A381139 a(1) = 1, a(2) = 2. Let j = a(n-1) and let d = A160995(j) be the smallest non-divisor of j which shares a prime factor with j. Then for n > 2 a(n) is the smallest multiple of d which is not yet in the sequence.
1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 9, 30, 28, 36, 32, 42, 40, 48, 27, 54, 44, 60, 56, 66, 52, 72, 10, 64, 78, 68, 84, 80, 90, 76, 96, 45, 102, 88, 108, 104, 114, 92, 120, 63, 126, 100, 132, 112, 138, 116, 144, 50, 124, 150, 128, 156, 136, 162, 140, 168, 81, 174
Offset: 1
Examples
a(3) = 4, the smallest non divisor of 2 which shares a divisor with 2. a(4) = 6, the smallest non divisor of 4 which shares a divisor with 4. a(5) = 8 because 4 is the smallest non divisor of 6 with a factor in common but 4 has been seen earlier and 8 is the least novel multiple of 4. a(10) = 24 implies a(11) = 9 since A160995(24) = 9, and 9 has not occurred earlier. This is the first odd number in the sequence. Thereafter we see a(19) = 27, a(36) = 45, a(44) = 63, a(61) = 81, etc (consecutive odd multiples of 9). It is not known if this behavior continues as the sequence extends.
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
- Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
- Michael De Vlieger, Prime power decomposition of a(n), n = 1..1000, (2025).
Programs
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Mathematica
nn = 120; c[] := False; m[] := 1; j = 2; {1, 2}~Join~ Reap[Do[k = 2; Until[Nor[Divisible[j, k], CoprimeQ[j, k]], k++]; While[Or[c[#], Divisible[j, #], CoprimeQ[j, #]] &[k*m[k]], m[k]++]; k *= m[k]; c[k] = True; j = Sow[k], {n, nn}] ][[-1, 1]] (* Michael De Vlieger, Feb 18 2025 *)
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