
FORMULA

An exhaustive search over forms of n having a prime value of tau(n)+3 finds that terms of this sequence satisfy the following congruences for tau(n)+3 < 60.
. p with prime p = 2 mod 5
. p^3*q with primes {p,q} == {5,6} mod 11
. p^3*q with primes {p,q} == {6,5} mod 11
. p*q*r with primes {p,q,r} == {3,5,6} mod 11
. p^4*q with primes {p,q} == {7,6} mod 13
. p^7*q with primes {p,q} == {9,10} mod 19
. p^7*q with primes {p,q} == {10,9} mod 19
. p^3*q*r with primes {p,q,r} == {5,9,10} mod 19
. p^3*q*r with primes {p,q,r} == {9,6,10} mod 19
. p^3*q*r with primes {p,q,r} == {10,6,9} mod 19
. p*q*r*s with primes {p,q,r,s} == {5,6,9,10} mod 19
. p^12*q with primes {p,q} == {15,14} mod 29
. p^16*q with primes {p,q} == {19,18} mod 37
. p^4*q*r*s with primes {p,q,r,s} == {14,13,15,22} mod 43
. p^4*q*r*s with primes {p,q,r,s} == {31,22,24,38} mod 43
. p^24*q with primes {p,q} == {27,26} mod 53
. p^4*q^4*r with primes {p,q,r} == {5,27,26} mod 53
. p^27*q with primes {p,q} == {29,30} mod 59
. p^27*q with primes {p,q} == {30,29} mod 59
. p^13*q*r with primes {p,q,r} == {15,29,30} mod 59
. p^13*q*r with primes {p,q,r} == {29,30,36} mod 59
. p^13*q*r with primes {p,q,r} == {30,29,36} mod 59
. p^6*q^3*r with primes {p,q,r} == {29,53,30} mod 59
. p^6*q^3*r with primes {p,q,r} == {30,6,29} mod 59
. p^6*q^3*r with primes {p,q,r} == {48,29,30} mod 59
. p^6*q^3*r with primes {p,q,r} == {48,30,29} mod 59
. p^6*q*r*s with primes {p,q,r,s} == {7,28,30,45} mod 59
. p^6*q*r*s with primes {p,q,r,s} == {15,29,30,36} mod 59
. p^6*q*r*s with primes {p,q,r,s} == {29,30,36,53} mod 59
. p^6*q*r*s with primes {p,q,r,s} == {30,6,29,36} mod 59
. p^6*q*r*s with primes {p,q,r,s} == {48,15,29,30} mod 59
Andrew Weimholt found some of these forms.
