A027856
Dan numbers: numbers m of the form 2^j * 3^k such that m +- 1 are twin primes.
Original entry on oeis.org
4, 6, 12, 18, 72, 108, 192, 432, 1152, 2592, 139968, 472392, 786432, 995328, 57395628, 63700992, 169869312, 4076863488, 10871635968, 2348273369088, 56358560858112, 79164837199872, 84537841287168, 150289495621632, 578415690713088, 1141260857376768
Offset: 1
a(14) = 243*4096 = 995328 and {995327, 995329} are twin primes.
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Select[#, Total@ Boole@ Map[PrimeQ, # + {-1, 1}] == 2 &] &@ Select[Range[10^7], PowerMod[6, #, #] == 0 &] (* Michael De Vlieger, Dec 31 2016 *)
seq[max_] := Select[Sort[Flatten[Table[2^i*3^j, {i, 1, Floor[Log2[max]]}, {j, 0, Floor[Log[3, max/2^i]]}]]], And @@ PrimeQ[# + {-1, 1}] &]; seq[2*10^15] (* Amiram Eldar, Aug 27 2024 *)
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 *)
A078884
Greater member p of a twin prime pair such that p-1 is 3-smooth.
Original entry on oeis.org
5, 7, 13, 19, 73, 109, 193, 433, 1153, 2593, 139969, 472393, 786433, 995329, 57395629, 63700993, 169869313, 4076863489, 10871635969, 2348273369089, 56358560858113, 79164837199873, 84537841287169, 150289495621633
Offset: 1
A000040(21)=73 and 73-1=72=2^3*3^2=A003586(17) and 73-2=71=A000040(20), therefore 73 is a term.
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N:= 10^100:
sort(select(t -> isprime(t) and isprime(t-2),
[seq(seq(1+2^i*3^j,i=1..ilog2(floor(N/3^j))),j=0..floor(log[3](N)))])); # Robert Israel, May 14 2018
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1 + Select[With[{n = 10^15}, Sort@ Flatten@ Table[2^p * 3^q, {p, 0, Log2@ n}, {q, 0, Log[3, n/(2^p)]}] ], AllTrue[# + {-1, 1}, PrimeQ] &] (* Michael De Vlieger, May 14 2018 *)
A284202
Numbers m such that phi(sum of divisors of m) = lambda(sum of distinct primes dividing m).
Original entry on oeis.org
3, 6, 10, 22, 34, 142, 214, 382, 862, 2302, 5182, 279934, 944782, 1572862, 1990654, 114791254, 127401982, 339738622, 8153726974, 21743271934, 4696546738174, 112717121716222, 158329674399742, 169075682574334, 300578991243262
Offset: 1
34 is in the sequence because A000010(A000203(34)) = A000010(54) = 18, and
A002322(A008472(34)) = A002322(19) = 18.
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Select[Range[10^6], EulerPhi@ DivisorSigma[1, #] == CarmichaelLambda[Total@ FactorInteger[#][[All, 1]]] &]
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lambda(n) = lcm(znstar(n)[2]); \\ after Charles R Greathouse IV in A002322
sopf(n) = vecsum(factor(n)[,1])
isok(n) = eulerphi(sigma(n)) == lambda(sopf(n)) \\ Indranil Ghosh, Mar 22 2017
A303436
Primes p such that all the composite numbers between p and its next prime have no more than 2 distinct prime factors.
Original entry on oeis.org
2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 43, 47, 53, 71, 79, 97, 107, 157, 191, 223, 431, 499, 673, 1151, 1213, 2591, 51199, 139967, 472391, 703123, 786431, 995327, 57395627, 63700991, 169869311, 4076863487, 10871635967
Offset: 1
157 is in the sequence since it is a prime, and the composite numbers between it and its next prime, 163, have only 2 distinct prime factors: 158 = 2*79, 159 = 3*53, 160 = 2^5*5, 161 = 7*23, and 162 = 2*3^4.
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b[n_] := Max[Map[PrimeNu, Range[n + 1, NextPrime[n] - 1]]]; c[n_] := b[Prime[n]]; a={}; Do[If[c[n] < 3, AppendTo[a, Prime[n]]], {n, 1, 10^7}]; a
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isok(p) = {if (isprime(p), for(c=p+1, nextprime(p+1)-1, if (omega(c) != 2, return(0));); return (1););} \\ Michel Marcus, Apr 26 2018
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