

A057335


a(0) = 1, and for n > 0, a(n) = A000040(A000120(n)) * a(floor(n/2)); essentially sequence A055932 generated using A000120, hence sorted by number of factors.


20



1, 2, 4, 6, 8, 12, 18, 30, 16, 24, 36, 60, 54, 90, 150, 210, 32, 48, 72, 120, 108, 180, 300, 420, 162, 270, 450, 630, 750, 1050, 1470, 2310, 64, 96, 144, 240, 216, 360, 600, 840, 324, 540, 900, 1260, 1500, 2100, 2940, 4620, 486, 810, 1350, 1890, 2250, 3150, 4410
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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 kth composition in standard order (graded reverselexicographic). 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).
A334032(a(n)) = n; a(A334032(n)) = A071364(n).  Gus Wiseman, Apr 19 2020
a(n) = A122111(A019565(n)); A019565(n) = A122111(a(n)).  Peter Munn, Jul 18 2020
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(i1)*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. A000120, A057334, A055932 and A056808.
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.
Cf. A029931, A048793, A052409, A056239, A066099, A112798, A124767, A228351, A233249, A333220.
Related to A019565 via A122111 and to A000079 via A336321.
Sequence in context: A050597 A288603 A324939 * A126907 A292994 A323114
Adjacent sequences: A057332 A057333 A057334 * A057336 A057337 A057338


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



