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%I A280864 #219 Mar 08 2025 04:03:49
%S A280864 1,2,4,3,6,8,5,10,12,9,7,14,16,11,22,18,15,20,24,21,28,26,13,17,34,30,
%T A280864 45,19,38,32,23,46,36,27,25,35,42,48,29,58,40,55,33,39,52,44,77,49,31,
%U A280864 62,50,65,78,54,37,74,56,63,51,68,60,75,41,82,64,43,86
%N A280864 Lexicographically earliest infinite sequence of distinct positive terms such that, for any prime p, any run of consecutive multiples of p has length exactly 2.
%C A280864 In other words, each multiple of a prime p has exactly one neighbor that is also a multiple of p.
%C A280864 This sequence is similar to A280866; the first difference occurs at n=42: a(42)=55 whereas A280866(42)=50.
%C A280864 Conjectured to be a permutation of the positive integers.
%C A280864 Sometimes referred to as the "cup of coffee" sequence, since it feels as if just one more cup of coffee is all it would take to prove that this is indeed a permutation of the positive integers. - _N. J. A. Sloane_, Nov 04 2020
%C A280864 There are several short cycles, and apparently at least two infinite cycles. For a list see the attached file "Properties of A280864". - _N. J. A. Sloane_, Feb 03 2017
%C A280864 Properties (For proofs, see the attached file "Properties of A280864")
%C A280864 Theorem 1: This sequence contains every prime and every even number. (Added by _N. J. A. Sloane_, Jan 15 2017)
%C A280864 Theorem 2: The sequence contains infinitely many odd composite numbers. (Added by _N. J. A. Sloane_, Feb 14 2017)
%C A280864 Theorem 3: If p is an odd prime, the sequence contains infinitely many odd multiples of p. (Added by _N. J. A. Sloane_, Mar 12 2017, with corrected proof Apr 03 2017)
%C A280864 There are two types of primes in this sequence: Type I, the first time a term a(n) is divisible by p is when a(n)=p for some n; Type II, the first time a term a(n) is divisible by p is when a(n)=k*p for some n and some k>1 (the Type II primes are listed in A280745).
%C A280864 Conjecture 4: If a prime p divides a(n) then p <= n. - _N. J. A. Sloane_, Apr 07 2017 and Apr 16 2017
%C A280864 Theorem 5: The sequence is a permutation of the natural numbers iff it contains every square. - _N. J. A. Sloane_, Apr 14 2017
%C A280864 From _Bob Selcoe_, Apr 03 2017: (Start)
%C A280864 Define the "radical class" C_R to be the set of numbers which have the same radical R (or the same largest squarefree divisor - i.e., the same product of their prime factors). These are the columns in A284311. So for example C_10 is the set of numbers with radical 10 or prime factors {2,5}: {10, 20, 40, 50, 80, 100, 160, ...}.
%C A280864 If the sequence contains any members of C_R, then those members must appear in order; so for example, if 160 has appeared, {10, 20, 40, 50, 80} will have already appeared, in that order. Naturally, this holds for prime powers; for example, C_5: if 3125 has appeared, {5, 25, 125, 625} will have appeared earlier, in that order.
%C A280864 After seeing a(n), let S be smallest missing number (A280740) and let prime(G) be largest prime already appearing in the sequence. Conjecture: Prime(G) < S <= prime(G+1), and a(35) = 25 = S is the only nonprime S term (following a(31) = 23, preceding a(39) = 29). (End)
%H A280864 N. J. A. Sloane, <a href="/A280864/b280864.txt">Table of n, a(n) for n = 1..100000</a> (First 10000 terms from Rémy Sigrist)
%H A280864 Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, <a href="https://arxiv.org/abs/2012.04625">Finding structure in sequences of real numbers via graph theory: a problem list</a>, arXiv:2012.04625, Dec 08, 2020
%H A280864 Rémy Sigrist, <a href="/A280864/a280864.gp.txt">PARI program for A280864</a>
%H A280864 N. J. A. Sloane, <a href="/A280864/a280864_5.txt">Properties of A280864</a> [Revised, Apr 25 2017]
%H A280864 N. J. A. Sloane, <a href="/A280864/a280864.txt">Table of n, a(n) for n = 1..1000000</a>, computed using Sigrist's PARI program.
%H A280864 N. J. A. Sloane, <a href="/A195264/a195264.pdf">Confessions of a Sequence Addict (AofA2017)</a>, slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.
%e A280864 The first terms, alongside their required and forbidden prime factors are:
%e A280864 n a(n) Required Forbidden
%e A280864 -- ---- -------- ---------
%e A280864 1 1 none none
%e A280864 2 2 none none
%e A280864 3 4 2 none
%e A280864 4 3 none 2
%e A280864 5 6 3 none
%e A280864 6 8 2 3
%e A280864 7 5 none 2
%e A280864 8 10 5 none
%e A280864 9 12 2 5
%e A280864 10 9 3 2
%e A280864 11 7 none 3
%e A280864 12 14 7 none
%e A280864 13 16 2 7
%e A280864 14 11 none 2
%e A280864 15 22 11 none
%e A280864 16 18 2 11
%e A280864 17 15 3 2
%e A280864 18 20 5 3
%e A280864 19 24 2 5
%e A280864 20 21 3 2
%e A280864 21 28 7 3
%e A280864 22 26 2 7
%e A280864 23 13 13 2
%e A280864 24 17 none 13
%e A280864 25 34 17 none
%e A280864 26 30 2 17
%e A280864 27 45 3, 5 2
%e A280864 28 19 none 3, 5
%e A280864 29 38 19 none
%e A280864 30 32 2 19
%e A280864 31 23 none 2
%e A280864 32 46 23 none
%e A280864 33 36 2 23
%e A280864 34 27 3 2
%e A280864 35 25 none 3
%e A280864 36 35 5 none
%e A280864 37 42 7 5
%e A280864 38 48 2, 3 7
%e A280864 39 29 none 2, 3
%e A280864 40 58 29 none
%e A280864 41 40 2 29
%e A280864 42 55 5 2
%p A280864 N:= 1000: # to get all terms until the first term > N
%p A280864 A[1]:= 1:
%p A280864 A[2]:= 2:
%p A280864 G:= {}:
%p A280864 Avail:= [$3..N]:
%p A280864 found:= true:
%p A280864 lastn:= 2:
%p A280864 for n from 3 while found and nops(Avail)>0 do
%p A280864 found:= false;
%p A280864 H:= G;
%p A280864 G:= numtheory:-factorset(A[n-1]);
%p A280864 r:= convert(G minus H,`*`);
%p A280864 s:= convert(G intersect H, `*`);
%p A280864 for j from 1 to nops(Avail) do
%p A280864 if Avail[j] mod r = 0 and igcd(Avail[j],s) = 1 then
%p A280864 found:= true;
%p A280864 A[n]:= Avail[j];
%p A280864 Avail:= subsop(j=NULL,Avail);
%p A280864 lastn:= n;
%p A280864 break
%p A280864 fi
%p A280864 od;
%p A280864 od:
%p A280864 seq(A[i],i=1..lastn); # _Robert Israel_, Mar 22 2017
%t A280864 terms = 100;
%t A280864 rad[n_] := Times @@ FactorInteger[n][[All, 1]];
%t A280864 A280864 = Reap[present = 0; p = 1; pp = 1; Do[forbidden = GCD[p, pp]; mandatory = p/forbidden; a = mandatory; While[BitGet[present, a] > 0 || GCD[forbidden, a] > 1, a += mandatory]; Sow[a]; present += 2^a; pp = p; p = rad[a], terms]][[2, 1]] (* _Jean-François Alcover_, Nov 23 2017, translated from _Rémy Sigrist_'s PARI program *)
%Y A280864 See A280738, A280740, A280741 (inverse), A280742, A280743, A280744, A280745, A280746, A280755, A280770, A280771, A280773, A280774, A283832, A284724, A284725, A284726, A284785, A285181 for various subsidiary sequences.
%Y A280864 A280754 gives fixed points.
%Y A280864 Cf. A280866.
%Y A280864 In the same spirit as A064413 and A098550.
%Y A280864 A338338, A338444, and A375029 are variants.
%Y A280864 A373797 is a finite version.
%K A280864 nonn,nice
%O A280864 1,2
%A A280864 _Rémy Sigrist_, Jan 09 2017
%E A280864 Added "infinite" to definition. - _N. J. A. Sloane_, Sep 28 2019