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A118914 Concatenation of the prime signatures (sorted lists of exponents of distinct prime factors) of the positive integers. 5
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).

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 sequence of prime signatures of positive integers

{{ }, {1}, {1}, {2}, {1}, {1, 1}, {1}, {3}, {2}, {1, 1}, {1}, {1, 2}, ...}

after concatenation gives (the current sequence)

{1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, ...}.

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

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 March 28 15:52 EDT 2017. Contains 284243 sequences.