cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A032178 Number of functions of n points with no fixed points and with no symmetries.

Original entry on oeis.org

0, 0, 1, 2, 6, 12, 32, 73, 180, 426, 1041, 2507, 6113, 14853, 36283, 88622, 217054, 532018, 1306281, 3210392, 7899770, 19456524, 47966169, 118346672, 292228449, 722098114, 1785513052, 4417719597, 10936705897
Offset: 1

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Author

Keywords

Crossrefs

Formula

WEIGH transform of A032177.

A124933 Number of prime divisors (counted with multiplicity) of number of endofunctions on n points (A001372).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 3, 3, 2, 2, 2, 2, 2, 4, 2, 3, 5, 3, 3, 3, 3, 5, 3, 2, 7, 9, 5, 3, 5, 5, 6, 3, 5, 6, 1, 2, 5, 4, 3, 4, 3, 3, 7, 7, 5, 7, 8, 4, 12, 7, 8, 1, 7, 4, 2, 4, 5, 4, 2, 5, 4, 3, 5, 6, 12, 2, 3, 5, 2, 3, 4, 4, 3, 5, 6, 2, 6, 3, 5, 3, 7, 2, 3, 7, 7, 8, 6, 5, 2, 7, 7, 4, 10, 11, 7, 7, 5, 4, 5, 6
Offset: 0

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Author

Jonathan Vos Post, Nov 12 2006

Keywords

Comments

Number of prime divisors (counted with multiplicity) of A001372 Number of mappings (or mapping patterns) from n points to themselves; number of endofunctions. {n: a(n) = 1} give the primes, beginning: A001372(2) = 3, A001372(3) = 7, A001372(4) = 19, A001372(2) = 47. {n: a(n) = 2} give the semiprimes, beginning: A001372(8) = 951 = 3 * 317, A001372(9) = 2615 = 5 * 523, A001372(10) = 7318 = 2 * 3659, A001372(11) = 20491 = 31 * 661, A001372(12) = 57903 = 3 * 19301, A001372(14) = 466199 = 107 * 4357, A001372(23) = 6218869389 = 3 * 2072956463. 3-almost primes begin: A001372(6) = 130 = 2 * 5 * 13, A001372(7) = 343 = 7^3, A001372(15) = 1328993 = 19 * 113 * 619, A001372(17) = 10884049 = 11 * 353 * 2803, A001372(18) = 31241170 = 2 * 5 * 3124117, A001372(19) = 89814958 = 2 * 5113 * 8783, A001372(20) = 258604642 = 2 * 101 * 1280221, A001372(22) = 2152118306 = 2 * 13 * 82773781, A001372(27) = 437571896993.

Crossrefs

Formula

a(n) = Omega(A001372(n)) = A001222(A001372(n)).

Extensions

More terms from R. J. Mathar, Sep 23 2007

A217896 Number of unlabeled functions on n nodes that have at least one fixed point.

Original entry on oeis.org

0, 1, 2, 5, 13, 34, 90, 243, 660, 1818, 5045, 14102, 39639, 111982, 317533, 903464, 2577724, 7372542, 21130127, 60672017, 174492633, 502568607, 1449360241, 4184719174, 12095325486, 34993693260, 101332159421, 293669741860, 851722291650, 2471948910379, 6824540110584
Offset: 0

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Author

Geoffrey Critzer, Oct 14 2012

Keywords

Comments

Crossrefs

Cf. A001372 and A001373.

Programs

  • Mathematica
    Needs["Combinatorica`"]; nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2 k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i] s[n-1,i] i,{i,1,n-1}]/(n-1);rt=Table[a[i],{i,1,nn}];cfd=Drop[Apply[Plus,Table[Take[CoefficientList[CycleIndex[CyclicGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]] x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,2,30}]],1];cf=Drop[Apply[Plus,Table[Take[CoefficientList[CycleIndex[CyclicGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]] x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,1,30}]],1];fd=CoefficientList[Series[Product[1/(1-x^i)^cfd[[i]],{i,1,nn-1}],{x,0,nn}],x];f=CoefficientList[Series[Product[1/(1-x^i)^cf[[i]],{i,1,nn-1}],{x,0,nn}],x];f-fd (* Geoffrey Critzer, Oct 14 2012, after code given by Robert A. Russell in A000081 *)
Previous Showing 11-13 of 13 results.