A092207 -id:A092207 - cp's OEIS frontend

A056809 Numbers k such that k, k+1 and k+2 are products of two primes.

33, 85, 93, 121, 141, 201, 213, 217, 301, 393, 445, 633, 697, 841, 921, 1041, 1137, 1261, 1345, 1401, 1641, 1761, 1837, 1893, 1941, 1981, 2101, 2181, 2217, 2305, 2361, 2433, 2461, 2517, 2641, 2721, 2733, 3097, 3385, 3601, 3693, 3865, 3901, 3957, 4285

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 04 2002

nonn

Each term is the beginning of a run of three 2-almost primes (semiprimes). No runs exist of length greater than three. For the same reason, each term must be odd: If k were even, then so would be k+2. In fact, one of k or k+2 would be divisible by 4, so must indeed be 4 to have only two prime factors. However, neither 2,3,4 nor 4,5,6 is such a run. - Rick L. Shepherd, May 27 2002
k+1, which is twice a prime, is in A086005. The primes are in A086006. - T. D. Noe, May 31 2006
The squarefree terms are listed in A039833. - Jianing Song, Nov 30 2021

			121 is in the sequence because 121 = 11^2, 122 = 2*61 and 123 = 3*41, each of which is the product of two primes.

D. W. Wilson, Table of n, a(n) for n = 1..10000

Intersection of A070552 and A092207.

Cf. A045939, A039833, A086005, A086006, A123255.

f[n_] := Plus @@ Transpose[ FactorInteger[n]] [[2]]; Select[Range[10^4], f[ # ] == f[ # + 1] == f[ # + 2] == 2 & ]
Flatten[Position[Partition[PrimeOmega[Range[5000]],3,1],{2,2,2}]] (* Harvey P. Dale, Feb 15 2015 *)
SequencePosition[PrimeOmega[Range[5000]],{2,2,2}][[;;,1]] (* Harvey P. Dale, Mar 03 2024 *)

forstep(n=1,5000,2, if(bigomega(n)==2 && bigomega(n+1)==2 && bigomega(n+2)==2, print1(n,",")))

is(n)=n%4==1 && isprime((n+1)/2) && bigomega(n)==2 && bigomega(n+2)==2 \\ Charles R Greathouse IV, Sep 08 2015

list(lim)=my(v=List(),t); forprime(p=2,(lim+1)\2, if(bigomega(t=2*p-1)==2 && bigomega(t+2)==2, listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Sep 08 2015

a(n) = A086005(n) - 1 = 2*A086006(n) - 1 = 4*A123255(n) + 1. - Jianing Song, Nov 30 2021

Edited and extended by Robert G. Wilson v, May 04 2002

A124936 Numbers k such that k - 1 and k + 1 are semiprimes.

Original entry on oeis.org

5, 34, 50, 56, 86, 92, 94, 120, 122, 142, 144, 160, 184, 186, 202, 204, 214, 216, 218, 220, 236, 248, 266, 288, 290, 300, 302, 304, 320, 322, 328, 340, 392, 394, 412, 414, 416, 446, 452, 470, 472, 516, 518, 528, 534, 536, 544, 552, 580, 582, 590, 634, 668

Offset: 1

Zak Seidov, Nov 13 2006

nonn

All but the first term are even.

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

Cf. A092207 (k and k+2 are semiprimes), A086005 (k-1, k, k+1 are semiprimes), A086006 (primes p such that 2*p-1 and 2*p+1 are semiprimes), A082130 (2*k-1 and 2*k+1 are semiprimes).

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [1..700] | IsSemiprime(n+1) and IsSemiprime(n-1)]; // Vincenzo Librandi, Mar 30 2015

lst={};Do[If[Plus@@Last/@FactorInteger[n-1]==2&&Plus@@Last/@FactorInteger[n+1]==2,AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 01 2009 *)
Select[Range[2, 700], PrimeOmega[# + 1] == PrimeOmega[# - 1] == 2 &] (* Vincenzo Librandi, Mar 30 2015 *)

list(lim)=if(lim<5,return([])); my(v=List([5]),x=1,y=1); forfactored(z=7,lim\1+1, if(vecsum(z[2][,2])==2 && vecsum(x[2][,2])==2, listput(v,z[1]-1)); x=y; y=z); Vec(v) \\ Charles R Greathouse IV, May 22 2018

from sympy import factorint
from itertools import count, islice
def agen(): # generator of terms
    yield 5
    nxt = 0
    for k in count(6, 2):
        prv, nxt = nxt, sum(factorint(k+1).values())
        if prv == nxt == 2: yield k
print(list(islice(agen(), 53))) # Michael S. Branicky, Nov 26 2022

a(n) = A092207(n) + 1; at n>=2, a(n) = 2*A082130(n-1).

A082919 Numbers k such that k, k+2, k+4, k+6, k+8, k+10, k+12 and k+14 are semiprimes.

Original entry on oeis.org

8129, 9983, 99443, 132077, 190937, 237449, 401429, 441677, 452639, 604487, 802199, 858179, 991289, 1471727, 1474607, 1963829, 1999937, 2376893, 2714987, 3111977, 3302039, 3869237, 4622087, 4738907, 6156137, 7813559, 8090759

Offset: 1

Hugo Pfoertner, Apr 22 2003

nonn

Start of a cluster of 8 consecutive odd semiprimes. Semiprimes in arithmetic progression. All terms are odd, see also A056809.
Note that there cannot exist 9 consecutive odd semiprimes. Out of any 9 consecutive odd numbers, one of them will be divisible by 9. The only multiple of 9 which is a semiprime is 9 itself and it is easy to see that's not part of a solution. - Jack Brennen, Jan 04 2006
For the first 500 terms, a(n) is roughly 40000*n^1.6, so the sequence appears to be infinite. Note that (a(n)+4)/3 and (a(n)+10)/3 are twin primes. - Don Reble, Jan 05 2006
All terms == 11 (mod 18). - Zak Seidov, Sep 27 2012
There is at least one even semiprime between k and k+14 for 1812 of the first 10000 terms. - Donovan Johnson, Oct 01 2012
All terms == {29,47,83} (mod 90). - Zak Seidov, Sep 13 2014
Among the first 10000 terms, from all 80000 numbers a(n)+m, m=0,2,4,6,8,10,12,14, the only square is a(4637) + 2 = 23538003241 = 153421^2 (153421 is prime, of course). - Zak Seidov, Dec 22 2014

			a(1)=8129 because 8129=11*739, 8131=47*173, 8133=3*2711, 8135=5*1627, 8137=79*103, 8139=3*2713, 8141=7*1163, 8143=17*479 are semiprimes.

Author of this sequence is Jack Brennen, who provided the terms up to 991289 in a posting to the seqfan mailing list on April 5, 2003.

Donovan Johnson, Table of n, a(n) for n = 1..10000 (terms a(1001) to a(2000) from Zak Seidov)
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A082130, A082131, A056809, A070552, A092207, A092125, A092126, A092127, A092128, A092129, A092209, A217222 (consecutive semiprimes).

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[3*10^6], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # + 2] == PrimeFactorExponentsAdded[ # + 4] == PrimeFactorExponentsAdded[ # + 6] == PrimeFactorExponentsAdded[ # + 8] == PrimeFactorExponentsAdded[ # + 10] == PrimeFactorExponentsAdded[ # + 12] == PrimeFactorExponentsAdded[ # + 14] == 2 &] (* Robert G. Wilson v and Zak Seidov, Feb 24 2004 *)

A092209 Smallest number k such that k, k+2, k+4, ..., k+2n are semiprimes.

Original entry on oeis.org

4, 4, 91, 213, 213, 1383, 3091, 8129

Offset: 0

Robert G. Wilson v and Zak Seidov, Feb 24 2004

Semiprimes in arithmetic progression. All terms are odd, except for the first two. See also A056809.

Eric Weisstein's World of Mathematics, Semiprime.

First entry in A001358, A092207, A092125, A092126, A092127, A092128, A092129, A082919 respectively.

A198327 Semiprimes k such that k-2 is also a semiprime.

Original entry on oeis.org

6, 35, 51, 57, 87, 93, 95, 121, 123, 143, 145, 161, 185, 187, 203, 205, 215, 217, 219, 221, 237, 249, 267, 289, 291, 301, 303, 305, 321, 323, 329, 341, 393, 395, 413, 415, 417, 447, 453, 471, 473, 517, 519, 529, 535, 537, 545, 553, 581, 583, 591, 635, 669, 671

Offset: 1

Michel Lagneau, Nov 25 2011

nonn

Omega(a(n)) = Omega(a(n) - Omega(a(n))) because Omega(a(n)) = 2, and a(n) - 2 is semiprime => this sequence is a subsequence of A200925.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Semiprime

Cf. A001222, A092207, A200925.

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[ 671], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # - 2] == 2 &]
SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; Select[Range[1000], SemiPrimeQ[#] && SemiPrimeQ[# - 2] &] (* T. D. Noe, Nov 27 2011 *)
#[[3,1]]&/@Select[Partition[Table[{n,PrimeOmega[n]},{n,700}],3,1], #[[1,2]]==#[[3,2]]==2&] (* Harvey P. Dale, Dec 10 2011 *)

a(n) = A092207(n) + 2.

A180150 Numbers n such that n and n+2 are both divisible by exactly 4 primes (counted with multiplicity).

Original entry on oeis.org

54, 88, 150, 196, 232, 248, 294, 306, 328, 340, 342, 348, 460, 488, 490, 568, 570, 664, 712, 738, 774, 850, 856, 858, 868, 870, 948, 1012, 1014, 1060, 1096, 1110, 1148, 1190, 1204, 1206, 1208, 1210, 1218, 1254, 1274, 1276, 1290, 1302, 1314, 1420, 1430, 1448

Offset: 1

Jonathan Vos Post, Aug 12 2010

"Quadruprimes" or "4-almost primes" that keep that property when incremented by 2. This sequence is to 4 as 3 is to A180117, as A092207 is to 2, and as A001359 is to 1. That is, this sequence is the 4th row of the infinite array A[k,n] = n-th natural number m such that m and m+2 are both divisible by exactly k primes (counted with multiplicity). The first row is the lesser of twin primes. The second row is the sequence such that m and m+2 are both semiprimes.

			a(1) = 54 because 54 = 2 * 3^3 is divisible by exactly 4 primes (counted with multiplicity), and so is 54+2 = 56 = 2^3 * 7.

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

Cf. A000040, A001222, A001358, A014614, A033987, A101637, A114106 (number of 4-almost primes <= 10^n).

SequencePosition[PrimeOmega[Range[1500]],{4,,4}][[;;,1]] (* _Harvey P. Dale, Jan 14 2024 *)

is(n)=bigomega(n)==4 && bigomega(n+2)==4 \\ Charles R Greathouse IV, Jan 31 2017

{m in A014613 and m+2 in A014613} = {m such that bigomega(m) = bigomega(m+2) = 4} = {m such that A001222(m) = A001222(m+2) = 4}.

More terms from R. J. Mathar, Aug 13 2010

A180151 Numbers k such that k and k + 2 are both divisible by exactly five primes (counted with multiplicity).

Original entry on oeis.org

270, 592, 700, 750, 918, 1168, 1240, 1638, 1648, 1672, 1710, 1750, 2070, 2310, 2392, 2548, 2550, 2608, 2728, 2860, 2862, 2896, 2898, 3184, 3330, 3568, 3630, 3822, 3848, 3850, 3942, 3976, 4230, 4264, 4648, 4662, 5070, 5080, 5236, 5238, 5390, 5550, 5560

Offset: 1

Jonathan Vos Post, Aug 12 2010

"5-almost primes" that keep that property when incremented by 2. This sequence is to 5 as 4 is to A180150, as 3 is to A180117, as A092207 is to 2, and as A001359 is to 1. That is, this sequence is the 5th row of the infinite array A[k,n] = n-th natural number m such that m and m+2 are both divisible by exactly k primes (counted with multiplicity). The first row is the lesser of twin primes. The second row is the sequence such that m and m+2 are both semiprimes.

			a(1) = 270 because 270 = 2 * 3^3 * 5 is divisible by exactly 5 primes (counted with multiplicity), and so is 270+2 = 272 = 2^4 * 17.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A001358, A001359, A014614, A033987, A101637, A180117, A180150.

f[n_] := Plus @@ (Last@# & /@ FactorInteger@n); fQ[n_] := f[n] == 5 == f[n + 2]; Select[ Range@ 10000, fQ] (* Robert G. Wilson v, Aug 15 2010 *)

for(x=2,10^4,if(bigomega(x)==5&&bigomega(x+2)==5,print1(x", "))) \\ Zak Seidov, Aug 12 2010

{m in A014614 and m+2 in A014614} = {m such that bigomega(m) = bigomega(m+2) = 5} = {m such that A001222(m) = A001222(m+2) = 5}.

Corrected and extended by Zak Seidov and R. J. Mathar, Aug 12 2010

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 *)

A241764 Semiprimes sp such that sp-3 is also semiprime.

Original entry on oeis.org

9, 25, 38, 49, 58, 65, 77, 85, 94, 118, 121, 122, 145, 146, 158, 161, 169, 205, 206, 209, 217, 218, 221, 262, 265, 298, 301, 302, 305, 326, 329, 358, 361, 365, 394, 398, 454, 469, 481, 485, 505, 514, 517, 529, 538, 545, 554, 562, 565, 586, 589, 614

Offset: 1

K. D. Bajpai, Apr 29 2014

nonn

Also semiprimes of the form 2^x - x.

The primes of the form 2^x - x are in A081296.

			a(3)= 38 = 2*19, which is semiprime: 38-3 = 35 = 5*7 is also semiprime.
a(5)= 58 = 2*29, which is semiprime: 58-3 = 55 = 5*11 is also semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..1100

Cf. A001358, A092207, A123017, A198327.

with(numtheory): A241764:= proc(); if bigomega(x)=2 and bigomega(x-3)=2 then RETURN (x); fi; end: seq(A241764 (), x=1..2000);

Flatten[Position[Partition[Table[If[PrimeOmega[n]==2,1,0],{n,700}],4,1], ?(First[ #] ==Last[#]==1&),{1},Heads->False]]+3 (* _Harvey P. Dale, Dec 21 2015 *)

for(k=1, 500, if(bigomega(k)==2 && bigomega(k-3)==2, print1(k, ", "))) \\ Colin Barker, May 07 2014

A241817 Semiprimes sp such that sp-3 is prime.

Original entry on oeis.org

6, 10, 14, 22, 26, 34, 46, 62, 74, 82, 86, 106, 134, 142, 166, 194, 202, 214, 226, 254, 274, 314, 334, 362, 382, 386, 422, 446, 466, 482, 502, 526, 566, 622, 634, 662, 694, 746, 842, 862, 866, 886, 914, 922, 974, 1042, 1094, 1126, 1154, 1174, 1226, 1234, 1262

Offset: 1

K. D. Bajpai, Apr 29 2014

nonn

Even numbers of the form 2p, p prime, that can be expressed as the sum of two primes in at least two ways as 2p = p + p = 3 + (2p-3). For example, 34 is in the sequence because 34 = 2*17 = 17 + 17 = 3 + 31. These are the only numbers that have Goldbach partitions with both a minimum and a maximum possible difference between their prime parts, i.e., |p-p| = 0 and |(2p-3)-3| = 2p-6 respectively. - Wesley Ivan Hurt, Apr 08 2018

			a(2) = 10 = 2*5, which is semiprime and 10-3 = 7 is a prime.
a(6) = 34 = 2*17, which is semiprime and 34-3 = 31 is a prime.

K. D. Bajpai, Table of n, a(n) for n = 1..2500

Cf. A001358, A063908, A092207, A123017, A198327.

with(numtheory): A241817:= proc(); if bigomega(x)=2 and isprime(x-3) then  RETURN (x); fi; end: seq(A241817 (), x=1..3000);

2 Select [Prime[Range[5!]], PrimeQ[2 # - 3] &] (* Vincenzo Librandi, Apr 10 2018 *)
Select[Range[1500],PrimeOmega[#]==2&&PrimeQ[#-3]&] (* Harvey P. Dale, Oct 14 2018 *)

a(n) = 2 * A063908(n). - Wesley Ivan Hurt, Apr 08 2018

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A056809 Numbers k such that k, k+1 and k+2 are products of two primes.

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A124936 Numbers k such that k - 1 and k + 1 are semiprimes.

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A082919 Numbers k such that k, k+2, k+4, k+6, k+8, k+10, k+12 and k+14 are semiprimes.

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A092209 Smallest number k such that k, k+2, k+4, ..., k+2n are semiprimes.

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A198327 Semiprimes k such that k-2 is also a semiprime.

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A180150 Numbers n such that n and n+2 are both divisible by exactly 4 primes (counted with multiplicity).

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A180151 Numbers k such that k and k + 2 are both divisible by exactly five primes (counted with multiplicity).

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