A379248 - OEIS

%I #34 Dec 22 2024 09:06:26

%S 1,2,4,6,8,10,12,9,3,18,15,25,5,50,16,14,20,22,24,26,28,30,27,21,36,

%T 32,34,40,38,44,42,45,33,54,39,63,48,46,52,56,49,7,98,35,75,55,100,58,

%U 60,62,64,66,68,70,72,51,81,57,90,69,99,78,76,74,80,82,84,86,88,92,94,96,104,102,108,87,117,93,126,111,135,114,112,106,116,110,121,11,242

%N a(1) = 1, a(2) = 2, for a(n) > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) while no exponent of each distinct prime factor of a(n) is the same as the exponent of the same prime factor of a(n-1).

%C For the terms studied the primes appear as terms in their natural order, and when a prime p appears as a term, the proceeding term is always p^2 and the following term is always 2*p^2; it is likely this is true for all primes. A similar pattern is seen in the EKG sequence A064413 except that there a prime is always proceeded by 2*p and followed by 3*p.

%C Unlike the EKG sequence a prime can appear as a factor of a proceeding term long before it appears as a term by itself - see A379291 for the indices where each prime first appears as a factor of a(n).

%C The indices where the primes appear show an interesting pattern of runs of consecutive primes that are separated by only 6 terms, with longer, sometimes much longer, gaps in between - see A379290 and A379296. These primes appear in regions where the terms overall show a strong oscillating pattern of jumping between terms containing a prime p and p^2 as a factor. The primes being oscillated between increase until a new prime q appears in a term q^2 which leads to the next term being q. The occurrence of a new prime q can start a run of consecutive primes appearing before these oscillations subside and the terms slowly grow again until the next oscillation. See the attached graphs which show the burst/oscillating behavior, with the primes appearing in these regions, followed by terms with a slow, more linear, growth.

%C In the first 500000 terms there are only six fixed points - see A379292. However, as the regions of oscillating terms crosses the a(n) = n line it is likely more exist for larger values of n.

%C The sequence is conjectured to be a permutation of the positive integers. See A379293 for the index where n first appears.

%H Scott R. Shannon, <a href="/A379248/b379248.txt">Table of n, a(n) for n = 1..20000</a>

%H Michael De Vlieger, <a href="/A379248/a379248_3.png">Log log scatterplot of a(n)</a>, n = 1..65536.

%H Michael De Vlieger, <a href="/A379248/a379248_4.png">Log log scatterplot of a(n)</a>, n = 1..16384, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither prime powers nor squarefree numbers in both blue and purple, where purple represents powerful numbers that are not prime powers.

%H Michael De Vlieger, <a href="/A379248/a379248_5.png">Plot p^m | a(n) at (x, y) = (n, pi(p))</a>, n = 1..2048, 4X vertical exaggeration, with a color function showing m = 1 in black, m = 2 in red, m = 3 in orange, ..., m = 11 in magenta.

%H Scott R. Shannon, <a href="/A379248/a379248_2.png">Image of the first 500000 terms</a>. The green line is a(n) = n.

%H Scott R. Shannon, <a href="/A379248/a379248_1.png">Colored image of the first 10000 terms</a>. The terms with one, two, three,... as their maximum prime factorization exponent are colored red, orange, yellow,... . The green line is a(n) = n.

%e a(3) = 4 as 4 is unused and shares a factor with a(2) = 2, while 4 = 2^2 which has 2 as the exponent of the prime 2, while a(2) = 2^1 which has exponent 1. As these are different 4 is acceptable.

%e a(5) = 8 as 8 is unused and shares a factor with a(4) = 6, while 8 = 2^3 which has 3 as the exponent of the prime 2, while a(4) = 2^1*3^1 which has exponent 1. As these are different 8 is acceptable. Note that although 3 shares a factor with 6, 3 = 3^1 which has the same exponent 1 on the prime 3 as 6 = 2^1*3^1, so 3 cannot be chosen. This is the first term to differ from A064413.

%t nn = 120; c[_] := False;

%t f[x_] := f[x] = FactorInteger[x]; j = 2; u = 3;

%t {1, 2}~Join~Reap[Do[

%t   k = u; While[Or[c[k], CoprimeQ[j, k], AnyTrue[f[k], MemberQ[f[j], #] &]], k++];

%t     Set[{j, c[k]}, {k, True}]; Sow[k];

%t If[k == u, While[c[u], u++]], {n, 3, nn}] ][[-1, 1]] (* _Michael De Vlieger_, Dec 21 2024 *)

%Y Cf. A124010, A027746, A051903, A064413, A348086, A375564, A375563, A373546, A373545.

%Y Cf. A379290 (index where prime n appears as a term), A379296 (differences between indices where prime terms appear), A379291 (index where prime n first appears as a factor of a(n)), A379293 (index where n appears as a term), A379292 (fixed points), A379294 (record high values), A379295 (indices of record high values).

%K nonn,look,new

%O 1,2

%A _Scott R. Shannon_, Dec 18 2024