The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A000688

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A000688

Number of Abelian groups of order n; number of factorizations of n into prime powers.
(history; published version)

#195 by Charles R Greathouse IV at Sun Feb 16 08:32:21 EST 2025

LINKS

Eric Weisstein's World of Mathematics, <a href="httphttps://mathworld.wolfram.com/AbelianGroup.html">Abelian Group</a>

Eric Weisstein's World of Mathematics, <a href="httphttps://mathworld.wolfram.com/FiniteGroup.html">Finite Group</a>

Eric Weisstein's World of Mathematics, <a href="httphttps://mathworld.wolfram.com/KroneckerDecompositionTheorem.html">Kronecker Decomposition Theorem</a>

Discussion

Sun Feb 16

08:32

OEIS Server: https://oeis.org/edit/global/3014

#194 by Michael De Vlieger at Sun Jul 14 20:49:34 EDT 2024

STATUS

proposed

approved

#193 by Charles R Greathouse IV at Sun Jul 14 20:09:21 EDT 2024

STATUS

editing

proposed

#192 by Charles R Greathouse IV at Sun Jul 14 01:09:23 EDT 2024

COMMENTS

Kendall & Rankin prove that the density of {n: a(n) = m} exists for each m. - Charles R Greathouse IV, Jul 14 2024

PROG

(PARI) A000688(n) = {local(f); f=factor(n); prod(i=1, matsize(f)[1], numbpart(f[i, 2]))} \\ Michael B. Porter, Feb 08 2010

STATUS

approved

editing

#191 by N. J. A. Sloane at Mon Feb 26 10:42:07 EST 2024

STATUS

proposed

approved

#190 by Miles Englezou at Sun Feb 18 02:30:53 EST 2024

STATUS

editing

proposed

#189 by Miles Englezou at Sun Feb 18 02:30:21 EST 2024

FORMULA

a(n) = A005361(n) except when n = is a term of A046101(k), , since A000041(x) = x for x <= 3. - Miles Englezou, Feb 17 2024

STATUS

proposed

editing

#188 by Miles Englezou at Sun Feb 18 02:02:00 EST 2024

STATUS

editing

proposed

#187 by Joerg Arndt at Sun Feb 18 01:49:52 EST 2024

STATUS

proposed

editing

Discussion

Sun Feb 18

02:00

Miles Englezou: Yes that's right. Perhaps it is clearer to say "if n is a term of A046101" instead?

#186 by Michel Marcus at Sat Feb 17 17:37:24 EST 2024

STATUS

editing

proposed

Discussion

Sun Feb 18

01:49

Joerg Arndt: "except when n = A046101(k)," ?  Do you mean "if n is a term of A046101" ?