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Powers of primes (A000961) whose neighbors have a prime number as their greatest odd divisor.
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%I #20 Feb 03 2024 10:14:16

%S 4,11,13,23,25,27,47,81,193,383,2187,1594323

%N Powers of primes (A000961) whose neighbors have a prime number as their greatest odd divisor.

%C From _Jon E. Schoenfield_, Jan 28 2024: (Start)

%C If it exists, a(13) > 10^2000.

%C Conjecture: a(12) = 1594323 is the final term of the sequence.

%C (End)

%e (prime = greatest odd divisor of a(n)-1; a(n); prime = greatest odd divisor of a(n)+1): (3; 4; 5), (5; 11; 3), (3; 13; 7), (11; 23; 3), (3; 25; 13), (13; 27; 7), (23; 47; 3), (5; 81; 41), (3; 193; 97), (191; 383; 3), (1093; 2187; 547), (797161; 1594323; 398581).

%t q[n_] := PrimeQ[n/2^IntegerExponent[n, 2]]; Select[Range[2*10^6], PrimePowerQ[#] && And @@ q /@ {# - 1, # + 1} &] (* _Amiram Eldar_, Jan 28 2024 *)

%o (Magma) [k: k in [2..1600000] | #Divisors(2*k-2)-#Divisors(k-1) eq 2 and

%o #PrimeDivisors(k) eq 1 and #Divisors(2*k+2)-#Divisors(k+1) eq 2];

%Y Intersection of A000961 and A369329.

%Y Cf. A038550.

%Y Comparable sequences: A275598, A343973.

%K nonn,more

%O 1,1

%A _Juri-Stepan Gerasimov_, Jan 27 2024