A127668 Concatenated indices of primes in prime factorization of n.

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

Offset: 2

Author

Wolfdieter Lang Jan 23 2007

Keywords

• nonn
• easy
• base

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.

Examples

			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.
		

Links

• W. Lang: Unique representation as lists.

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.

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_{k-1} times (k-1)... followed by n_1 times 1.

Extensions

Edited by Franklin T. Adams-Watters, May 21 2014