OFFSET
1,1
COMMENTS
See A167408 for information about orderly numbers. It appears that when n is in this sequence, then tau(n)+3 must be a prime p such that 2 is not a square mod p (A003629). For each one of those primes, it is possible to find all forms of n that are orderly. In particular, the form n=p^k*q is in this sequence when 2k+5 is in A001122. In that case, we have the congruences p=2+tau(n)/2 and q=1+tau(n)/2 (mod tau(n)+3). When tau(n) is a multiple of 8, then another pair of congruences is p=1+tau(n)/2 and q=2+tau(n)/2 (mod tau(n)+3).
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.
CROSSREFS
KEYWORD
nonn
AUTHOR
T. D. Noe, Nov 16 2009
EXTENSIONS
Comment corrected and congruences mod 43 added by T. D. Noe, Dec 02 2009
STATUS
approved