A082919 -id:A082919 - cp's OEIS frontend

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

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 *)

A241716 Primes p such that p^3 - 2 is semiprime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 41, 43, 47, 61, 79, 89, 101, 107, 139, 157, 181, 199, 239, 271, 307, 311, 331, 337, 347, 349, 379, 397, 409, 421, 431, 479, 487, 499, 521, 523, 541, 571, 607, 613, 641, 643, 661, 673, 701, 719, 761, 769, 811, 823, 829, 839, 877, 881, 883

Offset: 1

K. D. Bajpai, Apr 27 2014

nonn

			11 is prime and appears in the sequence because 11^3 - 2 = 1329 = 3 * 443, which is a semiprime.
17 is prime and appears in the sequence because 17^3 - 2 = 4911 = 3 * 1637, which is a semiprime.
23 is prime but does not appear in the sequence because 23^3 - 2 = 12165 =  3 * 5 * 811, which is not a semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..9380

Cf. A001358, A063637, A063638, A072381, A082919, A145292, A228183, A237627, A241483, A241493, A241659.

with(numtheory):A241716:= proc() local k; k:=ithprime(x); if bigomega(k^3-2)=2 then RETURN (k); fi; end: seq(A241716(), x=1..500);

A241716 = {}; Do[t = Prime[n]; If[PrimeOmega[t^3 - 2] == 2, AppendTo[A241716, t]], {n, 500}]; A241716
Select[Prime[Range[200]],PrimeOmega[#^3-2]==2&] (* Harvey P. Dale, Dec 09 2018 *)

A241732 Primes p such that p^3 + 2 and p^3 - 2 are semiprime.

Original entry on oeis.org

2, 11, 13, 17, 41, 89, 101, 239, 271, 331, 571, 641, 719, 1051, 1231, 1321, 1549, 1559, 1721, 1741, 1831, 1993, 1999, 2029, 2311, 2459, 2749, 2837, 2861, 2939, 3389, 3467, 3671, 4049, 4111, 4273, 4787, 4919, 4969, 5657, 5689, 5861, 6221, 6679, 6691, 6829, 7109

Offset: 1

K. D. Bajpai, Apr 27 2014

nonn

			11 is prime and appears in the sequence because 11^3 + 2 = 1333 = 31 * 43 and 11^3 - 2 = 1329 = 3 * 443, both are semiprime.
41 is prime and appears in the sequence because 41^3 + 2 = 68923 = 157 * 439 and 41^3 - 2 = 68919 = 3 * 22973, both are semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..2310

Cf. A001358, A063637, A063638, A072381, A082919, A145292, A228183, A237627, A241483, A241493, A241659.

with(numtheory): KD:= proc() local k; k:=ithprime(n); if bigomega(k^3+2)=2 and bigomega(k^3-2)=2 then k; fi; end: seq(KD(), n=1..2000);

A241732 = {}; Do[t = Prime[n]; If[PrimeOmega[t^3 + 2] == 2 && PrimeOmega[t^3 - 2] == 2, AppendTo[A241732, t]], {n, 500}]; A241732
Select[Prime[Range[1000]],PrimeOmega[#^3+2]==PrimeOmega[#^3-2]==2&] (* Harvey P. Dale, Jun 20 2019 *)

A289250 Primes p such that p + 4 is a semiprime.

Original entry on oeis.org

2, 5, 11, 17, 29, 31, 47, 53, 61, 73, 83, 89, 107, 137, 139, 151, 157, 173, 179, 181, 197, 199, 211, 233, 263, 283, 317, 331, 337, 367, 373, 389, 409, 433, 443, 449, 467, 523, 541, 547, 569, 577, 587, 593, 607, 619, 631, 677, 683, 691, 709, 719, 727, 733, 751, 787, 809, 811, 827

Offset: 1

Zak Seidov, Jun 29 2017

nonn

Except for case p=5, p+4 is never a perfect square.

For p = {2, 11, 31, 73, 139, 433, 1759, 2017} p+4 is a product of two consecutive primes.

			2+4=6=2*3, 5+4=9=3*3, 11+4=15=3*5 (all semiprimes).

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

Cf. A001358, A063637, A082919, A241484,

Select[Prime@ Range@ 150, PrimeOmega[# + 4] == 2 &] (* Michael De Vlieger, Jun 29 2017 *)

issemi(n)=bigomega(n)==2
is(n)=isprime(n) && issemi(n+4) \\ Charles R Greathouse IV, Jul 02 2017

A112888 Least semiprime of a cluster of just n semiprimes.

Original entry on oeis.org

9, 33, 91, 299, 213, 1383, 3091, 8129

Offset: 1

Robert G. Wilson v, Nov 30 2005

Clusters are sets composed of odd numbers.
If we include even numbers then the sequence would start 4,9,33 and terminates because in any group of four consecutive numbers greater than 4, 4 is a divisor to at least one member leaving a quotient greater than 1.
Any set of 9 consecutive odd numbers contain a multiple of 9, which not semiprime (unless it is equal to 9). Hence there are no 9 consecutive odd semiprimes.

			a(8)=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.

Cf. A001358, A097824, A082919.

spQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; f[n_] := Block[{k = 1}, While[ s[[k]] + 2n != s[[k + n]] || s[[k]] + 2n + 2 == s[[k + n + 1]], k++ ]; s[[k]]]; s = {}; Do[ If[ spQ[n], AppendTo[s, n]], {n, 9, 7*10^6, 2}]; Table[ f[n], {n, 0, 7}]
Join[{9},Module[{osps=Select[Range[9,10001,2],PrimeOmega[#]==2&]}, #[[2]]& /@ Table[ SelectFirst[Partition[osps,n+2,1],Union[ Differences[ Rest[ Most[#]]]]=={2}&&Last[#]-#[[-2]]!=2&&#[[2]]-#[[1]]!=2&],{n,2,8}]]] (* Harvey P. Dale, Jun 01 2016 *)

fini, full from Max Alekseyev, Feb 03 2010

A241607 Semiprimes generated by the polynomial (1/4)(n^5 - 133n^4 + 6729n^3 - 158379n^2 + 1720294*n - 6823316).

Original entry on oeis.org

5141923, 6084557, 11403823, 13201987, 17488411, 20017609, 33239291, 37446979, 42070423, 47139347, 72512623, 88747907, 118408673, 129881707, 169708339, 184952323, 201267887, 278376073, 324881567, 406044923, 436421497, 538566199, 616639427, 658920007, 750410069

Offset: 1

K. D. Bajpai, Apr 26 2014

nonn

(1/4)*(n^5 - 133*n^4 + 6729*n^3 - 158379*n^2 + 1720294*n - 6823316) is a well known prime producing polynomial found by Shyam Sunder Gupta, which generates 57 distinct primes for n = 0,1,...,55,56.

For n = 57, this polynomial yields the first semiprime: 5141923 = 821 * 6263.

			For n=57: (1/4)*(n^5 - 133*n^4 + 6729*n^3 - 158379*n^2 + 1720294*n - 6823316) = 5141923 = 821 * 6263, which is a semiprime and is included in the sequence.
For n=58: (1/4)*(n^5 - 133*n^4 + 6729*n^3 - 158379*n^2 + 1720294*n - 6823316) = 6084557 = 131 * 46447, which is a semiprime and is included in the sequence.

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

Cf. A007641 (for primes).

Cf. A001358, A072381, A082919, A121887, A145292, A228183, A237627.

with(numtheory): KD:= proc() local a,b,k; k:=(1/4)*(n^5 - 133*n^4 + 6729*n^3 - 158379*n^2 + 1720294*n - 6823316); a:=bigomega(k); if a=2 then RETURN (k); fi; end: seq(KD(), n=0..200);

A241607 = {}; Do[k= (1/4) * (n^5 - 133 * n^4 + 6729 * n^3 - 158379 * n^2 + 1720294 * n - 6823316); If[PrimeOmega[k] ==2, AppendTo[A241607, k]], {n,200}]; A241607
(*For the b-file:*) n=0;Do[t=((1/4) * (k^5 - 133 * k^4 + 6729 * k^3 - 158379 * k^2 + 1720294 * k - 6823316));If[PrimeOmega[t]==2, n++; Print[n," ",t]], {k,10^6}]

s=[]; for(n=1, 200, t=(1/4)*(n^5-133*n^4+6729*n^3-158379*n^2+1720294*n-6823316); if(bigomega(t)==2, s=concat(s, t))); s \\ Colin Barker, Apr 26 2014

A241959 Primes p such that p+2, p+4, p+6, p+8, p+10 are semiprimes.

Original entry on oeis.org

211, 1381, 3089, 5087, 10399, 18803, 26903, 27031, 31583, 41161, 47189, 49081, 53759, 62939, 63949, 76801, 87383, 93739, 98491, 107509, 109397, 113341, 128099, 143093, 158699, 182747, 186889, 193727, 197507, 201413, 204331, 209477, 239087, 252949, 255989, 256079

Offset: 1

K. D. Bajpai, May 03 2014

nonn

Each term in the sequence is prime p which yields 5 semiprimes in arithmetic progression with common difference of 2.

			a(1) = 211 is prime: 213, 215, 217, 219 and 221 are semiprimes.
a(2) = 1381 is prime: 1383, 1385, 1387, 1389 and 1391 are semiprimes.

K. D. Bajpai, Table of n, a(n) for n = 1..500

Cf. A001358, A082919, A092128, A241483.

with(numtheory): A241959:= proc() local p;p:=ithprime(x);if  bigomega(p+2)=2 and bigomega(p+4)=2 and bigomega(p+6)=2 and bigomega(p+8)=2 and bigomega(p+10)=2 then RETURN (p); fi; end: seq(A241959 (), x=1..100000);

A330508 Numbers k such that k + 6^t is semiprime for t = 0 to 9.

Original entry on oeis.org

61273, 109441, 160213, 274501, 275473, 311593, 360673, 394201, 477181, 486061, 514993, 522085, 617137, 620053, 715477, 725485, 803833, 812677, 847117, 1063585, 1146913, 1182577, 1215865, 1232917, 1409425, 1508113, 1587241, 1768993, 1863073, 1895413, 2085517, 2095177

Offset: 1

K. D. Bajpai, Dec 16 2019

nonn

a(2620) = 530079693 is the first multiple of 3 in this sequence; there are no multiples of 2. - Charles R Greathouse IV, Dec 20 2019

			a(1) = 61273:
  61273 + 6^0  =    61274 =   2 *  30637;
  61273 + 6^1  =    61279 = 233 *    263;
  61273 + 6^2  =    61309 =  37 *   1657;
  61273 + 6^3  =    61489 =  17 *   3617;
  61273 + 6^4  =    62569 =  13 *   4813;
  61273 + 6^5  =    69049 =  29 *   2381;
  61273 + 6^6  =   107929 =  37 *   2917;
  61273 + 6^7  =   341209 =  11 *  31019;
  61273 + 6^8  =  1740889 = 197 *   8837;
  61273 + 6^9  = 10138969 =  89 * 113921;
all ten results are semiprime.

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

Subsequence of A076274.

Cf. A001358, A082919, A096173, A104238, A105041.

f:=func; [k:k in [1..2100000]|forall{m:m in [0..9]|f(k+6^m)}]; // Marius A. Burtea, Dec 20 2019

fX[n_] = PrimeOmega[n] == 2; Select[Range[2000000], AllTrue[# + 6^{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, fX] &]

issemi(n)=bigomega(n)==2
is(n)=for(t=0,9, if(!issemi(n+6^t), return(0))); 1 \\ Charles R Greathouse IV, Dec 20 2019

cp's OEIS Frontend

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

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A241716 Primes p such that p^3 - 2 is semiprime.

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A241732 Primes p such that p^3 + 2 and p^3 - 2 are semiprime.

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A289250 Primes p such that p + 4 is a semiprime.

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A112888 Least semiprime of a cluster of just n semiprimes.

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A241607 Semiprimes generated by the polynomial (1/4)*(n^5 - 133*n^4 + 6729*n^3 - 158379*n^2 + 1720294*n - 6823316).

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A241959 Primes p such that p+2, p+4, p+6, p+8, p+10 are semiprimes.

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A330508 Numbers k such that k + 6^t is semiprime for t = 0 to 9.

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A241607 Semiprimes generated by the polynomial (1/4)(n^5 - 133n^4 + 6729n^3 - 158379n^2 + 1720294*n - 6823316).