A362631 - OEIS

%I #26 Jun 23 2023 07:51:00

%S 1,2,3,4,6,5,9,10,12,15,8,20,7,25,14,30,21,35,18,28,24,42,11,49,22,56,

%T 33,63,44,70,55,77,40,66,45,88,50,99,60,110,27,121,36,132,13,143,16,

%U 26,17,39,34,52,51,65,68,78,85,91,102,104,119,117,136,130,153,156,170,169,187,182,204,195,221,75,238,80,255,32,272,19

%N Lexicographically earliest infinite sequence of distinct positive integers with a(n) = n for n <= 3, and for n > 3  a(n) is the least novel multiple of the greatest prime divisor of a(n-2) which does not divide a(n-1).

%C The definition reflects that of A098550 in that it places a condition on a(n-2) which does not apply to a(n-1).

%C If there is no prime divisor of a(n-2) which does not divide a(n-1), then by empty product convention a(n) = u the least unused number.

%C Some primes (23,29,31,47,...) enter because of dividing a(n-2) but not a(n-1), whereas others (5,7,11,13,17,19,...) enter as least u; see Example.

%C With the exception of 16 all least u terms (up to a(2^28)) are primes, so it seems likely that a(47) = 16 is a one-off (fluke) term.

%C The scatterplot resembles a fine-toothed comb similar to those of A361629, A361133 and A361534, in which each "tooth" starts with a novel prime p and continues with a run of (mostly) alternate multiples of p and the greatest prime less than p until interrupted by the arrival of the next prime, and so forth.

%C The sequence, after a(1) = 1 can be represented as an irregular table in which the n-th row starts with prime(n), see Example.

%C Conjectured to be a permutation of the positive integers with the primes in order.

%H Michael De Vlieger, <a href="/A362631/b362631.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A362631/a362631.png">Scatterplot of a(n)</a> n = 1..10000.

%H Michael De Vlieger, <a href="/A362631/a362631_1.png">Log log scatterplot of a(n)</a> n = 1..2^20.

%H Michael De Vlieger, <a href="/A362631/a362631_2.png">Log log scatterplot of a(n)</a>, n = 1..10000, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue. We accentuate powerful numbers that are not prime powers in light blue.

%e a(4,5) = 4, 6 and since rad(4)|rad(6) a(6) = least u = 5.

%e a(11,12) = 8, 20 and since rad(8)|rad(20) a(13) = least u = 7.

%e a(44,45) = 132, 13 and gpd(132) = 11 does not divide 13, and since it is the 13th occurrence of p = 11, a(46) = 13*11 = 143.

%e a(45,46) = 13, 143 which forces a(47) = least u = 16 (see Comment).

%e a(90,91) = 69, 114 and 23 is the greatest prime dividing 69 which does not divide 114. Since 23 has not appeared earlier in the sequence a(92) = 23.

%e There is as yet no known formula for the row lengths of the table below. Whereas most rows terminate with a multiple of the prime they start with, there are exceptions, e.g., 47, 109. This behavior is open to explanation.

%e The table starts:

%e 2;

%e 3,4,6;

%e 5,9,10,12,15,8,20;

%e 7,25,14,30,21,35,18,28,24,42;

%e 11,49,22,56,33,63,44,70,55,77,40,66,45,88,50,99,60,110,27,121,36,132;

%e 13,143,16,26;

%e 17,39,34,52,51,65,68,78,85,91,102,104,119,117,136,130,153,156,170,169,187,182,204,195,221,75,238,80,255,32,272;

%e 19,....

%t mm = 3; nn = 10^4; c[_] := False; q[_] := 1;

%t Array[Set[{a[#], c[#]}, {#, True}] &, mm]; q[2]++; u = mm + 1;

%t i = a[mm - 1]; j = a[mm]; s = {1};

%t Do[t = FactorInteger[i][[All, 1]];

%t   If[SubsetQ[s, t], q[u]++; Set[k, u],

%t    p = SelectFirst[Reverse@ t, CoprimeQ[j, #] &];

%t    If[! IntegerQ[p], Set[k, u], While[c[p*q[p]], q[p]++]; k = p*q[p]] ];

%t   Set[{a[n], c[k], i, j, s}, {k, True, j, k, t}];

%t   If[k == u, While[c[u], u++]], {n, mm + 1, nn}];

%t Array[a, nn]

%Y Cf. A007947, A098550, A361629, A361133, A361534, A362855.

%K nonn

%O 1,2

%A _David James Sycamore_, May 06 2023