A355893 - OEIS

%I #33 Aug 24 2022 12:01:47

%S 0,1,10,100,2,1000,20,10000,100000,1000000,3,10000000,100000000,200,

%T 1010,1000000000,10000000000,100000000000,1000000000000,

%U 10000000000000,100000000000000,1000000000000000,4,10000000000000000

%N Let A090252(n) = Product_{i >= 1} prime(i)^e(i); then a(n) is the concatenation, in reverse order, of e_1, e_2, ..., ending at the exponent of the largest prime factor of A090252(n); a(1)=0 by convention.

%C A090252 and A354169 are similar in many ways. This sequence and A355892 illustrate this.

%C This compressed format only make sense if all e_i are less than 10, that is, for n <= 24574.

%C It is believed that 6 does not appear in A090252, so 11 is missing from the present sequence.

%H Michael De Vlieger, <a href="/A355893/b355893.txt">Table of n, a(n) for n = 1..1073</a>

%F a(n) = A054841(A090252(n)). - _Stefano Spezia_, Aug 24 2022

%e The initial terms of A090252 are:

%e 1 -> 0

%e 2 = 2^1 ->1

%e 3 = 2^0 3^1 -> 10

%e 5 = 2^0 3^0 5^1 -> 100

%e 4 = 2^2 -> 2

%e 7 = 2^0 3^0 5^0 7^1 -> 1000

%e 9 = 2^0 3^2 -> 20

%e ...

%e The terms, right-justified, for comparison with A355892, are:

%e .1 ...................................0

%e .2 ...................................1

%e .3 ..................................10

%e .4 .................................100

%e .5 ...................................2

%e .6 ................................1000

%e .7 ..................................20

%e .8 ...............................10000

%e .9 ..............................100000

%e 10 .............................1000000

%e 11 ...................................3

%e 12 ............................10000000

%e 13 ...........................100000000

%e 14 .................................200

%e 15 ................................1010

%e 16 ..........................1000000000

%e 17 .........................10000000000

%e 18 ........................100000000000

%e 19 .......................1000000000000

%e 20 ......................10000000000000

%e 21 .....................100000000000000

%e 22 ....................1000000000000000

%e 23 ...................................4

%e 24 ...................10000000000000000

%e ...

%t nn = 24, s = Import["https://oeis.org/A090252/b090252.txt", "Data"][[1 ;; nn, -1]]; f[n_] := If[n == 1, 0, Function[g, FromDigits@ Reverse@ ReplacePart[Table[0, {PrimePi[g[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, g]]@ FactorInteger@ n]; Array[f[s[[#]]] &, nn] (* _Michael De Vlieger_, Aug 24 2022 *)

%Y Cf. A054841, A090252, A354169.

%Y See A354150 for indices of powers of 2 in A090252.

%K nonn

%O 1,3

%A _N. J. A. Sloane_, Aug 23 2022