A059960
Smaller term of a pair of twin primes such that prime factors of their average are only 2 and 3.
Original entry on oeis.org
5, 11, 17, 71, 107, 191, 431, 1151, 2591, 139967, 472391, 786431, 995327, 57395627, 63700991, 169869311, 4076863487, 10871635967, 2348273369087, 56358560858111, 79164837199871, 84537841287167, 150289495621631, 578415690713087, 1141260857376767
Offset: 1
a(11)+1 = 2*2*2*3*3*3*3*3*3*3*3*3*3 = 472392.
Apart from initial terms, same as
A078883.
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nn=10^15; Sort[Reap[Do[n=2^i 3^j; If[n<=nn && PrimeQ[n-1] && PrimeQ[n+1], Sow[n-1]], {i, Log[2, nn]}, {j, Log[3, nn]}]][[2, 1]]]
Select[Select[Partition[Prime[Range[38*10^5]],2,1],#[[2]]-#[[1]]==2&][[All,1]],FactorInteger[#+1][[All,1]]=={2,3}&] (* The program generates the first 15 terms of the sequence. *)
seq[max_] := Select[Sort[Flatten[Table[2^i*3^j - 1, {i, 1, Floor[Log2[max]]}, {j, 1, Floor[Log[3, max/2^i]]}]]], And @@ PrimeQ[# + {0, 2}] &]; seq[2*10^15] (* Amiram Eldar, Aug 27 2024 *)
A075580
Smallest prime p(k) such that the number of distinct prime divisors of all composite numbers between p(k) and p(k+1) is n.
Original entry on oeis.org
2, 3, 5, 7, 13, 19, 31, 53, 73, 89, 359, 139, 401, 181, 113, 211, 293, 661, 863, 773, 523, 1933, 1831, 1069, 1381, 887, 1637, 1129, 1669, 1951, 4027, 3469, 4177, 6397, 2477, 2971, 5531, 1327, 4297, 4831, 5351, 5591, 9973, 11743, 13187, 8467, 27851, 18803
Offset: 0
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Table[i=1; While[Length[Union[Flatten[Table[First/@FactorInteger[j],{j,(x=Prime[i])+1,NextPrime[x]-1}]]]]!=n,i++]; x,{n,0,47}] (* Jayanta Basu, May 25 2013 *)
A075583
Primes p such that the composite numbers between p and the next prime together contain exactly three distinct prime factors.
Original entry on oeis.org
7, 29, 41, 59, 101, 137, 149, 179, 197, 227, 239, 269, 281, 311, 347, 521, 599, 617, 641, 809, 821, 827, 881, 1031, 1061, 1277, 1451, 1487, 1607, 1619, 1667, 1697, 1787, 1871, 1877, 1997, 2027, 2081, 2087, 2111, 2237, 2267, 2381, 2657, 2687, 2711, 2801, 2999
Offset: 1
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a:=[]; for k in PrimesInInterval(2,3000) do b:={}; for s in [k..NextPrime(k)-1] do if not IsPrime(s) then b:=b join Set(PrimeDivisors(s)); end if; end for; if #Set(b) eq 3 then Append(~a,k); end if; end for; a; // Marius A. Burtea, Sep 26 2019
A075585
Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 5.
Original entry on oeis.org
19, 23, 37, 43, 97, 127, 223, 499, 673, 1213, 2309, 2729, 6089, 6269, 7589, 8969, 9239, 9281, 10709, 11549, 11969, 12539, 13397, 14321, 15329, 16829, 17489, 18059, 19139, 19379, 19469, 19889, 20747, 21317, 21839, 22109, 22619, 23369, 23561, 24179
Offset: 1
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a:=[]; for k in PrimesInInterval(2,25000) do b:={}; for s in [k..NextPrime(k)-1] do if not IsPrime(s) then b:=b join Set(PrimeDivisors(s)); end if; end for; if #Set(b) eq 5 then Append(~a,k); end if; end for; a; // Marius A. Burtea, Sep 26 2019
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dpd5Q[p_]:=Length[Union[Flatten[FactorInteger[#][[All,1]]&/@Range[ p+1,NextPrime[ p]-1]]]]==5; Select[Prime[Range[3000]],dpd5Q] (* Harvey P. Dale, Aug 11 2021 *)
A075586
Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 6.
Original entry on oeis.org
31, 47, 67, 103, 109, 163, 193, 277, 313, 349, 379, 397, 457, 463, 487, 877, 1087, 1093, 1279, 1303, 1567, 1873, 2269, 2347, 2473, 2797, 3697, 4447, 4789, 4999, 5077, 5413, 5503, 5923, 6007, 6217, 6469, 6997, 7603, 7639, 7723, 7933, 8779, 9277, 10159
Offset: 1
Between 31 and the next prime 37, there are 5 composite numbers whose prime divisors are respectively for 32: {2}, 33: {3,11}, 34: {2,17}, 35: {5,7} and 36: {2,3}; hence, these distinct prime divisors are {2,3,5,7,11,17}, the number of these distinct prime divisors is 6, so 31 is a term. - _Bernard Schott_, Sep 26 2019
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a:=[]; for k in PrimesInInterval(2,10000) do b:={}; for s in [k..NextPrime(k)-1] do if not IsPrime(s) then b:=b join Set(PrimeDivisors(s)); end if; end for; if #Set(b) eq 6 then Append(~a,k); end if; end for; a; // Marius A. Burtea, Sep 26 2019
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Select[Partition[Prime[Range[1250]],2,1],Length[Union[Flatten[ FactorInteger/@ Range[ #[[1]]+1,#[[2]]-1],1][[All,1]]]]==6&][[All,1]] (* Harvey P. Dale, May 25 2020 *)
A075587
Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 7.
Original entry on oeis.org
53, 61, 157, 229, 307, 439, 613, 757, 769, 823, 853, 859, 883, 907, 937, 967, 1009, 1297, 1423, 1429, 1447, 1483, 1489, 1549, 1597, 1663, 1693, 1993, 2083, 2137, 2203, 2239, 2389, 2437, 2659, 2689, 2707, 2749, 2833, 2857, 2953, 3019, 3037, 3163, 3187
Offset: 1
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pd[n_]:=Transpose[FactorInteger[n]][[1]]; Transpose[Select[ Partition[ Prime[ Range[500]],2,1],Length[Union[Flatten[ pd/@Range[First[#]+1, Last[#]-1]]]] == 7&]][[1]] (* Harvey P. Dale, Jun 15 2013 *)
Select[Prime@ Range@ 500, Length@ Union@ Flatten@ Map[First /@ FactorInteger@ # &, Select[Range[#, NextPrime@ #], CompositeQ]] == 7 &] (* Michael De Vlieger, May 27 2016 *)
A075588
Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 8.
Original entry on oeis.org
73, 83, 131, 167, 173, 251, 331, 383, 443, 563, 643, 739, 971, 1123, 1223, 1367, 1579, 1609, 1783, 1867, 1999, 2293, 2539, 2617, 2683, 3083, 3217, 3253, 3343, 3457, 3847, 4003, 4513, 4783, 4813, 4969, 5167, 5233, 5527, 5737, 5779, 5839, 5857, 6199, 6733
Offset: 1
For p = 131, the next prime number is 137. The numbers between 131 and 137 and the prime factors are respectively 132 { 2, 3, 11 }, 133 { 7, 19 }, 134 { 2, 67 }, 135 { 3, 5 }, 136 { 2, 17 }. The set of prime divisors is { 2, 3, 5, 7, 11, 17, 19, 67 } and has 8 elements, so 131 is a term. - _Marius A. Burtea_, Sep 26 2019
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a:=[]; for p in PrimesInInterval(2,7000) do b:={}; for s in [p..NextPrime(p)-1] do if not IsPrime(s) then b:=b join Set(PrimeDivisors(s)); end if; end for; if #Set(b) eq 8 then Append(~a,p); end if; end for; a; // Marius A. Burtea, Sep 26 2019
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Select[Partition[Prime[Range[1000]],2,1],Length[Union[ Flatten[ FactorInteger[ Range[ #[[1]]+1,#[[2]]-1]],1][[All,1]]]]==8&][[All,1]] (* Harvey P. Dale, Dec 26 2019 *)
A075589
Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 9.
Original entry on oeis.org
89, 151, 233, 257, 263, 271, 353, 367, 373, 503, 541, 571, 587, 601, 647, 653, 727, 733, 751, 977, 991, 1013, 1181, 1291, 1321, 1433, 1453, 1621, 1753, 1861, 2281, 2371, 2377, 2671, 3061, 3079, 3203, 3323, 3793, 4051, 4073, 4283, 4357, 4519, 4591, 4639
Offset: 1
For p = 233, the next prime number is 239. The numbers between 233 and 237 and the prime divisors are respectively 234 {2, 3, 13}, 235 {5, 47}, 236 {2, 59}, 237 {3, 79 }, 238 {2, 7, 17}. The set of prime divisors is {2, 3, 5, 7, 13, 17, 47, 59, 79} and has 9 elements, so 233 is a term.
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a:=[]; for p in PrimesInInterval(2,5000) do b:={}; for s in [p..NextPrime(p)-1] do if not IsPrime(s) then b:=b join Set(PrimeDivisors(s)); end if; end for; if #Set(b) eq 9 then Append(~a,p); end if; end for; a; // Marius A. Burtea, Sep 26 2019
A332740
Prime numbers p such that the set of composite numbers in the range [p+1, nextprime(p)-1] has more than one element and all the elements have the same number of divisors.
Original entry on oeis.org
229, 8293, 9829, 14887, 16087, 20389, 21493, 44983, 50581, 53887, 57943, 63463, 64663, 72223, 81547, 93253, 108343, 134917, 138727, 143239, 157207, 192613, 199669, 203653, 206407, 210853, 218839, 244837, 248749, 251287, 255049, 262693, 280183, 296437, 300319
Offset: 1
229 is a term since between 229 and its next prime, 233, there are 3 composite numbers, 230, 231 and 232 and all of them have the same number of divisors, 8.
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seqQ[n_] := PrimeQ[n] && (nx=NextPrime[n]) > n + 2 && Length @ Union @ DivisorSigma[0, Range[n+1, nx-1]] == 1; Select[Range[10^6], seqQ]
Showing 1-9 of 9 results.
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