cp's OEIS Frontend

A127668 Concatenated indices of primes in prime factorization of n.

%I A127668 #14 Aug 29 2019 17:39:15
%S A127668 1,2,11,3,21,4,111,22,31,5,211,6,41,32,1111,7,221,8,311,42,51,9,2111,
%T A127668 33,61,222,411,10,321,11,11111,52,71,43,2211,12,81,62,3111,13,421,14,
%U A127668 511,322,91,15,21111,44,331,72,611,16,2221,53,4111,82,101,17,3211,18,111
%N A127668 Concatenated indices of primes in prime factorization of n.
%C A127668 For each n>=2 the indices i of primes p(i), i>=1, in the prime number decomposition of n are ordered from right to left.
%C A127668 The mapping n->a(n) is from {2,3,...} onto {1,2,3,...}=N but not injective; hence not invertible.
%C A127668 There are at most pa(k):=A000041(k) (partition numbers) different numbers which map to any a(n) with k digits. 10 and 12 are the smallest numbers for which this is not equality; 10 because 1,0 is not a partition, and 12 because 1,2 lists partition parts in the wrong order.
%C A127668 For the invertible map onto lists of prime number indices see the W. Lang link; also A112798.
%H A127668 W. Lang: <a href="/A127668/a127668.txt">Unique representation as lists.</a>
%F A127668 If n=p_1^(n_1) p_2^(n_2)...p_k^(n_k), with n_j>=0, then a(n) = n_k times k followed by n_{k-1} times (k-1)... followed by n_1 times 1.
%e A127668 111=a(2*2*2)=a(31*2)=a(607). 111 has k=3 digits, hence pa(3)=3 different numbers are mapped to it.
%e A127668 a(5)=3 because 5=p(3). a(4)=11 because 4=2*2=p(1)*p(1). Also a(31)=11 because p(11)=31.
%Y A127668 For numbers with no prime divisor > 23, the sum of digits gives A056239(n), n>=2.
%Y A127668 For numbers with no prime divisor > 23, the length of the digits gives A001222(n), n>=2, (number of prime divisors of n).
%Y A127668 The number of numbers mapped to a(n) gives A127669.
%Y A127668 Cf. A054841(n), n>=2: exponents in prime decomposition of n.
%Y A127668 See A112798 for another version of this data.
%K A127668 nonn,easy,base
%O A127668 2,2
%A A127668 _Wolfdieter Lang_ Jan 23 2007
%E A127668 Edited by _Franklin T. Adams-Watters_, May 21 2014