A171241
Numbers k such that A169611(k) = 3.
Original entry on oeis.org
8, 12, 18, 27, 40, 56, 60, 84, 88, 90, 104, 126, 132, 135, 136, 152, 156, 184, 189, 198, 200, 204, 228, 232, 234, 248, 276, 280, 296, 297, 300, 306, 328, 342, 344, 348, 351, 372, 376, 392, 414, 420, 424, 440, 444, 450, 459, 472, 488, 492, 513, 516, 520, 522
Offset: 1
Corrected (200, 232, 280 inserted) and extended by
R. J. Mathar, Jun 04 2010
A382491
a(n) is the numerator of the asymptotic density of the numbers whose number of 3-smooth divisors is n.
Original entry on oeis.org
1, 5, 13, 71, 97, 1355, 793, 19163, 53473, 292355, 60073, 13102907, 535537, 78584915, 790859641, 3523099499, 43112257, 99646519235, 387682633, 2764285630427, 7604811750289, 7337148996275, 31385253913, 2226944658077771, 3656440886376673, 2341258386360995, 80539587570991081
Offset: 1
Fractions begin with 1/3, 5/18, 13/108, 71/648, 97/3888, 1355/23328, 793/139968, 19163/839808, 53473/5038848, 292355/30233088, 60073/181398528, 13102907/1088391168, ...
a(1) = 1 since a(1)/A081341(1) = 1/3 is the asymptotic density of the numbers with a single 3-smooth divisor, 1, i.e., the numbers that are congruent to 1 or 5 mod 6 (A007310).
a(2) = 5 since a(2)/A081341(2) = 5/18 is the asymptotic density of the numbers with exactly two 3-smooth divisors, either 1 and 2 or 1 and 3, i.e., A171126.
-
a[n_] := DivisorSum[n, 2^(n-#) * 3^(n-n/#) &]; Array[a, 30]
-
a(n) = sumdiv(n, d, 2^(n-d)*3^(n-n/d));
A171837
Array g(n,k) read by antidiagonals: the k-th integer with prime factorization 2^i * 3^(n-i) * 5^e_5 *7^e_7 * (... higher primes).
Original entry on oeis.org
1, 2, 5, 4, 3, 7, 8, 6, 10, 11, 16, 12, 9, 14, 13, 32, 24, 18, 20, 15, 17, 64, 48, 36, 27, 28, 21, 19, 128, 96, 72, 54, 40, 30, 22, 23, 256, 192, 144, 108, 80, 56, 42, 26, 25, 512, 384, 288, 216, 160, 81, 60, 44, 33, 29, 1024, 768, 576, 432, 320, 162, 112, 84, 45, 34, 31
Offset: 1
The array starts in row n=0 as:
1, 5, 7, 11, 13, 17, 19, 23, 25, 29: not divisible by 2 or 3
2, 3, 10, 14, 15, 21, 22, 26, 33, 34: divisible by 2^i*3^(1-i), i<=1
4, 6, 9, 20, 28, 30, 42, 44, 45, 52: divisible by 2^i*3^(2-i), i<=2
8, 12, 18, 27, 40, 56, 60, 84, 88, 90: divisible by 2^i*3^(3-i): i<=3
16, 24, 36, 54, 80, 81, 112, 120, 168, 176
32, 48, 72, 108, 160, 162, 224, 240, 243, 336
64, 96, 144, 216, 320, 324, 448, 480, 486, 672
-
f[n_] := Plus @@ Last /@ Select[FactorInteger@n, 1 < #[[1]] < 4 &]; g[n_, k_] := Select [Range@ 1100, f@# == n &][[k]]; Table[g[n - k, k], {n, 11}, {k, n}] // Flatten
Showing 1-3 of 3 results.
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