A378233 a(1) = 1, a(2) = 2, let i = a(n-2), j = a(n-1). For n > 2, If i*j is a term in A055932, a(n) is the least novel multiple of the smallest prime which does not divide i*j. Otherwise a(n) is the least novel multiple of the greatest prime < Gpf(i*j) which does not divide i*j.
1, 2, 3, 5, 4, 6, 10, 7, 9, 15, 8, 14, 20, 12, 21, 25, 16, 18, 30, 28, 11, 35, 24, 22, 42, 40, 33, 49, 45, 26, 44, 56, 50, 27, 63, 55, 32, 70, 36, 66, 77, 60, 13, 88, 84, 65, 99, 91, 75, 110, 98, 39, 121, 105, 34, 52, 132, 112, 80, 48, 119, 78, 143, 126, 85, 104
Offset: 1
Examples
Since rad(1*2) = 2 = A002110(1), a(3) = prime(2) = 3. Then rad(2*3) = 6 = A002110(2), so a(4) = prime(3) = 5. Rad(3*5) = 15, not a term in A055932, 2 is the greatest prime < 5 = Gpf(15) which does not divide 15, so a(5) = 4, the least novel multiple of 2. Rad(5*4) = 10 so a(6) is 6, the least novel multiple of 3, the greatest prime < 5 which does not divide 10. a(8) = 7, a(9) = 9, 7*9 = 63 is not in A055932 so a(10) = 15, the least novel multiple of 5, the greatest prime < 7 = Gpf(63) which does not divide 63. a(10) = 15 is the first departure from A359804.
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
- Michael De Vlieger, Log log scatterplot of a(n), n = 1..1048576.
- Michael De Vlieger, Log log scatterplot of a(n), n = 1..16384, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple also shows numbers that are powerful but not prime powers.
Programs
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Mathematica
c[] := False; m[] := 1; nn = 120; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; Set[{i, j}, Range[2]]; Do[s = FactorInteger[i*j][[All, 1]]; h = Times @@ s; If[Or[IntegerQ@ Log2[h], And[EvenQ[h], Union@ Differences@ PrimePi[s] == {1}] ], k = NextPrime[s[[-1]]]; While[c[k*m[k]], m[k]++]; k *= m[k], k = s[[-1]]; While[Divisible[h, k = NextPrime[k, -1] ] ]; While[c[k*m[k]], m[k]++]; k *= m[k] ]; Set[{a[n], c[k], i, j}, {k, True, j, k}], {n, 3, nn}]; Array[a, nn]
Comments