OFFSET
0,2
COMMENTS
Note that for n>0 the prime divisors of a(n) are consecutive primes starting with 2. All of the least prime signatures (A025487) are included; with the other values forming A056808.
Using the formula, terms of b(n)= a(n)/A057334(n) are: 1, 1, 2, 2, 4, 4, 6, 6, 8, ..., indeed a(n) repeated. - Michel Marcus, Feb 09 2014
a(n) is the unique normal number whose unsorted prime signature is the k-th composition in standard order (graded reverse-lexicographic). This composition (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. A number is normal if its prime indices cover an initial interval of positive integers. Unsorted prime signature is the sequence of exponents in a number's prime factorization. - Gus Wiseman, Apr 19 2020
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..10000
FORMULA
a(n) = A057334(n) * a (repeated).
a(n) = A336321(2^n). - Peter Munn, Mar 04 2022
EXAMPLE
From Gus Wiseman, Apr 19 2020: (Start)
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
12: {1,1,2}
18: {1,2,2}
30: {1,2,3}
16: {1,1,1,1}
24: {1,1,1,2}
36: {1,1,2,2}
60: {1,1,2,3}
54: {1,2,2,2}
90: {1,2,2,3}
150: {1,2,3,3}
210: {1,2,3,4}
32: {1,1,1,1,1}
48: {1,1,1,1,2}
For example, the 27th composition in standard order is (1,2,1,1), and the normal number with prime signature (1,2,1,1) is 630 = 2*3*3*5*7, so a(27) = 630.
(End)
MATHEMATICA
Table[Times @@ Map[If[# == 0, 1, Prime@ #] &, Accumulate@ IntegerDigits[n, 2]], {n, 0, 54}] (* Michael De Vlieger, May 23 2017 *)
PROG
(PARI) mg(n) = if (n==0, 1, prime(hammingweight(n))); \\ A057334
lista(nn) = {my(v = vector(nn)); v[1] = 1; for (i=2, nn, v[i] = mg(i-1)*v[(i+1)\2]; ); v; } \\ Michel Marcus, Feb 09 2014
(PARI) A057335(n) = if(0==n, 1, prime(hammingweight(n))*A057335(n\2)); \\ Antti Karttunen, Jul 20 2020
CROSSREFS
Cf. A324939.
Unsorted prime signature is A124010.
Numbers whose prime signature is aperiodic are A329139.
The reversed version is A334031.
A partial inverse is A334032.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Aperiodic compositions are A328594.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
KEYWORD
easy,nonn
AUTHOR
Alford Arnold, Aug 27 2000
EXTENSIONS
More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003
New primary name from Antti Karttunen, Jul 20 2020
STATUS
approved