, 1, 138415, 49, 51, 3160, 3403, 1, 3916, 1, 25637001, 65, 67, 69, 477191, 1, 75, 77, 657359, 1, 7750, 1, 8515, 8911, 91, 1, 132563501, 97, 11026, 101, 11935, 1, 1633355

Offset: 1

Michel Lagneau, May 17 2010

nonn

Omega(.) = A001222(.) is the number of prime divisors of n (counted with multiplicity).
binomial(nk,k)= n*binomial(nk-1,k-1) ensures that all entries are integers.
Subcases for this sequence:
If n is prime, Omega(n) = 1, and a(n) = binomial (n,1) / n = 1.
If n and n+1 are products of two primes (A070552), then Omega(n) = Omega(n+1) = 2, and binomial(n*Omega(n), Omega(n)) / n = binomial(2*n, 2) / n = 2*n-1 and binomial(2*(n+1), 2) / (n+1) = 2*n+1, and we obtain two consecutive numbers of the form (x, x+2), for example (17,19), (27,29), (41,43),... at n =9, 14...
Chaining this property: If n, n+1, and n+2 are semiprimes (A056809) , we find three consecutive numbers of the form (x, x+2,x+4), for example (65, 67, 69), (169, 171, 173), at n=33, 85.
At places where Omega(n)=3, we find the subsequence A060544, for example a(8) = A060544(8).
At places where Omega(n)=4, we find the subsequence A015219.

			a(8) = binomial(8*Omega(8),Omega(8))/8 = binomial(8*3,3)/8 = 2024/8 = 253.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A038456

A178034 := proc(n)
        local o ;
        o := numtheory[bigomega](n) ;
        binomial(n*o,o)/n ;
end proc: # R. J. Mathar, Jul 08 2012

bon[n_]:=Module[{o=PrimeOmega[n]},Binomial[n*o,o]/n]; Array[bon,60] (* Harvey P. Dale, Jul 22 2014 *)

a(n)=my(b=bigomega(n));binomial(n*b,b)/n \\ Charles R Greathouse IV, Oct 25 2012

A268588 Numbers n such that n, n + 1, n + 2, n + 3 and n + 4 are products of exactly three primes.

Original entry on oeis.org

602, 2522, 2523, 4202, 4921, 4922, 5034, 5282, 7730, 18241, 18242, 18571, 19129, 21931, 23161, 23305, 25203, 25553, 25554, 27290, 27291, 29233, 30354, 30793, 32035, 33843, 34561, 35714, 36001, 36835, 40313, 40314, 40394, 45265, 55361, 67609, 69667, 70202, 72721

Offset: 1

K. D. Bajpai, Feb 07 2016

nonn

Subsequence of A045941. - Zak Seidov, Jan 29 2017

			a(1) = 602: 602 = 2 * 7 * 43; 603 = 3 * 3 * 67; 604 = 2 * 2 * 151; 605 = 5 * 11 * 11; 606 = 2 * 3 * 101 are all products of three primes.
a(4) = 4202 : 4202 = 2 * 11 * 191; 4203 = 3 * 3 * 467; 4204 = 2 * 2 * 1051; 4205 = 5 * 29 * 29; 4206 = 2 * 3 * 701 are all products of three primes.

Chai Wah Wu, Table of n, a(n) for n = 1..11023

Cf. A056809, A113789, A124057, A045941.

IsP3:=func< n | &+[k[2]: k in Factorization(n)] eq 3 >; [ n: n in [2..50000] | IsP3(n) and IsP3(n+1) and IsP3(n+2) and IsP3(n+3) and IsP3(n+4)];

with(numtheory): A268588:= proc() if bigomega(n)=3 and bigomega(n+1)=3 and bigomega(n+2)=3 and bigomega(n+3)=3 and bigomega(n+4)=3 then RETURN (n); fi; end: seq(A268588(), n=1..100000);

Select[Range[100000], PrimeOmega[#] == 3 && PrimeOmega[# + 1] == 3 && PrimeOmega[# + 2] == 3 && PrimeOmega[# + 3] == 3 && PrimeOmega[# + 4] == 3 &]
SequencePosition[PrimeOmega[Range[73000]],{3,3,3,3,3}][[All,1]] (* Harvey P. Dale, Sep 03 2021 *)

for(n = 1,50000, bigomega(n)==3 & bigomega(n+1)==3 & bigomega(n+2)==3 & bigomega(n+3)==3 & bigomega(n+4)==3 & print1(n,","))

Comment removed by Zak Seidov, Jan 29 2017

A283527 First of three consecutive Sophie Germain semiprimes: n, n+1 and n+2 are all terms of A111153.

Original entry on oeis.org

15117, 17245, 34413, 93453, 143101, 157713, 190621, 208293, 233097, 294301, 323281, 346497, 470341, 501477, 1306113, 1337221, 1346401, 1655853, 1682313, 1774801, 1877613, 1879021, 1933233, 1976041

Offset: 1

Zak Seidov, Mar 09 2017

nonn

All terms are 1 mod 4, see A056809.

Zak Seidov, Table of n, a(n) for n = 1..408

Subsequence of A056809 and of A111153. Cf. A001358.

  po[x_] := PrimeOmega[x];   Select[Range[15117, 200000, 2],
2 == po[#] == po[2*# + 1] ==po[# + 1] == po[2*# + 3] == po[# + 2] ==
po[2*# + 5] &]

{bo(x)=bigomega(x)
forstep(n=15117,2000000,2, if(
2 == bo(n) && 2 == bo(n+1) && 2 == bo(n+2) && 2 == bo(2*n+1) &&
2 == bo(2*n+3) && 2 == bo(2*n+5), print1(n",")))}

list(lim)=lim\=1; my(v=List(),x=2*lim+5,u=vectorsmall(x)); forprime(p=2,x\2, forprime(q=2,min(lim\p,p), u[p*q]=1)); forstep(n=15117,lim,4, if(u[n] && u[n+1] && u[n+2] && u[2*n+1] && u[2*n+3] && u[2*n+5], listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Mar 10 2017

A368670 Numbers k such that k, k + 1, k + 2, and k + 4 are all semiprimes.

Original entry on oeis.org

141, 201, 213, 217, 301, 1137, 1345, 1401, 1761, 1837, 1893, 1941, 1981, 2101, 3097, 3865, 3957, 4413, 4533, 4881, 5997, 6157, 6241, 7113, 7141, 7165, 7401, 7977, 8185, 8257, 8913, 9753, 9985, 10117, 11013, 11181, 11377, 11757, 12057, 13953, 14037, 14253, 14917, 14977, 14997, 16177, 16293, 16437, 16593

Offset: 1

Robert Israel, Jan 02 2024

nonn

k, k + 1, k + 2 and k + 3 can't all be semiprimes, as one of them is divisible by 4.

All terms == 1 (mod 4).

			a(3) = 213 is a term because 213 = 3 * 71, 214 = 2 * 107, 215 = 5 * 43 and 217 = 7 * 31 are all semiprimes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A056809, A070552.

select(t -> numtheory:-bigomega(t)=2 and numtheory:-bigomega(t+1)=2 and numtheory:-bigomega(t+2)=2 and numtheory:-bigomega(t+4)=2, 4 * [$1..10000] +~ 1);

Select[Range[17000], PrimeOmega[#] == PrimeOmega[#+1] == PrimeOmega[#+2] == PrimeOmega[#+4] == 2 &] (* Stefano Spezia, Jan 02 2024 *)

A086007 Number of semiprime triples <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Reinhard Zumkeller, Jul 07 2003

nonn

A semiprime triple is <= n if its least member is <= n;

if a(n-1)A086005(a(n))=n+1.

Cf. A001358, A056809, A086005.

cp's OEIS Frontend

A202319 Lesser of two semiprimes sandwiched each between semiprimes thus forming a twin semiprime-triple.

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A217222 Initial terms of sets of 8 consecutive semiprimes with gap 2.

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A113784 Difference between semiprime(n) and semiprime(n+2).

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A115394 List of triples of semiprimes: each three numbers are consecutive semiprimes.

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A115402 Difference between 3-almostprime(n) and 3-almostprime(n+3).

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A178034 a(n) = binomial(n*Omega(n),Omega(n)) / n.

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A268588 Numbers n such that n, n + 1, n + 2, n + 3 and n + 4 are products of exactly three primes.

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A283527 First of three consecutive Sophie Germain semiprimes: n, n+1 and n+2 are all terms of A111153.

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