A290308 - OEIS

A290308

Decimal encoding of the prime factorization of n: for n > 0 with prime factorization Product_{i=1..k} prime(i)^e_i, let E_n = (e_k, ..., e_1), replace each nonzero e_i with A052382(e_i) and each zero e_i with "" in E_n to obtain F_n, concatenate the elements of F_n with a "0" inserted after every element except for the last, and interpret in decimal base.

0, 1, 10, 2, 100, 101, 1000, 3, 20, 1001, 10000, 102, 100000, 10001, 1010, 4, 1000000, 201, 10000000, 1002, 10010, 100001, 100000000, 103, 200, 1000001, 30, 10002, 1000000000, 10101, 10000000000, 5, 100010, 10000001, 10100, 202, 100000000000, 100000001

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OFFSET

1,3

COMMENTS

This sequence is an analog of A156552 for the decimal base.

This sequence establishes a bijection between the positive numbers and the nonnegative numbers; see A290389 for the inverse sequence.

The number of runs of consecutive nonzero digits in the decimal representation of a(n) corresponds to the number of distinct prime factors of n.

a(A003961(n)) = 10 * a(n) for any n > 0.

a(n) = 0 mod 10 iff n is odd.

a(prime(n)^k) = A052382(k) * 10^(n-1) for any n > 0 and k > 0 (where prime(n) is the n-th prime).

a(prime(n)#) = Sum_{k=1..n} 100^(k-1) for any n > 0 (where prime#(n) = A002110(n)).

LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..5000

Index entries for sequences that are permutations of the natural numbers

EXAMPLE

For n = 5120 = 5^1 * 3^0 * 2^10:

- E_5120 = (1, 0, 10),

- F_5120 = ("1", "", "11"),

- a(5120) = 10011.

For n = 5040 = 7^1 * 5^1 * 3^2 * 2^4:

- E_5040 = (1, 1, 2, 4),

- F_5040 = ("1", "1", "2", "4"),

- a(5040) = 1010204.

MATHEMATICA

f[n_] := Function[m, Sum[(1 + Mod[Floor[(8 n + 1 - 9^m)/(8*9^j)], 9]) 10^j, {j, 0, m - 1}]]@ Floor@ Log[9, 8 n + 1]; Table[If[n == 1, 0, With[{s = FactorInteger[n] /. {p_, e_} /; p > 0 :> If[p > 1, PrimePi@ p -> f@ e]}, Function[t, FromDigits@ Flatten@ Reverse@ Riffle[#, ConstantArray[0, Length@ #]] &[ReplacePart[t, s] /. 0 -> {}]]@ConstantArray[0, Max[s[[All, 1]] ]]]], {n, 38}] (* Michael De Vlieger, Jul 31 2017 *)

PROG

(PARI) a(n) = {

my (f = factor(n), v = 0, nz = 0);

for (i=1, #f~,

my (x = A052382(f[i, 2]));

v += x * 10^(nz + primepi(f[i, 1]) - 1);

nz += #digits(x);

);

return (v)

}

CROSSREFS

Cf. A002110, A003961, A052382, A156552, A290389 (inverse).