| Displaying 1-5 of 5 results found.
page
1
Primes p of the form 6n-1 such that p-1 is a semiprime and p+2 is prime or prime squared.
+10
6
5, 11, 23, 47, 59, 107, 167, 179, 227, 347, 359, 839, 1019, 1319, 1367, 1487, 1619, 2027, 2207, 2999, 3119, 3167, 3467, 4127, 4259, 4547, 4787, 4799, 5099, 5639, 5879, 6659, 6779, 6827, 7559, 8819, 10007, 10607, 11699, 12107, 12539, 14387, 14867, 15287, 15647
MATHEMATICA
Select[6*Range[3000]-1, PrimeQ[#]&&PrimeOmega[#-1]==2&&AnyTrue[ {#+2, Sqrt[ #+2]}, PrimeQ]&] (* Harvey P. Dale, Jul 01 2022 *)
3, 7, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 173, 181, 191, 193, 197, 199, 211, 223, 229, 233, 239, 241, 251, 257, 263, 269, 271
COMMENTS
Except for term 5, the sequence contains all greater of twin primes
7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609, 1621, 1669, 1699, 1723, 1789, 1873, 1879, 1933, 1951, 1999
COMMENTS
For a(n) > 5, first difference of the sequence is divisible by 6. (Conjectured or proved?)
Also for a(n)>5, a(n)-1 is divisible by 6, if a(n)-2 is prime p such that p+1 is divisible by 6.
MAPLE
isA006512 := proc(p) isprime(p) and isprime(p-2) ; end proc:
isA000430 := proc(p) if isprime(p) then true; else if issqr(p) then isprime(sqrt(p)) ; else false; end if; end if; end proc:
isA181602 := proc(p) if isprime(p) then if numtheory[bigomega](p-1) =2 and isA000430(p+2) then true; else false; end if; else false; end if ; end proc:
isA181669 := proc(p) isA181602(p) and (p mod 6)= 5 ; end proc:
isA172240 := proc(n) isprime(n) and not isA181669(n) ; end proc:
isA173176 := proc(n) isA172240(n) and isA006512(n) ; end proc:
for n from 2 to 2000 do if isA173176(n) then printf("%d, ", n) ; end if; end do:
3, 17, 29, 41, 71, 101, 137, 149, 191, 197, 239, 269, 281, 311, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1427, 1451, 1481, 1607, 1667, 1697, 1721, 1787, 1871, 1877, 1931, 1949, 1997
COMMENTS
For a(n) > 3, the first differences of the sequence are divisible by 6. (Is this a conjecture or a theorem?)
MAPLE
isA001359 := proc(p) isprime(p) and isprime(p+2) ; end proc:
isA000430 := proc(p) if isprime(p) then true; else if issqr(p) then isprime(sqrt(p)) ; else false; end if; end if; end proc:
isA181669 := proc(p) if isprime(p) and (p mod 6)= 5 then if numtheory[bigomega](p-1) =2 and isA000430(p+2) then true; else false; end if; else false; end if ; end proc:
isA172240 := proc(n) isprime(n) and not isA181669(n) ; end proc:
isA172487 := proc(n) isA172240(n) and isA001359(n) ; end proc:
for n from 2 to 2000 do if isA172487(n) then printf("%d, ", n) ; end if; end do:
Numbers m such that m-2, m-1, m+1, m+2 cannot all be represented in the form x*y + x + y for values x, y with x >= y > 1.
+10
3
2, 3, 4, 5, 8, 11, 59, 1319, 1619, 4259, 5099, 6659, 6779, 11699, 12539, 21059, 66359, 83219, 88259, 107099, 110879, 114659, 127679, 130199, 140759, 141959, 144539, 148199, 149519, 157559, 161339, 163859, 175079, 186479, 204599, 230939, 249539, 267959, 273899, 312839
COMMENTS
Indices of terms surrounded by pairs of zeros in A255361. Conjectures:
2. All terms > 8 are primes.
For n > 4, a(n) is not a term of A254636. This means that a(n)-2, a(n)-1, a(n)+1 and a(n)+2 are adjacent terms in A254636. Number of terms < 10^k: 5, 7, 7, 13, 19, 96, 441, 2552, ...
Conjecture 2 would follow if we establish the equivalence "t is in sequence" <=> "t is a term of b(n): lesser of twin primes pair p and q such that (p - 1)/2 and (q + 1)/2 are also a pair of twin primes ( A077800)". It appears that b(n) = a(n) for n > 5. Verified for all terms < 10^9. (End)
EXAMPLE
9, 10, 12, 13 cannot be represented as x*y + x + y, where x >= y > 1. Therefore 11 is in the sequence.
Search completed in 0.005 seconds
|