A118914 - OEIS

A118914

Table of the prime signatures (sorted lists of exponents of distinct prime factors) of the positive integers.

240

1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 4, 2, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 6, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,3

COMMENTS

Since the prime factorization of 1 is the empty product (i.e., the multiplicative identity, 1), it follows that the prime signature of 1 is the empty multiset { }. (Cf. http://oeis.org/wiki/Prime_signature)

MathWorld wrongly defines the prime signature of 1 as {1}, which is actually the prime signature of primes.

The sequences A025487, A036035, A046523 consider the prime signatures of 1 and 2 to be distinct, implying { } for 1 and {1} for 2.

Since the prime signature of n is a partition of Omega(n), also true for Omega(1) = 0, the order of exponents is only a matter of convention (using reverse sorted lists of exponents would create a different sequence).

Here the multisets of nonzero exponents are sorted in increasing order; it is slightly more common to order them, as the parts of partitions, in decreasing order. This yields A212171. - M. F. Hasler, Oct 12 2018

LINKS

Reinhard Zumkeller, Rows n = 2..1000 of table, flattened

Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 73.

Eric Weisstein's World of Mathematics, Prime Signature

OEIS Wiki, Prime signatures

OEIS Wiki, Ordered prime signatures

EXAMPLE

The table starts:

n : prime signature of n (factorization of n)

1 : {}, (empty product)

2 : {1}, (2^1)

3 : {1}, (3^1)

4 : {2}, (2^2)

5 : {1}, (5^1)

6 : {1, 1}, (2^1 * 3^1)

7 : {1}, (5^1)

8 : {3}, (2^3)

9 : {2}, (3^2)

10 : {1, 1}, (2^1 * 5^1)

11 : {1}, (11^1)

12 : {1, 2}, (2^2 * 3^1, but exponents are sorted increasingly)

etc.

MATHEMATICA

primeSignature[n_] := Sort[ FactorInteger[n] , #1[[2]] < #2[[2]]&][[All, 2]]; Flatten[ Table[ primeSignature[n], {n, 2, 65}]](* Jean-François Alcover, Nov 16 2011 *)

PROG

(Haskell)

import Data.List (sort)

a118914 n k = a118914_tabf !! (n-2) !! (k-1)

a118914_row n = a118914_tabf !! (n-2)

a118914_tabf = map sort $ tail a124010_tabf

-- Reinhard Zumkeller, Mar 23 2014

(PARI) A118914_row(n)=vecsort(factor(n)[, 2]~) \\ M. F. Hasler, Oct 12 2018

CROSSREFS

Cf. A025487, A036035, A046523, A095904.

Cf. A124010.

Cf. A001221 (row lengths), A001222 (row sums).