A140803
Numbers of the form (2^(p*q)-1) /((2^p-1)*(2^q-1)), where p>q are primes.
Original entry on oeis.org
3, 11, 43, 151, 683, 2359, 2731, 43691, 174763, 599479, 2796203, 8727391, 9588151, 178956971, 715827883, 2454285751, 39268347319, 45812984491, 567767102431, 733007751851, 2932031007403, 10052678938039, 46912496118443, 145295143558111, 3002399751580331, 41175768098368951, 192153584101141163
Offset: 1
Entry 3 from (q=2,p=3), entry 11 from (q=2,p=5), entry 43 from (q=2,p=7), entry 151 from (q=3,p=5), entry 683 from (q=2,p=11).
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N:= 100: # to use all (p,q) with p*q < N
Primes:= select(isprime,[$2..floor(N/2)]):
A:= {}:
for i from 1 to nops(Primes) do
p:= Primes[i];
Qs:= select(q -> q < N/p, [seq(Primes[j],j=1..i-1)]);
A:= A union {seq((2^(p*q)-1)/(2^p-1)/(2^q-1),q=Qs)};
od:
A; # Robert Israel, Sep 02 2014
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terms = 27; Clear[seq]; seq[m_] := seq[m] = Table[(2^(p q)-1)/((2^p-1) (2^q-1)), {q, Prime[Range[m]]}, {p, Prime[Range[PrimePi[q]+1, terms]]}] // Flatten // Union // PadRight[#, terms]&;
seq[1]; seq[m=2]; While[seq[m] != seq[m-1], m++]; seq[m] (* Jean-François Alcover, Sep 17 2018 *)
A143012
Numbers of the form (4^p + 2^p + 1)/7, where p > 3 is prime.
Original entry on oeis.org
151, 2359, 599479, 9588151, 2454285751, 39268347319, 10052678938039, 41175768098368951, 658812288653553079, 2698495133088002829751, 690814754065816531725751, 11053036065049294753459639, 2829577232652317876553477559, 11589948344943812957569751412151
Offset: 1
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p:=ithprime: seq((4^p(n)+2^p(n)+1)*1/7, n=3..14); # Emeric Deutsch, Aug 16 2008
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(4^#+2^#+1)/7&/@Prime[Range[3,30]] (* Harvey P. Dale, Feb 19 2013 *)
A127959
Nonprime numbers of the form 1 + Sum_{k=1..m} 2^(2*k - 1).
Original entry on oeis.org
171, 10923, 699051, 11184811, 44739243, 178956971, 2863311531, 11453246123, 45812984491, 183251937963, 733007751851, 11728124029611, 46912496118443, 187649984473771, 750599937895083, 3002399751580331, 12009599006321323
Offset: 1
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a = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c] == False, AppendTo[a, c]], {x, 1, 50}]; a
Select[Table[Sum[2^(2k-1),{k,n}]+1,{n,50}],!PrimeQ[#]&] (* Harvey P. Dale, Dec 23 2017 *)
A243979
Indices of Wagstaff primes.
Original entry on oeis.org
2, 5, 14, 124, 399, 4552, 15898, 203095, 37029521, 105973558438, 19140185454656173, 3827634977577891833517
Offset: 1
For n = 3 the third Wagstaff prime is A000979(3) = 43 and 43 is also the 14th prime number, so a(3) = 14.
- Andrew R. Booker, The Nth Prime Page.
- Chris K. Caldwell, Wagstaff, The Top Twenty, The PrimePages.
- Xavier Gourdon and Pascal Sebah, Counting primes.
- Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x).
- Samuel S. Wagstaff, Jr., The Cunningham Project.
- Kim Walisch, Fast C++ prime counting function implementation (primecount).
- Wikipedia, Wagstaff prime.
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default(primelimit, 10^9); forprime(p=3, 31, q=(2^p+1)/3; if(isprime(q), print1(primepi(q)", "))) \\ Jens Kruse Andersen, Jun 22 2014
a(12) calculated using Kim Walisch's primecount and added by
Amiram Eldar, Sep 05 2024
A360475
Smallest prime factor of (2^prime(n) + 1) / 3.
Original entry on oeis.org
3, 11, 43, 683, 2731, 43691, 174763, 2796203, 59, 715827883, 1777, 83, 2932031007403, 283, 107, 2833, 768614336404564651, 7327657, 56409643, 1753, 201487636602438195784363, 499, 179, 971, 845100400152152934331135470251, 415141630193, 643, 104124649, 227
Offset: 2
a(2)=3 since for prime(2)=3, (2^3+1)/3 = 3;
a(3)=11 since for prime(3)=5, (2^5+1)/3 = 11;
a(10)=59 since for prime(10)=29, (2^29+1)/3 = 59*3033169.
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a:= n-> min(numtheory[factorset]((2^ithprime(n)+1)/3)):
seq(a(n), n=2..30); # Alois P. Heinz, Feb 28 2023
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a[n_] := FactorInteger[(2^Prime[n]+1)/3][[1, 1]];
Table[a[n], {n, 2, 30}] (* Jean-François Alcover, Jan 27 2025 *)
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forprime(p=3, 100, An=(2^p+1)/3; if(isprime(An), print1(An,", "), forprime(div=3, 2^((p-1)/2), if(An%div==0, print1(div,", "); next(2)))))
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