A082919 -id:A082919 - cp's OEIS frontend

A092208 Duplicate of A082919.

8129, 9983, 99443, 132077, 190937, 237449, 401429, 441677, 452639, 604487

Offset: 1

dead

A092207 Semiprimes k such that k+2 is also a semiprime.

4, 33, 49, 55, 85, 91, 93, 119, 121, 141, 143, 159, 183, 185, 201, 203, 213, 215, 217, 219, 235, 247, 265, 287, 289, 299, 301, 303, 319, 321, 327, 339, 391, 393, 411, 413, 415, 445, 451, 469, 471, 515, 517, 527, 533, 535, 543, 551, 579, 581, 589, 633, 667

Offset: 1

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

nonn

Starting with 33 all terms are odd. First squares are 4, 49, 169, 361, 529, 961, 1369, 2209, 2809, 4489, ... - Zak Seidov, Feb 17 2017

Zak Seidov, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Semiprime

Cf. A056809, A070552, A092125, A092126, A092127, A092128, A092129, A082919, A092209.

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[ 668], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # + 2] == 2 &]
Select[Range[700],PrimeOmega[#]==PrimeOmega[#+2]==2&] (* Harvey P. Dale, Aug 20 2011 *)
SequencePosition[Table[If[PrimeOmega[n]==2,1,0],{n,700}],{1,,1}] [[All,1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, May 29 2017 *)

is(n)=if(n%2==0, return(n==4)); bigomega(n)==2 && bigomega(n+2)==2 \\ Charles R Greathouse IV, Feb 21 2017

from sympy import factorint
from itertools import count, islice
def agen(): # generator of terms
    yield 4
    nxt = 0
    for k in count(5, 2):
        prv, nxt = nxt, sum(factorint(k+2).values())
        if prv == nxt == 2: yield k
print(list(islice(agen(), 53))) # Michael S. Branicky, Nov 26 2022

A241483 Primes p such that p+2, p+4, p+6, p+8, p+10 and p+12 are all semiprime.

Original entry on oeis.org

1381, 3089, 10399, 49081, 53759, 63949, 76801, 98491, 107509, 109397, 113341, 143093, 182747, 204331, 209477, 239087, 252949, 255989, 313409, 396983, 426287, 500341, 602779, 677333, 812281, 832801, 1516531, 1574939, 1599151, 1619507, 1678639, 1866737, 2046449

Offset: 1

K. D. Bajpai, Apr 23 2014

nonn

			1381 is prime and appears in the sequence because 1381+2 = 1383 = 3*461, 1381+4 = 1385 = 5*277, 1381+6 = 1387 = 19*73, 1381+8 = 1389 = 3*463, 1381+10 = 1391 = 13*107 and  1381+12 = 1393 = 7*199, which are all semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..510

Cf. A072381, A082919.

with(numtheory): KD:= proc() local a,b,d,e,f,g,k; k:=ithprime(n); a:=bigomega(k+2); b:=bigomega(k+4); d:=bigomega(k+6);  e:=bigomega(k+8); f:=bigomega(k+10); g:=bigomega(k+12);  if a=2 and  b=2 and  d=2 and  e=2 and  f=2 and  g=2then RETURN (k);  fi; end: seq(KD(), n=1..200000);

KD = {};  Do[t = Prime[n]; If[PrimeOmega[t + 2] == 2 && PrimeOmega[t + 4] == 2 && PrimeOmega[t + 6] == 2 && PrimeOmega[t + 8] == 2 && PrimeOmega[t + 10] == 2 && PrimeOmega[t + 12] == 2, AppendTo[KD, t]], {n, 200000}]; KD
Select[Prime[Range[155000]],Union[PrimeOmega/@(#+2Range[6])]=={2}&] (* Harvey P. Dale, Dec 13 2018 *)

is(n)=if(n%3==1, isprime((n+2)/3) && isprime((n+8)/3) && bigomega(n+4)==2 && bigomega(n+10)==2, isprime((n+4)\3) && isprime((n+10)\3) && bigomega(n+2)==2 && bigomega(n+8)==2) && isprime(n) && bigomega(n+6)==2 && bigomega(n+12)==2
forprime(p=2,1e7,if(is(p),print1(p", "))) \\ Charles R Greathouse IV, Aug 25 2014

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.

A097824 Gaps associated with the arithmetic progressions of semiprimes in A096003.

Original entry on oeis.org

12, 12, 4, 52, 52, 72, 96, 198, 198, 114, 594, 48, 354, 1860, 3942, 2574, 2574, 2574, 20910, 20910, 9600, 9600, 152250, 152250, 152250, 152250, 1887270, 4667040, 4094790

Offset: 4

Hugo Pfoertner, Aug 27 2004

The terms a(1),a(2) and a(3) are omitted to avoid the ambiguity caused by the two progressions of length 3 ending at A096003(3)=14: a(3)=5 for (4,9,14) or a(3)=4 for (6,10,14).

			a(6)=4 because the 6 semiprimes in the progression ending at A096003(6)=221 are separated by an increment of 4: 201=3*67, 205=5*41, 209=11*19, 213=3*71, 217=7*31, 221=13*17.

Cf. A096003, A093364 gaps in arithmetic progressions of primes, A082919 clusters of 8 consecutive semiprimes.

a(26)-a(30) from Hugo Pfoertner, Sep 07 2004

a(31)-a(32) from Donovan Johnson, Jun 03 2012

A124570 Array read by antidiagonals: T(d,k) (k >= 1, d = 1,2,3,4,5,6,...) = smallest semiprime s of k (not necessarily consecutive) semiprimes in arithmetic progression with common difference d, or 0 if there is no such arithmetic progression.

Original entry on oeis.org

4, 4, 4, 4, 9, 4, 4, 4, 33, 4, 4, 6, 91, 0, 4, 4, 6, 115, 213, 0, 4, 4, 4, 6, 0, 213, 0, 4, 4, 4, 4, 111, 0, 1383, 0, 4, 4, 14, 9, 0, 201, 0, 3091, 0, 4, 4, 6, 51, 203, 0, 201, 0, 8129, 0, 4, 4, 6, 6, 0, 1333, 0, 481, 0, 0, 0, 4, 4, 4, 77, 69, 0, 1333, 0, 5989, 0, 0, 0, 4

Offset: 1

Jonathan Vos Post, Nov 04 2006

Comment from Hugo van der Sanden Aug 14 2021: (Start)
Row d=12 starts 4 9 9 10 10 469 3937 7343 7343 44719 78937 78937 78937 78937 55952333 233761133 597191343199.
Row d=18 starts 4 4 15 15 15 695 695 1727 7711 13951 13951 46159 400847 400847 400847 65737811 13388955301 934046384293.
Row d=24 starts 4 9 9 10 10 793 4819 6415 7271 14069 14069 14069 31589 67344271 616851797 48299373047 48299373047 20302675273219.
Row d=30 starts 4 4 9 25 25 2779 2779 6347 6347 6347 10811 10811 87109 87109 87109 1513723 15009191 15009191 316612697 316612697 1275591688621.
Row d=36 starts 4 10 10 10 15 1333 3161 4997 6865 34885 142171 834863 1327447 35528747 720945097 63389173477 63389173477 16074207679897 41728758250241.
Row d=42 starts 4 4 9 35 35 2701 2987 2987 7729 26995 26995 185795 307553 708385 708385 708385 1090198367 1819546069 20263042201 5672249016001.
Later terms in these rows are always >10^14. (End)
If p is the least prime that does not divide d, then T(d,k) <= p^2 if k >= p^2 (i.e. any a.p. of length >= p^2 with difference d contains a term divisible by p^2, and the only semiprime divisible by p^2 is p^2). Thus every row is eventually 0. - Robert Israel, Aug 11 2024

			Array begins:
d.\...k=1.k=2.k=3.k=4.k=5..k=6..k=7..k=8....k=9..k=10.k=11..k=12.
0..|..4...4...4...4...4....4....4....4......4....4.....4.....4...
1..|..4...9...33..0...0....0....0....0......0....0.....0.....0....
2..|..4...4...91..213.213..1383.3091.8129...0....0.....0.....0.....
3..|..4...6...115.0...0....0....0....0......0....0.....0.....0.....
4..|..4...6...6...111.201..201..481..5989...0....0.....0.....0....
5..|..4...4...4...0...0....0....0....0......0....0.....0.....0.....
6..|..4...4...9...203.1333.1333.1333.2159...8309.18799.60499.60499
7..|..4...14..51..0...0....0....0....0......0....0.....0.....0.....
8..|..4...6...6...69..473..511..511..112697.0....0.....0.....0.....
9..|..4...6...77..0...0....0....0....0......0....0.....0.....0.....
10.|..4...4...15..289.289..289..1631.13501..0....0.....0.....0.....
11.|..4...4...4...0...0....0....0....0......0....0.....0.....0.....
Example for row 3: 115 = 5 * 23 is semiprime, 115+3 = 118 = 2 * 59 is semiprime and 115+3+3 = 121 = 11^2 is semiprime, so T(3,3) = 115.

R. J. Mathar, Table for d <= 999 (PDF)

Semiprime analog of A124064.

Cf. A125025 (row lengths), A001358, A056809, A070552, A092125, A092126, A092127, A092128, A092129, A124064, A092209 (row d=2), A091016 (row d=6).

T(1,2)=A070552(1). T(1,3)=A056809(1). T(2,4)=A092126(1). T(2,5)=A092127(1). T(2,6)=A092128(1). T(2,7)=A092129(1). T(2,8)=A082919(1). T(3,2)=A123017(1). T(d,1)=A001358(1). - R. J. Mathar, Aug 05 2021

Corrected and extended by R. J. Mathar, Nov 06 2006

Definition clarified by Robert Israel, Aug 11 2024

A241493 Primes p such that p + 4, p + 16, p + 64, p + 256 and p + 1024 are all semiprimes.

Original entry on oeis.org

1627, 2917, 3583, 4603, 5581, 6367, 6379, 8263, 9697, 12517, 12763, 13339, 14197, 15289, 16339, 16993, 17539, 17737, 18199, 19267, 19531, 20023, 28057, 28879, 29587, 32647, 33427, 34033, 34537, 35353, 35617, 37039, 37087, 37657, 37663, 42337, 43093, 47533, 48049

Offset: 1

K. D. Bajpai, Apr 24 2014

nonn

The constants in the definition (4, 16, 64, 256 and 1024 ) are in geometric progression.

			1627 is prime and appears in the sequence because 1627+4 = 1631 = 7*233, 1627+16 = 1643 = 31*53, 1627+64 = 1691 = 19*89, 1627+256 = 1883 = 7*269 and 1627+1024 = 2651 = 11*241, which are all semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A072381, A082919, A241483, A241484.

with(numtheory): KD:= proc() local a,b,d,e,f,k; k:=ithprime(n); a:=bigomega(k+4); b:=bigomega(k+16); d:=bigomega(k+64); e:=bigomega(k+256); f:=bigomega(k+1024); if a=2 and  b=2 and d=2 and  e=2 and f=2 then RETURN (k); fi; end: seq(KD(), n=1..10000);

KD = {}; Do[t = Prime[n]; If[PrimeOmega[t + 4] == 2 && PrimeOmega[t + 16] == 2 && PrimeOmega[t + 64] == 2 && PrimeOmega[t + 256] == 2 && PrimeOmega[t + 1024] == 2, AppendTo[KD, t]], {n, 10000}]; KD
(* For the b-file *) c = 0; Do[t = Prime[n]; If[PrimeOmega[t + 4] == 2 && PrimeOmega[t + 16] == 2 && PrimeOmega[t + 64] == 2 && PrimeOmega[t + 256] == 2 && PrimeOmega[t + 1024] == 2, c++; Print[c, "  ", t]], {n, 1,5*10^6}];
Select[Prime[Range[5000]],Union[PrimeOmega[#+{4,16,64,256,1024}]] == {2}&] (* Harvey P. Dale, Nov 28 2017 *)

A241484 Primes p such that p+2 and p+4 are semiprime.

Original entry on oeis.org

2, 31, 47, 53, 83, 89, 139, 157, 181, 199, 211, 233, 263, 317, 337, 389, 409, 443, 449, 467, 541, 577, 587, 631, 677, 683, 719, 751, 787, 811, 839, 919, 947, 991, 1039, 1097, 1117, 1163, 1187, 1201, 1259, 1367, 1381, 1399, 1559, 1637, 1669, 1709, 1759, 1777, 1847

Offset: 1

K. D. Bajpai, Apr 23 2014

nonn

			31 is prime and appears in the sequence because 31+2 = 33 = 3*11 and 31+4 = 35 = 5*7, which are semiprime.
53 is prime and appears in the sequence because 53+2 = 55 = 5*11 and 53+4 = 57 = 3*19, which are semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A072381, A082919.

IsSemiprime:=func< p | &+[ k[2]: k in Factorization(p)] eq 2 >; [p: p in PrimesUpTo(2000)| IsSemiprime(p+2) and IsSemiprime(p+4)]; // Vincenzo Librandi, Apr 24 2014

with(numtheory): KD:= proc() local a,b,d,k; k:=ithprime(n); a:=bigomega(k+2);b:=bigomega(k+4); if a=2 and  b=2 then RETURN (k); fi; end: seq(KD(), n=1..1000);

KD = {}; Do[t = Prime[n];If[PrimeOmega[t + 2] == 2 && PrimeOmega[t + 4] == 2,AppendTo[KD, t]], {n, 1000}]; KD

A241554 Semiprimes generated by the polynomial 2 * n^2 + 29.

Original entry on oeis.org

1711, 1829, 2077, 2479, 3071, 3901, 5029, 6527, 6757, 7471, 7967, 8479, 10397, 10981, 11581, 14141, 15167, 15517, 15871, 16591, 16957, 17701, 18079, 18847, 19631, 20837, 22927, 23791, 25567, 26941, 27877, 28829, 29797, 30287, 31279, 31781, 32287, 35941, 38117

Offset: 1

K. D. Bajpai, Apr 25 2014

nonn

2 * n^2 + 29 is a well-known Legendre prime-producing polynomial which generates 29 distinct primes for n = 0, 1, ..., 28. For n = 29, it yields the first semiprime, 1711 = 29 * 59.

The number n = 185 is the least positive integer for which 2*n^2 + 29 = 68479 = 31 * 47 * 47 is not squarefree.

			2 * 30^2 + 29 = 1829 = 31 * 59, which is a semiprime and is a term.
2 * 35^2 + 29 = 2479 = 37 * 67, which is a semiprime and is a term.

K. D. Bajpai, Table of n, a(n) for n = 1..10000
R. A. Mollin, Prime-Producing Quadratics, The American Mathematical Monthly, Vol. 104, No. 6 (Jun. - Jul., 1997), pp. 529-544.

Cf. A007641 (for primes).

Cf. A001358, A072381, A082919, A145292, A228183, A237627, A353004, A353388.

with(numtheory):A241554:= proc() local k; k:=2*x^2+29;if bigomega(k)=2 then RETURN (k); fi; end: seq(A241554(), x=0..500);

A241554 = {}; Do[k = 2 * n^2 + 29; If[PrimeOmega[k] == 2, AppendTo[A241554, k]], {n,200}]; A241554

s=[]; for(n=1, 200, t=2*n^2+29; if(bigomega(t)==2, s=concat(s, t))); s \\ Colin Barker, Apr 26 2014

A091016 a(n) is the least x such that the n values x+0, x+6, x+12, ..., x+6*(n-1) are all products of exactly two primes (i.e., semiprimes).

Original entry on oeis.org

4, 4, 9, 203, 1333, 1333, 1333, 2159, 8309, 18799, 60499, 60499, 186779, 186779, 186779, 12015573923, 12015573923, 6763513182767

Offset: 1

Don Reble, Feb 25 2004

a(19) > 1.e14 if it exists. - Hugo van der Sanden, Aug 06 2021

			a(4)=203 because 203 = 7*29, 209 = 11*19, 215 = 5*43, 221 = 13*17.

Cf. A082919, A092129. Row 6 of A124570.

a(8) and a(9) corrected and a(18) from Donovan Johnson, Oct 01 2010

cp's OEIS Frontend

A092208 Duplicate of A082919.

Views

Author

Keywords

A092207 Semiprimes k such that k+2 is also a semiprime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

A241483 Primes p such that p+2, p+4, p+6, p+8, p+10 and p+12 are all semiprime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

A097824 Gaps associated with the arithmetic progressions of semiprimes in A096003.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A124570 Array read by antidiagonals: T(d,k) (k >= 1, d = 1,2,3,4,5,6,...) = smallest semiprime s of k (not necessarily consecutive) semiprimes in arithmetic progression with common difference d, or 0 if there is no such arithmetic progression.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A241493 Primes p such that p + 4, p + 16, p + 64, p + 256 and p + 1024 are all semiprimes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A241484 Primes p such that p+2 and p+4 are semiprime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

A241554 Semiprimes generated by the polynomial 2 * n^2 + 29.

Views

Author

Keywords

Comments

Examples