

A127668


Concatenated indices of primes in prime factorization of n.


4



1, 2, 11, 3, 21, 4, 111, 22, 31, 5, 211, 6, 41, 32, 1111, 7, 221, 8, 311, 42, 51, 9, 2111, 33, 61, 222, 411, 10, 321, 11, 11111, 52, 71, 43, 2211, 12, 81, 62, 3111, 13, 421, 14, 511, 322, 91, 15, 21111, 44, 331, 72, 611, 16, 2221, 53, 4111, 82, 101, 17, 3211, 18, 111
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OFFSET

2,2


COMMENTS

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.
The mapping n>a(n) is from {2,3,...} onto {1,2,3,...}=N but not injective; hence not invertible.
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.
For the invertible map onto lists of prime number indices see the W. Lang link; also A112798.


LINKS

Table of n, a(n) for n=2..62.
W. Lang: Unique representation as lists.


FORMULA

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_{k1} times (k1)... followed by n_1 times 1.


EXAMPLE

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.
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.


CROSSREFS

For numbers with no prime divisor > 23, the sum of digits gives A056239(n), n>=2.
For numbers with no prime divisor > 23, the length of the digits gives A001222(n), n>=2, (number of prime divisors of n).
The number of numbers mapped to a(n) gives A127669.
Cf. A054841(n), n>=2: exponents in prime decomposition of n.
See A112798 for another version of this data.
Sequence in context: A113625 A279237 A090323 * A261300 A068743 A322761
Adjacent sequences: A127665 A127666 A127667 * A127669 A127670 A127671


KEYWORD

nonn,easy,base


AUTHOR

Wolfdieter Lang Jan 23 2007


EXTENSIONS

Edited by Franklin T. AdamsWatters, May 21 2014


STATUS

approved



