A355892 -id:A355892 - cp's OEIS frontend

A355893 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.

0, 1, 10, 100, 2, 1000, 20, 10000, 100000, 1000000, 3, 10000000, 100000000, 200, 1010, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 4, 10000000000000000

Offset: 1

N. J. A. Sloane, Aug 23 2022

nonn

A090252 and A354169 are similar in many ways. This sequence and A355892 illustrate this.
This compressed format only make sense if all e_i are less than 10, that is, for n <= 24574.
It is believed that 6 does not appear in A090252, so 11 is missing from the present sequence.

			The initial terms of A090252 are:
1 -> 0
2 = 2^1 ->1
3 = 2^0 3^1 -> 10
5 = 2^0 3^0 5^1 -> 100
4 = 2^2 -> 2
7 = 2^0 3^0 5^0 7^1 -> 1000
9 = 2^0 3^2 -> 20
...
The terms, right-justified, for comparison with A355892, are:
.1 ...................................0
.2 ...................................1
.3 ..................................10
.4 .................................100
.5 ...................................2
.6 ................................1000
.7 ..................................20
.8 ...............................10000
.9 ..............................100000
10 .............................1000000
11 ...................................3
12 ............................10000000
13 ...........................100000000
14 .................................200
15 ................................1010
16 ..........................1000000000
17 .........................10000000000
18 ........................100000000000
19 .......................1000000000000
20 ......................10000000000000
21 .....................100000000000000
22 ....................1000000000000000
23 ...................................4
24 ...................10000000000000000
...

Michael De Vlieger, Table of n, a(n) for n = 1..1073

Cf. A054841, A090252, A354169.

See A354150 for indices of powers of 2 in A090252.

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 *)

a(n) = A054841(A090252(n)). - Stefano Spezia, Aug 24 2022

A355894 Let A354790(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 A354790(n); a(1)=0 by convention.

Original entry on oeis.org

0, 1, 10, 100, 1000, 10000, 11, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 1100, 10001, 100000000000, 100010, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 10000000000000000, 100000000000000000, 1000000000000000000, 10000000000000000000

Offset: 0

N. J. A. Sloane, Aug 25 2022

nonn

The terms of A354790 are squarefree, so here the exponents e_i are 0 or 1.

This bears the same relation to A354790 as A355893 does to A090252.

			The terms, right-justified, for comparison with A355892 and A355893, are:
   1 ...................................0
   2 ...................................1
   3 ..................................10
   4 .................................100
   5 ................................1000
   6 ...............................10000
   7 ..................................11
   8 ..............................100000
   9 .............................1000000
  10 ............................10000000
  11 ...........................100000000
  12 ..........................1000000000
  13 .........................10000000000
  14 ................................1100
  15 ...............................10001
  16 ........................100000000000
  17 ..............................100010
  18 .......................1000000000000
  19 ......................10000000000000
  20 .....................100000000000000
  21 ....................1000000000000000
  22 ...................10000000000000000
  23 ..................100000000000000000
  24 .................1000000000000000000
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..1051

Cf. A090252, A354790, A355893.

cp's OEIS Frontend

A355893 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.

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