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%I #4 Nov 09 2018 15:37:44
%S 7,37,67,97,127,157,255,277,307,337,367,397,457,487,547,577,607,727,
%T 757,787,877,907,915,937,967,997,1087,1117,1237,1245,1297,1327,1447,
%U 1567,1597,1627,1657,1747,1777,1867,1905,1987,2017,2125,2137,2235,2287,2347
%N Orderly numbers (mod tau(n)+3).
%C 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).
%F 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.
%F . p with prime p = 2 mod 5
%F . p^3*q with primes {p,q} == {5,6} mod 11
%F . p^3*q with primes {p,q} == {6,5} mod 11
%F . p*q*r with primes {p,q,r} == {3,5,6} mod 11
%F . p^4*q with primes {p,q} == {7,6} mod 13
%F . p^7*q with primes {p,q} == {9,10} mod 19
%F . p^7*q with primes {p,q} == {10,9} mod 19
%F . p^3*q*r with primes {p,q,r} == {5,9,10} mod 19
%F . p^3*q*r with primes {p,q,r} == {9,6,10} mod 19
%F . p^3*q*r with primes {p,q,r} == {10,6,9} mod 19
%F . p*q*r*s with primes {p,q,r,s} == {5,6,9,10} mod 19
%F . p^12*q with primes {p,q} == {15,14} mod 29
%F . p^16*q with primes {p,q} == {19,18} mod 37
%F . p^4*q*r*s with primes {p,q,r,s} == {14,13,15,22} mod 43
%F . p^4*q*r*s with primes {p,q,r,s} == {31,22,24,38} mod 43
%F . p^24*q with primes {p,q} == {27,26} mod 53
%F . p^4*q^4*r with primes {p,q,r} == {5,27,26} mod 53
%F . p^27*q with primes {p,q} == {29,30} mod 59
%F . p^27*q with primes {p,q} == {30,29} mod 59
%F . p^13*q*r with primes {p,q,r} == {15,29,30} mod 59
%F . p^13*q*r with primes {p,q,r} == {29,30,36} mod 59
%F . p^13*q*r with primes {p,q,r} == {30,29,36} mod 59
%F . p^6*q^3*r with primes {p,q,r} == {29,53,30} mod 59
%F . p^6*q^3*r with primes {p,q,r} == {30,6,29} mod 59
%F . p^6*q^3*r with primes {p,q,r} == {48,29,30} mod 59
%F . p^6*q^3*r with primes {p,q,r} == {48,30,29} mod 59
%F . p^6*q*r*s with primes {p,q,r,s} == {7,28,30,45} mod 59
%F . p^6*q*r*s with primes {p,q,r,s} == {15,29,30,36} mod 59
%F . p^6*q*r*s with primes {p,q,r,s} == {29,30,36,53} mod 59
%F . p^6*q*r*s with primes {p,q,r,s} == {30,6,29,36} mod 59
%F . p^6*q*r*s with primes {p,q,r,s} == {48,15,29,30} mod 59
%F _Andrew Weimholt_ found some of these forms.
%K nonn
%O 1,1
%A _T. D. Noe_, Nov 16 2009
%E Comment corrected and congruences mod 43 added by _T. D. Noe_, Dec 02 2009
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