A175648 -id:A175648 - cp's OEIS frontend

A175668 First differences of A175648.

4, 11, 1, 12, 1, 16, 7, 7, 17, 5, 4, 20, 4, 3, 1, 10, 12, 1, 13, 28, 18, 1, 3, 4, 4, 1, 1, 2, 32, 25, 13, 4, 4, 3, 1, 2, 4, 14, 4, 12, 23, 3, 16, 5, 9, 3, 9, 4, 4, 2, 34, 7, 15, 9, 3, 4, 4, 4, 4, 4, 10, 4, 14, 4, 5, 24, 17, 43, 7, 38, 14, 4, 9, 1, 4, 4, 10, 4, 28, 4, 14, 4, 14, 4, 4, 10, 4, 10

Offset: 1

Juri-Stepan Gerasimov, Aug 05 2010

nonn

Distance between twin semiprime pairs.

Cf. A053319, A175612, A175648.

A175648 := proc(n) option remember; if n = 1 then 6; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 2 and numtheory[bigomega](a+4) = 2 then return a; end if; end do: end if; end proc:
A175668 := proc(n) A175648(n+1)-A175648(n) ; end proc:
seq(A175668(n),n=1..100) ; # R. J. Mathar, Aug 07 2010

Terms from a(33) on corrected by R. J. Mathar, Aug 07 2010

A352137 a(n) is the first start of a sequence of exactly n members of A175648 under the map k -> 3*k+4.

Original entry on oeis.org

21, 6, 155, 1111, 77635, 25877, 16392913, 78494323

Offset: 1

J. M. Bergot and Robert Israel, Mar 05 2022

a(n), 3*a(n)+4, 3*(3*a(n)+4)+4, ..., 3^(n-1)*a(n)+2*3^(n-1)-2 are in A175648 but 3^n*a(n)+2*3^n-2 is not.

			a(3) = 155 because 155, 3*155+4 = 469 and 3*469+4 = 1411 are in A175648 but 3*1411+4 = 4237 is not (155 = 5*31, 159 = 3*53, 469 = 7*67, 473 - 11*43, 1411 = 17*83, and 1415 = 5*283 are semiprimes, but 4241 is prime).

Cf. A001358, A175648, A352132.

f:= proc(n) option remember;
   if numtheory:-bigomega(n)=2 and numtheory:-bigomega(n+4)=2 then 1 + procname(3*n+4) else 0 fi
end proc:
V:= Vector(7): count:= 0:
for nn from 1 while count < 7 do
   v:= f(nn);
   if v > 0 and V[v] = 0 then count:= count+1; V[v]:= nn; fi
od:
convert(V,list);

A242804 Integers k such that each of k, k+1, k+2, k+4, k+5, k+6 is the product of two distinct primes.

Original entry on oeis.org

213, 143097, 194757, 206133, 273417, 684897, 807657, 1373937, 1391757, 1516533, 1591593, 1610997, 1774797, 1882977, 1891761, 2046453, 2051493, 2163417, 2163957, 2338053, 2359977, 2522517, 2913837, 3108201, 4221753

Offset: 1

Daniel Constantin Mayer, May 23 2014

nonn

A remarkable gap occurs between the initial two members, and the sequence seems to be rather sparse compared to the related A242805.
Here, the first member k of the sextet is the reference, whereas in A068088 the center k+3 is selected as reference. Observe that k+3 must be divisible by the square 4.
All terms are congruent to 9 (mod 12). - Zak Seidov, Apr 14 2015
From Robert Israel, Apr 15 2015: (Start)
All terms are congruent to 33 (mod 36).
Numbers k in A039833 such that k+4 is in A039833. (End)
From Robert G. Wilson v, Apr 15 2015: (Start)
k is congruent to 33 (mod 36) so one of its factors is 3 and the other is == 11 (mod 12);
k+1 is congruent to 34 (mod 36) so one of its factors is 2 and the other is == 17 (mod 18);
k+2 is congruent to 35 (mod 36) so its factors are == +-1 (mod 6);
k+4 is congruent to 1 (mod 36) so its factors are == +-1 (mod 6);
k+5 is congruent to 2 (mod 36) so one of its factors is 2 and the other is == 1 (mod 18);
k+6 is congruent to 3 (mod 36) so one of its factors is 3 and the other is == 1 (mod 12). (End).
Number of terms < 10^m: 0, 0, 1, 1, 1, 7, 39, 169, 882, 4852, 27479, ...,. - Robert G. Wilson v, Apr 15 2015
Or, numbers k such that k, k+1 and k+2 are terms in A175648. - Zak Seidov, Dec 08 2015

			213=3*71, 214=2*107, 215=5*43, 217=7*31, 218=2*109, 219=3*73.

Zak Seidov and Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A242793 (minima for two, three and more prime divisors) and A068088 (arbitrary squarefree integers).

Cf. A242805, A039833, A175648, A202319.

f:= t -> numtheory:-issqrfree(t) and (numtheory:-bigomega(t) = 2):
select(t -> andmap(f, [t,t+1,t+2,t+4,t+5,t+6]), [seq(36*k+33,k=0..10^6)]); # Robert Israel, Apr 15 2015

fQ[n_] := PrimeQ[n/3] && PrimeQ[(n + 1)/2] && PrimeQ[(n + 5)/2] && PrimeQ[(n + 6)/3] && PrimeNu[{n + 2, n + 4}] == {2, 2} == PrimeOmega[{n + 2, n + 4}]; k = 33; lst = {}; While[k < 10^8, If[fQ@ k, AppendTo[lst, k]]; k += 36]; lst (* Robert G. Wilson v, Apr 14 2015 and revised Apr 15 2015 after Zak Seidov and Robert Israel *)

default(primelimit, 1000M); i=0; j=0; k=0; l=0; m=0; loc=0; lb=2; ub=9*10^9; o=2; for(n=lb, ub, if(issquarefree(n)&&(o==omega(n)), loc=loc+1; if(1==loc, i=n; ); if(2==loc, if(i+1==n, j=n; ); if(i+1

forstep(x=213,4221753,12, if( isprime(x/3) && isprime((x+1)/2) && 2==omega(x+2) && 2==bigomega(x+2) && 2==omega(x+4) && 2==bigomega(x+4) && isprime((x+5)/2) && isprime((x+6)/3), print1(x", "))) \\ Zak Seidov, Apr 14 2015

a(n) = A202319(n) - 1. - Jon Maiga, Jul 10 2021

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

A175663 Maximal run length of primes of the form n, n+2, n+23, n+23*5,..

Original entry on oeis.org

0, 1, 2, 0, 3, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 0, 5, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 9, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 6, 0, 1, 0, 0

Offset: 1

Vladislav-Stepan Malakovsky & Juri-Stepan Gerasimov, Aug 04 2010

nonn

			a(107)=8 because 107=prime, 107+2=109=prime, 107+2*3=113=prime, 107+2*3*5=137=prime, 107+2*3*5*7=317=prime, 107+2*3*5*7*11=2417=prime, 107+2*3*5*7*11*13=30137=prime, 107+2*3*5*7*11*13*17=510617=prime.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A006512 (greater of twin primes), A175612 (list of twin semiprimes), A175648 (lesser of twin semiprimes).

Cf. also A175682.

A002110 := proc(n) option remember; mul(ithprime(i),i=1..n) ; end proc:
A175663 := proc(n) if isprime(n) then for p from 1 do if not isprime(n+A002110(p)) then return p ; end if; end do: else return 0 ; end if; end proc:
seq(A175663(n),n=1..120) ; # R. J. Mathar, Aug 07 2010

Array[If[PrimeQ@ #, Block[{s = {1}}, While[PrimeQ[# + Times @@ Prime@ s], AppendTo[s, s[[-1]] + 1]]; Last@ s], 0] &, 105] (* Michael De Vlieger, Jan 03 2019 *)

A175663(n) = if(!isprime(n),0,my(pr=2); for(k=1, oo, if(!isprime(pr+n), return(k)); pr *= prime(1+k))); \\ Antti Karttunen, Jan 03 2019

a(n) <= A175682(n). - Antti Karttunen, Jan 03 2019

A175664 Greater of twin semiprimes.

Original entry on oeis.org

10, 14, 25, 26, 38, 39, 55, 62, 69, 86, 91, 95, 115, 119, 122, 123, 133, 145, 146, 159, 187, 205, 206, 209, 213, 217, 218, 219, 221, 253, 278, 291, 295, 299, 302, 303, 305, 309, 323, 327, 339, 362, 365, 381, 386, 395, 398, 407, 411, 415, 417, 451, 458, 473

Offset: 1

Juri-Stepan Gerasimov, Aug 04 2010

nonn

Semiprimes m such that m-4 is also semiprime.

			a(1)=10 because 10 (semiprime) - 4 = 6 (semiprime);
a(2)=14 because 14 (semiprime) - 4 = 10 (semiprime).

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

Cf. A006512 (greater of twin primes), A175612 (list of twin semiprimes), A175648 (lesser of twin semiprimes).

A175664 := proc(n) option remember; if n = 1 then 10; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 2 and numtheory[bigomega](a-4) = 2 then return a; end if; end do: end if; end proc: seq(A175664(n),n=1..100) ; # R. J. Mathar, Aug 07 2010

SequencePosition[Table[If[PrimeOmega[n]==2,1,0],{n,500}],{1,,,_,1}][[All,2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 16 2017 *)

a(n) = 4 + A175648(n). - R. J. Mathar, Aug 07 2010

Corrected (313 removed) by R. J. Mathar, Aug 07 2010

A255746 Squarefree semiprimes n such that n+4 is also a squarefree semiprime.

Original entry on oeis.org

6, 10, 22, 34, 35, 51, 58, 65, 82, 87, 91, 111, 115, 118, 119, 129, 141, 142, 155, 183, 201, 202, 205, 209, 213, 214, 215, 217, 249, 274, 287, 291, 295, 298, 299, 301, 305, 319, 323, 335, 358, 377, 382, 391, 394, 403, 407, 411, 413, 447, 454, 469, 478, 481

Offset: 1

Vladimir Shevelev, Jul 11 2015

nonn

Conjecturally, the sequence is infinite.

			65 = 5*13; 65 + 4 = 69 = 3*23. So 65 is in the sequence.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000005, A001358, A005117, A175648.

Select[Range@ 500, And[SquareFreeQ@ #, PrimeOmega@ # == 2, SquareFreeQ[# + 4], PrimeOmega[# + 4] == 2] &] (* Michael De Vlieger, Jul 12 2015 *)

main(size)={ v=vector(size); i=0; m=1; while(iAnders Hellström, Jul 11 2015 */

A000005(a(n)) = A000005(a(n)+4) = 4.

More terms from Peter J. C. Moses, Jul 11 2015

A352132 Numbers k such that k, k+4, 3k+4 and 3k+8 are all semiprimes.

Original entry on oeis.org

6, 10, 118, 119, 129, 155, 287, 295, 299, 319, 377, 413, 447, 469, 511, 538, 629, 681, 699, 717, 785, 831, 865, 913, 1003, 1073, 1077, 1111, 1115, 1137, 1141, 1145, 1267, 1343, 1345, 1379, 1393, 1437, 1469, 1509, 1687, 1817, 1835, 1919, 1923, 1981, 2167, 2173, 2177, 2195, 2245, 2429, 2479, 2569

Offset: 1

J. M. Bergot and Robert Israel, Mar 05 2022

nonn

Numbers k such that k and 3*k+4 are both in A175648.

Even terms are 2*k for k in A174920.

			a(4) = 119 is a term because 119 = 7*17, 119+4 = 123 = 3*41, 3*119+4 = 361 = 19^2 and 3*119+8 = 365 = 5*73 are semiprimes.

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

Cf. A001358, A174920, A175648.

filter:= proc(x)
numtheory:-bigomega(x) = 2 and numtheory:-bigomega(x+4) = 2 and numtheory:-bigomega(3*x+4) = 2 and numtheory:-bigomega(3*x+8)=2
end proc:
select(filter, [$1..3000]);

okQ[k_] := AllTrue[{k, k+4, 3k+4, 3k+8}, PrimeOmega[#] == 2&];
Select[Range[3000], okQ] (* Jean-François Alcover, May 16 2023 *)

A175710 Numbers k such that k-4, k and k+4 are all semiprimes.

Original entry on oeis.org

10, 91, 115, 119, 205, 209, 213, 217, 291, 295, 299, 305, 323, 407, 411, 485, 489, 493, 497, 501, 515, 533, 685, 699, 703, 717, 749, 767, 785, 789, 803, 917, 955, 989, 1007, 1077, 1115, 1137, 1141, 1145, 1195, 1199, 1203, 1207, 1257, 1267, 1333, 1343, 1347

Offset: 1

Juri-Stepan Gerasimov, Aug 12 2010

nonn

If k is a term, one of k-4, k and k+4 is 3 times a prime. - Robert Israel, Mar 25 2025

			a(1)=10 because 10-4=6, 10 and 10+4=14 are all semiprimes.

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

Cf. A001358, A175612, A175648, A175664.

SP:= select(t -> numtheory:-bigomega(t) = 2, {$1..10000}):
sort(convert(SP intersect (SP +~ 4) intersect (SP -~ 4),list)); # Robert Israel, Mar 25 2025

Corrected (299, 411 etc inserted) by R. J. Mathar, Aug 13 2010

Name edited by Robert Israel, Mar 25 2025

cp's OEIS Frontend

A175668 First differences of A175648.

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A352137 a(n) is the first start of a sequence of exactly n members of A175648 under the map k -> 3*k+4.

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A242804 Integers k such that each of k, k+1, k+2, k+4, k+5, k+6 is the product of two distinct primes.

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

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A175663 Maximal run length of primes of the form n, n+2, n+2*3, n+2*3*5,..

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A175664 Greater of twin semiprimes.

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A255746 Squarefree semiprimes n such that n+4 is also a squarefree semiprime.

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A175663 Maximal run length of primes of the form n, n+2, n+23, n+23*5,..

A352132 Numbers k such that k, k+4, 3k+4 and 3k+8 are all semiprimes.