A131109
a(n) is the smallest semiprime such that difference between a(n) and next semiprime, b(n), is n.
Original entry on oeis.org
9, 4, 6, 10, 69, 15, 26, 169, 146, 237, 95, 1082, 818, 597, 1603, 2705, 2078, 4511, 1418, 2681, 14545, 13863, 37551, 6559, 16053, 55805, 26707, 17965, 308918, 32777, 41222, 35103, 393565, 219509, 153263, 87627, 2263057, 35981, 1789339, 741841, 797542
Offset: 1
n, b(n)-a(n): 1=10-9, 2=6-4, 3=9-6, 4=14-10, 5=74-69, 6=21-15, 7=33-26, 8=177-169, 9=155-146, 10=247-237, 11=106-95, 12=1094-1082, 13=831-818, 14=611-597, 15=1618-1603, 16=2721-2705, 17=2095-2078, 18=4529-4511, 19=1437-1418, 20=2701-2681, 21=14566-14545, 22=13885-13863, 23=37574-37551, 24=6583-6559, 25=16078-16053, 26=55831-55805, 27=26734-26707, 28=17993-17965, 29=308947-308918, 30=32807-32777, 31=41253-41222, 32=35135-35103, 33=393598-393565, 34=219543-219509, 35=153298-153263, 36=87663-87627, 37=2263094-2263057, 38=36019-35981.
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SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; NextSemiPrime[n_] := Module[{m = n + 1}, While[! SemiPrimeQ[m], m++]; m]; nn = 30; t = Table[0, {nn}]; found = 0; sp0 = 4; While[found < nn, sp1 = NextSemiPrime[sp0]; d = sp1 - sp0; If[d <= nn && t[[d]] == 0, t[[d]] = sp0; found++]; sp0 = sp1]; t (* T. D. Noe, Oct 02 2012 *)
A228170
The least semiprime (A001358) such that between it and the next n semiprimes, but not the next n+1 semiprimes, there are no primes.
Original entry on oeis.org
9, 33, 91, 141, 115, 213, 1382, 1639, 1133, 2558, 2973, 1329, 15685, 16143, 9974, 35678, 34063, 43333, 19613, 107381, 162145, 44294, 404599, 461722, 838259, 155923, 535403, 492117, 396737, 2181739, 370262, 1468279, 6034249, 3933601, 1671783, 25180174, 1357203
Offset: 1
a(1) = 9 because between 9 and 10 there are no primes;
a(2) = 33 because between 33 and 35 (the second semiprime past 33) there are no primes;
a(3) = 91 because between 91 and 95 (the third semiprime past 91 with 93 & 94 also semiprimes) there are no primes;
a(4) = 141 because between 141 and 146 (the fourth semiprime past 141 with 142, 143 & 145 also being semiprimes) there are no primes;
the reason a(4) is not 115 is because although there are no primes between 115 and 121, the string "2, 3, 3, 2, 2, 5, 2, 2" is a substring of the string generated by 115 through 123. See the next line.
a(5) = 115 because between 115 and 123 (the fifth semiprime past 115 with 118, 119, 121, and 122 also being semiprimes) there are no primes;
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NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; t = Table[0, {100}]; p=3; While[p < 3100000000, q = NextPrime[p]; a = Count[ PrimeOmega[ Range[p, q]], 2]; If[ t[[a]] == 0, t[[a]] = p; Print[{p, a}]]; p = q]; NextSemiPrime@# & /@ t
A228171
Least prime such that between it and the next prime there are exactly n semiprimes.
Original entry on oeis.org
2, 3, 7, 31, 89, 139, 113, 211, 1381, 1637, 1129, 2557, 2971, 1327, 15683, 16141, 9973, 35677, 34061, 43331, 19609, 107377, 162143, 44293, 404597, 461717, 838249, 155921, 535399, 492113, 396733, 2181737, 370261, 1468277, 6034247, 3933599, 1671781, 25180171
Offset: 0
a(2) = 7 since between 7 and the next prime, 11, there are 2 semiprimes (9, 10).
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t = Table[0, {100}]; p=3; While[p < 3100000000, q = NextPrime[p]; a = Count[ PrimeOmega[ Range[p, q]], 2]; If[ t[[a]] == 0, t[[a]] = p; Print[{p, a}]]; p = q]; t
A136344
Least primes p1 such that gap between p1 and p2=nextprime(p1) contains no semiprimes. Gap is 1 for n=0 (first term) and 2n for n=1..30.
Original entry on oeis.org
2, 11, 97, 601, 6473, 2521, 35117, 63113, 39343, 104659, 705053, 512821, 1123279, 4357781, 4942711, 4808987, 5922317, 32995057, 105484481, 37212113, 108525829, 335433389, 598559963, 789565537, 864056573, 1913815493, 2311939579
Offset: 1
a(1)=2 because gap=3-2=1 and between 2 and 3 there is no semiprimes,
a(2)=11 because gap=11-13=2 and between 11 and 13 there is no semiprimes,
a(3)=97 because gap=97-101=4 and between 97 and 101 there is no semiprimes,
a(31)=9042022037 because gap=9042022037-9042022097=60 and between 9042022037 and 9042022097 there is no semiprimes.
Showing 1-4 of 4 results.
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