[seqfan] Orderly Numbers (and related sequences) submitted

Tue Nov 3 12:06:56 CET 2009

I've submitted the "Orderly Numbers", "Very Orderly Numbers",
"Disorderly Numbers", and "Minimal K Values for Orderly Numbers"
sequences.

%I A167408
%S A167408 1,2,5,7,8,9,11,12,13,17,19,20,23,27,29,31,37,38,41,43,47,52,53,57,58,
%T A167408 59,61,67,68,71,72,73,76,79,83,87,89,97,101,103,107,109,113,117,118,124,
%U A167408 127,131,133,137,139,149,151,157,158,162,163,164,167,173,177,178,179
%N A167408 Orderly Numbers: a number, N, is orderly if there exists
some number K > tau(N), such that the set of the divisors of N is
congruent to the set {1,2,...,tau(N)} mod K.
%C A167408 . N: {divisors(N)} == {1,2,...,tau(N)} mod K
%C A167408 . -------------------------------------------
%C A167408 . 1: {1} == {1} mod 2
%C A167408 . 2: {1,2} == {1,2} mod 3
%C A167408 . 5:  {1,5} == {1,2} mod 3
%C A167408 . 7:  {1,7} == {1,2} mod 5
%C A167408 . 8:  {1,2,8,4} == {1,2,3,4} mod 5
%C A167408 . 9:  {1,9,3} == {1,2,3} mod 7
%C A167408 . 11:  {1,11} == {1,2} mod 3 or 9
%C A167408 . 12:  {1,2,3,4,12,6} == {1,2,3,4,5,6} mod 7
%C A167408 . 13:  {1,13} == {1,2} mod 11
%C A167408 . 17:  {1,17} == {1,2} mod 3,5, or 15
%C A167408 . 19:  {1,19} == 1,2 mod 17
%C A167408 . 20:  {1,2,10,4,5,20} == {1,2,3,4,5,6} mod 7
%C A167408 . 23:  {1,23} == {1,2} mod 3,7, or 21
%C A167408 . 27:  {1,27,3,9} == {1,2,3,4} mod 5
%C A167408 . 29:  {1,29} == {1,2} mod 3,9, or 27
%C A167408 . 31:  {1,31} == {1,2} mod 29
%C A167408 . 37:  {1,37} == 1,2 mod 5,7, or 35
%C A167408 . 38:  {1,2,38,19} == {1,2,3,4} mod 5
%C A167408 . 41:  {1,41} == {1,2} mod 3,13, or 39
%C A167408 . 43:  {1,43} == {1,2} mod 41
%C A167408 . 47:  {1,47} == {1,2} mod 3,5,9,15, or 45
%C A167408 . 52:  {1,2,52,4,26,13} == {1,2,3,4,5,6} mod 7
%C A167408 . 53:  {1,53} == {1,2} mod 3,17, or 51
%C A167408 . 57:  {1,57,3,19} == {1,2,3,4} mod 5
%C A167408 . 58:  {1,2,58,29} == {1,2,3,4} mod 5
%C A167408 . 59:  {1,59} == {1,2} mod 3,19, or 57
%C A167408 . 61:  {1,61} == {1,2} mod 59
%C A167408 . 67:  {1,67} == {1,2} mod 5,13, or 65
%C A167408 . 68:  {1,2,17,4,68,34} == {1,2,3,4,5,6} mod 7
%C A167408 . 71:  {1,71} == {1,2} mod 3,23, or 69
%C A167408 . 72:  {1,2,3,4,18,6,72,8,9,36,24,12} ==
{1,2,3,4,5,6,7,8,9,10,11,12} mod 13
%C A167408 . 73:  {1,73} == {1,2} mod 71
%C A167408 . 76:  {1,2,38,4,19,76} == {1,2,3,4,5,6} mod 7
%C A167408 . 79:  {1,79} == {1,2} mod 7,11, or 77
%C A167408 . 83:  {1,83} == {1,2} mod 3,9,27, or 81
%C A167408 . 87:  {1,87,3,29} == {1,2,3,4} mod 5
%C A167408 . 89:  {1,89} == {1,2} mod 3,29, or 87
%C A167408 . 97:  {1,97} == {1,2} mod 5,19, or 95
%C A167408
%C A167408 The primes, except for 3, are orderly.
%C A167408 Numbers of the form 4p are orderly when p is an odd prime
congruent to 3,5, or 6 mod 7.
%C A167408 For primes, K values can be p-2, or a divisor of p-2 other than 1.
%C A167408
%C A167408 T. D. Noe observed that for composite orderly numbers, N,
%C A167408 K seems to be one of the three values: tau(n)+1, tau(n)+3, tau(n)+4.
%C A167408
%C A167408 The composite numbers with K = tau(N)+4 are of the form
%C A167408 . p^2, where prime p == 3 mod 7.
%C A167408
%C A167408 The composite orderly numbers with K = tau(N)+3, come in
the following forms for K <= 67
%C A167408
%C A167408 . p*q*r with primes {p,q,r} == {3,5,6} mod 11
%C A167408 . p^3*q with primes {p,q} == {5,6} mod 11
%C A167408 . p^3*q with primes {p,q} == {6,5} mod 11
%C A167408 . p^4*q with primes {p,q} == {7,6} mod 13
%C A167408 . p*q*r*s with primes {p,q,r,s} == {5,6,9,10} mod 19
%C A167408 . p^3*q*r with primes {p,q,r} == {5,9,10} mod 19
%C A167408 . p^3*q*r with primes {p,q,r} == {9,6,10} mod 19
%C A167408 . p^3*q*r with primes {p,q,r} == {10,6,9} mod 19
%C A167408 . p*q*r*s*t*u with primes {p,q,r,s,t,u} == {17,21,33,34,39,47} mod 67
%C A167408 . p*q*r*s*t*u with primes {p,q,r,s,t,u} == {19,34,35,36,49,56} mod 67
%C A167408
%C A167408 Note that 11, 19, and 67, are primes of the form 2^x+3.
%C A167408
%C A167408 The forms for composite orderly numbers with K = tau(N)+1
are too numerous to list here, but seem to occur for any prime K > 3.
%C A167408
%e A167408 12 is an orderly number because 12's divisors are 1,2,3,4,5,6 and
%e A167408 . 1 == 1 mod 7
%e A167408 . 2 == 2 mod 7
%e A167408 . 3 == 3 mod 7
%e A167408 . 4 == 4 mod 7
%e A167408 .12 == 5 mod 7
%e A167408 . 6 == 6 mod 7
%e A167408
%Y A167408 Cf. A167409 - Very Orderly Numbers ( K = tau(N)+1 )
%Y A167408 Cf. A167410 - Disorderly Numbers - numbers not in this sequence
%Y A167408 Cf. A167411 - Minimal K Values for the Orderly Numbers
%Y A167408
%K A167408 nonn
%O A167408 1,2
%A A167408 Andrew Weimholt (andrew(AT)weimholt.com), Nov 03 2009

%I A167409
%S A167409 1,2,5,8,11,12,17,20,23,27,29,38,41,47,52,53,57,58,59,68,71,72,76,83,87,
%T A167409 89,101,107,113,117,118,124,131,133,137,149,158,162,164,167,173,177,178,
%U A167409 179,188,191,197,203,218,227,233,236,237,239,243,244,247,251,257
%N A167409 Very Orderly Numbers: a number, N, is "very orderly" if the
set of the divisors of N is congruent to the set {1,2,...,tau(N)} mod
tau(N)+1.
%C A167409 The very orderly numbers are orderly numbers (Cf. A167408)
with K = tau(N)+1
%o A167409 (PARI)
%o A167409 vo(n)=#(n=divisors(n))==#(n=Set(n%(1+#n))) & n[1]!="0"
%o A167409 for(n=1,999,vo(n)&print1(n", "))
%o A167409 --Maximilian Hasler
%o A167409
%Y A167409 Cf. A167408 - Orderly Numbers
%Y A167409 Cf. A167410 - Disorderly Numbers - numbers not in A167408
%Y A167409 Cf. A167411 - Minimal K Values for the Orderly Numbers
%Y A167409
%K A167409 nonn
%O A167409 1,2
%A A167409 Andrew Weimholt (andrew(AT)weimholt.com), Nov 03 2009

%I A167410
%S A167410 3,4,6,10,14,15,16,18,21,22,24,25,26,28,30,32,33,34,35,36,39,40,42,44,
%T A167410 45,46,48,49,50,51,54,55,56,60,62,63,64,65,66,69,70,74,75,77,78,80,81,
%U A167410 82,84,85,86,88,90,91,92,93,94,95,96,98,99,100,102,104,105,106,108
%N A167410 Disorderly Numbers: numbers not in A167408 (orderly numbers).
%e A167410 3 is disorderly because there exists no K > 2=tau(3),
such that {1,3} == {1,2} mod K.
%Y A167410 Cf. A167408 - Orderly Numbers
%Y A167410 Cf. A167409 - Very Orderly Numbers ( K = tau(N)+1 )
%Y A167410 Cf. A167411 - Minimal K Values for the Orderly Numbers
%Y A167410
%K A167410 nonn
%O A167410 1,1
%A A167410 Andrew Weimholt (andrew(AT)weimholt.com), Nov 03 2009

%I A167411
%S A167411 2,3,3,5,5,7,3,7,11,3,17,7,3,5,3,29,5,5,3,41,3,7,3,5,5,3,59,5,7,3,13,71,
%T A167411 7,7,3,5,3,5,3,101,3,107,3,7,5,7,5,3,5,3,137,3,149,5,5,11,7,7,3,3,5,5,3,
%U A167411 179,7,3,191,3,197,5,11,5,13,3,227,3,7,5,3,239,7,7,5,3,11,3,3,5,3
%N A167411 a(n) = the minimal K value for the orderly number A167408(n).
%e A167411 a(6) = 7, because A167408(6) = 9, and divisors of 9 =
{1,9,3} == {1,2,3} mod 7.
%Y A167411 Cf. A167408 - Orderly Numbers
%Y A167411 Cf. A167409 - Very Orderly Numbers ( K = tau(N)+1 )
%Y A167411 Cf. A167410 - Disorderly Numbers - numbers not in A167408
%Y A167411
%K A167411 nonn
%O A167411 1,1
%A A167411 Andrew Weimholt (andrew(AT)weimholt.com), Nov 03 2009

Andrew