A361376 - OEIS

A361376

Rewrite A129912(n), a product of distinct primorials P(i) = A002110(i) instead as a sum of powers 2^(i-1).

%I #22 Jun 11 2023 12:28:54

%S 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,

%T 22,24,64,23,36,25,65,37,26,66,38,27,67,40,128,39,41,28,68,129,29,69,

%U 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

%N Rewrite A129912(n), a product of distinct primorials P(i) = A002110(i) instead as a sum of powers 2^(i-1).

%C Permutation of nonnegative numbers.

%H Michael De Vlieger, <a href="/A361376/b361376.txt">Table of n, a(n) for n = 1..15303</a> (a(15303) = 2^29.)

%H Michael De Vlieger, <a href="/A361376/a361376.png">Log log scatterplot of a(n)</a>, n = 1..10^6.

%H Michael De Vlieger, <a href="/A361376/a361376_1.png">Plot terms S(n) = A272011(a(n)) at (x,y) = (n,S(n,k))</a> for n = 1..2^11.

%F 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).

%e a(1) = 0 by convention.

%e 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.

%e Table relating a(n), n=1..19 with the set S(n) of indices of distinct primorial factors of A129912(n):

%e n A129912(n) S(n) a(n) A272011(a(n))

%e -----------------------------------------

%e 1 1 0

%e 2 2 1 1 0

%e 3 6 2 2 1

%e 4 12 2,1 3 1,0

%e 5 30 3 4 2

%e 6 60 3,1 5 2,0

%e 7 180 3,2 6 2,1

%e 8 210 4 8 3

%e 9 360 3,2,1 7 2,1,0

%e 10 420 4,1 9 3,0

%e 11 1260 4,2 10 3,1

%e 12 2310 5 16 4

%e 13 2520 4,2,1 11 3,1,0

%e 14 4620 5,1 17 4,0

%e 15 6300 4,3 12 3,2

%e 16 12600 4,3,1 13 3,2,0

%e 17 13860 5,2 18 4,1

%e 18 27720 5,2,1 19 4,1,0

%e 19 30030 6 32 5

%e ...

%t a6939[n_] := Product[Prime[n + 1 - i]^i, {i, n}];

%t g[m_] := Block[{f, j = 1},

%t f[n_, i_, e_] :=

%t If[n < m, Block[{p = Prime[i + 1]}, If[e == 1, Sow@ n];

%t f[n p^e, i + 1, e];

%t If[e > 1, f[n p^(e - 1), i + 1, e - 1]]]];

%t Sort@ Reap[While[a6939[j] < m, f[2^j, 1, j]; j++]][[-1, 1]] ];

%t Map[Total@

%t Map[2^(# - 1) &,

%t Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} ] &[

%t FactorInteger[#][[All, -1]]] &, g[2^31]] (* _Michael De Vlieger_, Jun 08 2023, after _Giovanni Resta_ at A129929 *)

%Y Cf. A002110, A129912, A272011, A283477.

%K nonn

%O 1,3

%A _Michael De Vlieger_, Jun 08 2023