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A118914 Table of the prime signatures (sorted lists of exponents of distinct prime factors) of the positive integers. 9
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

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-Fran├žois 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).

Sequence in context: A254613 A129265 A030358 * A135063 A124010 A212171

Adjacent sequences:  A118911 A118912 A118913 * A118915 A118916 A118917

KEYWORD

nonn,tabf

AUTHOR

Eric W. Weisstein, May 05 2006

EXTENSIONS

Corrected and edited by Daniel Forgues, Dec 22 2010

STATUS

approved

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Last modified November 14 09:49 EST 2018. Contains 317182 sequences. (Running on oeis4.)