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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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 + prime pi(f[i, 1]) - 1);

                   nz += #digits(x);

           );

           return (v)

       }

CROSSREFS

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

Sequence in context: A185076 A178643 A038304 * A159005 A144859 A280519

Adjacent sequences:  A290305 A290306 A290307 * A290309 A290310 A290311

KEYWORD

nonn,base

AUTHOR

Rémy Sigrist, Jul 27 2017

STATUS

approved

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Last modified June 19 10:32 EDT 2019. Contains 324219 sequences. (Running on oeis4.)