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%I #27 Apr 11 2022 21:45:19
%S 5,17,65537,9632244737,20892967937,127831991297,149255504897,
%T 159667373057,351108391937,542497063937,1650957730817,2270398022657,
%U 2322380932097,2747956028417,2888694547457,3516735087617,6029264167937,6122338640897,6705696695297,11125266727937
%N Primes p such that tau(p - 1) - 1 = tau(p - 2) = tau(p - 3), where tau(k) is the number of divisors of k (A000005).
%C Corresponding values of tau(a(n)-1): 3, 5, 17, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, ...
%C Corresponding values of tau(a(n)-2) = tau(a(n)-3): 2, 4, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, ...
%C Quadruples of [tau(a(n)-3), tau(a(n)-2), tau(a(n)-1), tau(a(n))]: [2, 2, 3, 2], [4, 4, 5, 2], [16, 16, 17, 2], [32, 32, 33, 2], [32, 32, 33, 2], [32, 32, 33, 2], [32, 32, 33, 2], [32, 32, 33, 2], [32, 32, 33, 2], ...
%C Quadruple [32, 32, 33, 2] holds for all 128 terms 65537 < a(n) < 10^15.
%C Number p-1 is a perfect square as its number of divisors is odd.
%C The first 3 terms are Fermat primes from A019434.
%C Term 103565955613697 is the smallest primes p such that tau(p - 1) - 1 = tau(p - 2) = tau(p - 3) = tau(p - 4).
%e Quadruple of [tau(65534), tau(65535), tau(65536), tau(65537)]: [16, 16, 17, 2].
%o (Magma) [m: m in [4..10^6] | IsPrime(m) and #Divisors(m - 1) eq #Divisors(m - 2) + 1 and #Divisors(m - 2) eq #Divisors(m - 3)]
%Y Subsequence of A347078.
%Y Cf. A000005 (tau), A019434.
%K nonn
%O 1,1
%A _Jaroslav Krizek_, Mar 03 2022