A168003 Orderly numbers (mod tau(n)+3).

7, 37, 67, 97, 127, 157, 255, 277, 307, 337, 367, 397, 457, 487, 547, 577, 607, 727, 757, 787, 877, 907, 915, 937, 967, 997, 1087, 1117, 1237, 1245, 1297, 1327, 1447, 1567, 1597, 1627, 1657, 1747, 1777, 1867, 1905, 1987, 2017, 2125, 2137, 2235, 2287, 2347

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).

Links

Table of n, a(n) for n=1..48.

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

Sequence in context: A123084 A123085 A128471 * A132231 A289353 A221982

Adjacent sequences: A168000 A168001 A168002 * A168004 A168005 A168006

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