cp's OEIS Frontend

A299400 a(n) = concatenation of all (i, e_i) with e_i > 0, when n = Product_{i >= 1} prime(i)^e_i.

%I A299400 #24 Dec 11 2019 08:53:34
%S A299400 0,11,21,12,31,1121,41,13,22,1131,51,1221,61,1141,2131,14,71,1122,81,
%T A299400 1231,2141,1151,91,1321,32,1161,23,1241,101,112131,111,15,2151,1171,
%U A299400 3141,1222,121,1181,2161,1331,131,112141,141,1251,2231,1191,151,1421,42,1132,2171
%N A299400 a(n) = concatenation of all (i, e_i) with e_i > 0, when n = Product_{i >= 1} prime(i)^e_i.
%C A299400 The conventional a(1) = 0 represents the empty concatenation.
%C A299400 Due to simple concatenation, this encoding of the positive integers becomes ambiguous from n = 613 = prime(112)^1 on, which has the same encoding a(n) = 1121 as 6 = prime(1)^1*prime(2)^1. To get a unique encoding, one could use, e.g., the digit 9 as delimiter to separate indices and exponents, written in base 9 as to use only digits 0..8, as soon as a term would be the duplicate of an earlier term (or for all n >= 613). Then one would have, e.g., a(613) = prime(134_9)^1 = 13491.
%C A299400 Sequence A067599 is based on the same idea, but uses the primes instead of their indices. In A037276 the prime factors are repeated, instead of giving the exponent. In A080670 exponents 1 are omitted. In A124010 only the prime signature is given. In A054841 the sum e_i*10^(i-1) is given, i.e., exponents are used as digits in base 10, while they are listed individually in the rows of A067255.
%e A299400 2 = prime(1)^1 => a(2) = 11,
%e A299400 3 = prime(2)^1 => a(3) = 21,
%e A299400 4 = prime(1)^2 => a(4) = 12,
%e A299400 5 = prime(3)^1 => a(5) = 31,
%e A299400 6 = prime(1)^1*prime(2)^1 => a(1) = 1121,
%e A299400 7 = prime(3)^1 => a(7) = 41,
%e A299400 8 = prime(1)^3 => a(8) = 13, and so on.
%p A299400 a:= n-> `if`(n=1, 0, parse(cat(seq([numtheory[pi]
%p A299400        (i[1]), i[2]][], i=sort(ifactors(n)[2]))))):
%p A299400 seq(a(n), n=1..60);  # _Alois P. Heinz_, Mar 16 2018
%t A299400 Array[FromDigits@ Flatten@ Map[{PrimePi@ #1, #2} & @@ # &, FactorInteger@ #] &, 51] (* _Michael De Vlieger_, Mar 16 2018 *)
%o A299400 (PARI) A299400(n)=if(n=factor(n),eval(concat(apply(f->Str(primepi(f[1]),f[2]), Col(n)~))))
%Y A299400 Cf. A067599 (decimal encoding of prime factorization).
%K A299400 nonn,base
%O A299400 1,2
%A A299400 _M. F. Hasler_, Mar 08 2018