[seqfan] Re: Orderly Numbers (and related sequences) submitted

Fri Nov 6 07:31:56 CET 2009

For what it's worth, I think A167408 ought to have the keyword "nice." It is
amazing that for all the years the OEIS has existed, no one had thought of
sending this sequence in before. Also, the more I think about the concept,
the more tantalizing questions that arise. For instance, what is the
smallest orderly number with precisely six values of k? (My calculations
suggest it's either a very large number or does not exist).

Al

On Tue, Nov 3, 2009 at 6:06 AM, Andrew Weimholt
<andrew.weimholt at gmail.com>wrote:

> 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
>
>
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