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A220890
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a(n) = smallest m such that A187824(m) = n, or -1 if A187824 never takes the value n.
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4
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-1, -1, -1, 2, 3, 4, 5, 29, 41, 55, 71, 881, 791, 9360, 10009, 1079, 30239, -1, 246960, -1, 636481, 1360800, 3160079, -1, 2162161
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OFFSET
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0,4
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COMMENTS
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a(17) = -1. Proof: If x mod 9 and x mod 12 are both in {-1, 0, 1} then so is x mod 18. So if x is a number which is congruent to -1, 0 or 1 mod k for k=1..17, then also x mod 18 is congruent to -1, 0 or 1. So there is no x such that A187824(x) = 17. QED
Similarly, a(19) = -1. Indeed, if x = 0, 1 or -1 (mod 15) and (mod 12), then also (mod 60). [Proof: Write x = 15(4k+d)+e, |e|<2, then d = 1, 2, 3 all give impossible x (mod 12).] Therefore A187824 cannot have the value 19 (nor 29, nor 59).
Also, a(23) = -1, because x = 0, 1 or -1 (mod 8) and (mod 12) implies the same (mod 24). [To see this, write x = 12(2k+d)+e, |e|<2, then d = 1 gives impossible x (mod 8).] Therefore A187824 cannot have the value 23.
From A220891 one may deduce the values for n = 26, 28, 31, 36, 40, 42, 46, 48, 52, 58, 60, 61 to be a(n) = 39412801, 107881201, 3625549201, 170918748000, 2355997644001, 237662810985599, 4614209634434399, 7522575180120001, 362645725505263201, 10684484093105222399, 442709913651892286399, 5205240636387758366399. (End)
Don Reble shows that a(n) > -1 iff n + 1 is either 12, 2p, 3p or p^k > 3, where p is a prime, k >= 1. - M. F. Hasler, Mar 17 2020
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LINKS
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MAPLE
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N:= 70: # maximum m
V[0]:= -1: V[1]:= -1: V[2]:= -1:
S[3]:= {$0..5}: M[3]:= 6:
# M[m] is the lcm of 1..m
# S[m] is the set of residues mod M[m] for numbers n with A187824(n)>=m
# A[m] is the set of residues mod M[m] for numbers n with A187824(n)=m-1
for m from 4 to N+1 do
M[m]:= ilcm(M[m-1], m); p:= M[m]/M[m-1];
if p = 1 then T:= S[m-1]
else T:= {seq(seq(a+b*M[m-1], a=S[m-1]), b=0..p-1)}
end if;
S[m], A[m]:= selectremove(t -> member(mods(t, m), {1, 0, -1}), T);
if A[m] = {} then V[m-1]:= -1
else V[m-1]:= min(A[m])
end if;
end do:
seq(V[j], j=0..N);
# Robert Israel, Dec 31 2012
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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a(26) = 39412801. Double-checked all lower given values. - M. F. Hasler, Dec 30 2012
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STATUS
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approved
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