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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. 4
 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 * A354941 A352689 A159005 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 August 9 11:35 EDT 2024. Contains 375042 sequences. (Running on oeis4.)