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A067889 Primes sandwiched between two numbers having same number of divisors.

%I A067889 #30 Jun 28 2022 02:57:11
%S A067889 7,19,41,103,137,199,307,349,491,739,823,919,1013,1061,1193,1277,1289,
%T A067889 1409,1433,1447,1481,1543,1609,1667,1721,1747,2153,2357,2441,2617,
%U A067889 2683,2777,3259,3319,3463,3581,3593,3769,3797,3911,3943,4013,4217,4423,4457
%N A067889 Primes sandwiched between two numbers having same number of divisors.
%C A067889 Primes p such that tau(p+1) = tau(p-1) where tau(k) = A000005(k).
%C A067889 These are the primes in sequence A067888 of numbers n such that tau(n+1) = tau(n-1). - _M. F. Hasler_, Aug 06 2015
%H A067889 T. D. Noe, <a href="/A067889/b067889.txt">Table of n, a(n) for n=1..10000</a>
%F A067889 a(n) seems curiously to be asymptotic to 25*n*log(n). [From the number of terms up to 10^8, 10^9, 10^10 and 10^11, i.e., 306147, 2616930, 22835324 and 202105198, this constant can be estimated by 25.858..., 25.858..., 25.845... and 25.872..., respectively. - _Amiram Eldar_, Jun 28 2022]
%e A067889 7 is a member as 6 and 8 both have 4 divisors; 19 is a member as 18 and 20 both have 6 divisors each.
%p A067889 with(numtheory):j := 0:for i from 1 to 10000 do b := ithprime(i): if nops(divisors(b-1))=nops(divisors(b+1)) then j := j+1:a[j] := b:fi:od:seq(a[k],k=1..j);
%t A067889 Prime[ Select[ Range[ 700 ], Length[ Divisors[ Prime[ #1 ] - 1 ]] == Length[ Divisors[ Prime[ #1 ] + 1 ]] & ]]
%t A067889 Select[Prime[Range[1000]],DivisorSigma[0,#-1]==DivisorSigma[0,#+1]&] (* _Harvey P. Dale_, Jun 08 2018 *)
%o A067889 (PARI) is_A067889(p)=numdiv(p-1)==numdiv(p+1)&&isprime(p) \\ _M. F. Hasler_, Jul 31 2015
%Y A067889 Cf. A067888, A067891 (analog with sigma).
%K A067889 easy,nonn
%O A067889 1,1
%A A067889 _Benoit Cloitre_, Mar 02 2002