A361376 Rewrite A129912(n), a product of distinct primorials P(i) = A002110(i) instead as a sum of powers 2^(i-1).
0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 16, 11, 17, 12, 13, 18, 19, 32, 14, 33, 20, 15, 21, 34, 35, 22, 24, 64, 23, 36, 25, 65, 37, 26, 66, 38, 27, 67, 40, 128, 39, 41, 28, 68, 129, 29, 69, 42, 130, 48, 43, 30, 70, 72, 131, 49, 31, 71, 44, 73, 256, 132, 45, 50, 257, 133, 74, 51, 46, 80, 75, 258, 134, 136
Offset: 1
Keywords
Examples
a(1) = 0 by convention. a(8) = 8 comes before a(9) = 7, since we interpret 8 = 2^3 instead as P(4) = 210, while for a(9), 7 = 2^2 + 2^1 + 2^0 becomes P(3)*P(2)*P(1) = 30*6*2 = 360. Because 210 < 360, 8 appears before 7 in this sequence. Table relating a(n), n=1..19 with the set S(n) of indices of distinct primorial factors of A129912(n): n A129912(n) S(n) a(n) A272011(a(n)) ----------------------------------------- 1 1 0 2 2 1 1 0 3 6 2 2 1 4 12 2,1 3 1,0 5 30 3 4 2 6 60 3,1 5 2,0 7 180 3,2 6 2,1 8 210 4 8 3 9 360 3,2,1 7 2,1,0 10 420 4,1 9 3,0 11 1260 4,2 10 3,1 12 2310 5 16 4 13 2520 4,2,1 11 3,1,0 14 4620 5,1 17 4,0 15 6300 4,3 12 3,2 16 12600 4,3,1 13 3,2,0 17 13860 5,2 18 4,1 18 27720 5,2,1 19 4,1,0 19 30030 6 32 5 ...
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..15303 (a(15303) = 2^29.)
- Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^6.
- Michael De Vlieger, Plot terms S(n) = A272011(a(n)) at (x,y) = (n,S(n,k)) for n = 1..2^11.
Programs
-
Mathematica
a6939[n_] := Product[Prime[n + 1 - i]^i, {i, n}]; g[m_] := Block[{f, j = 1}, f[n_, i_, e_] := If[n < m, Block[{p = Prime[i + 1]}, If[e == 1, Sow@ n]; f[n p^e, i + 1, e]; If[e > 1, f[n p^(e - 1), i + 1, e - 1]]]]; Sort@ Reap[While[a6939[j] < m, f[2^j, 1, j]; j++]][[-1, 1]] ]; Map[Total@ Map[2^(# - 1) &, Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} ] &[ FactorInteger[#][[All, -1]]] &, g[2^31]] (* Michael De Vlieger, Jun 08 2023, after Giovanni Resta at A129929 *)
Formula
Let S(n) be the set of indices of primorials P(i), reverse sorted, such that A129912(n) = Product_{k=1..m} S(n,k), where m = | S(n) |. Then a(n) = Sum_{k=1..m} 2^(S(n,k)-1).
Comments