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

7, 19, 41, 103, 137, 199, 307, 349, 491, 739, 823, 919, 1013, 1061, 1193, 1277, 1289, 1409, 1433, 1447, 1481, 1543, 1609, 1667, 1721, 1747, 2153, 2357, 2441, 2617, 2683, 2777, 3259, 3319, 3463, 3581, 3593, 3769, 3797, 3911, 3943, 4013, 4217, 4423, 4457

Offset: 1

Author

Benoit Cloitre, Mar 02 2002

Keywords

• easy
• nonn

Comments

Primes p such that tau(p+1) = tau(p-1) where tau(k) = A000005(k).

These are the primes in sequence A067888 of numbers n such that tau(n+1) = tau(n-1). - M. F. Hasler, Aug 06 2015

Examples

			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.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..10000

Crossrefs

Cf. A067888, A067891 (analog with sigma).

Programs

Maple

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);

Mathematica

Prime[ Select[ Range[ 700 ], Length[ Divisors[ Prime[ #1 ] - 1 ]] == Length[ Divisors[ Prime[ #1 ] + 1 ]] & ]]
Select[Prime[Range[1000]],DivisorSigma[0,#-1]==DivisorSigma[0,#+1]&] (* Harvey P. Dale, Jun 08 2018 *)

PARI

is_A067889(p)=numdiv(p-1)==numdiv(p+1)&&isprime(p) \\ M. F. Hasler, Jul 31 2015

Formula

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]