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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A171919 Number of solutions to n=x*y*z, x+y-z=1 with ordered triples (x,y,z), x,y,z>=1.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2
Offset: 1

Views

Author

Georgi Guninski, Oct 23 2010

Keywords

Comments

Record values start a(1)=1, a(4)=2, a(112)=3, a(144)=6, a(23400)=8, a(28224)=10.
If n is a perfect square, a(n)>0.
Larger record indices are listed in A181485, and associated values in A181486. - M. F. Hasler, Oct 23 2010
First occurrence of k: 2, 1, 4, 112, 480, 43120, 144, 218880, 23400, ??, 28224, ??, 373464, ??, 247104, ??, 604800, ??, 83010312, ??, 26812800, ..., . - Robert G. Wilson v, Oct 30 2010
a(388778796252000) = 38.

Examples

			For n=4, the a(4)=2 solutions are (x,y,z)=(1,2,2) and (2,1,2).
For n=12, the a(12)=1 solution is (x,y,z)=(2,2,3).

Links

Crossrefs

Cf. A181485, A181486.

Programs

  • Maple
    A := proc(n) local a,dvs,x,y,z,dvsyz; a :=0 ; dvs := numtheory[divisors](n) ; for x in dvs do yz := n/x ; dvsyz := numtheory[divisors](yz) ; for y in dvsyz do z := yz/y ; if x+y-z=1 then a := a+1 ; fi; end do; end do:
return a; end proc: seq(A(n),n=1..100) ; # R. J. Mathar, Oct 23 2010
  • Mathematica
    f[n_] := Block[{c = 0, cong = {0, 1, 4, 9, 12, 16, 21, 24, 25, 36, 37, 40, 45, 49, 52, 57}, dvs = Divisors@ n, dvt, j = 1, k, lmt1, lmt2}, If[ MemberQ[ cong, Mod[n, 60]], lmtj = Length@ dvs + 1; While[j < lmtj, dvt = Divisors[ n/dvs[[j]]]; k = 1; lmtk = Length@ dvt + 1; While[k < lmtk, If[ dvs[[j]] + dvt[[k]] == n/(dvs[[j]]*dvt[[k]]) + 1, c++ ]; k++ ]; j++ ]]; c]; Array[f, 105] (* Robert G. Wilson v, Oct 24 2010 *)
  • PARI
    A171919(n)={ my(c=0,t); fordiv(n, z, fordiv( t=n/z, y, y>z & break; y*(z+1-y)==t & c++) ); c} /* can be improved by restricting to x<=y and counting twice if xM. F. Hasler, Oct 23 2010

Extensions

Some more values from M. F. Hasler, Oct 23 2010

A181485 Indices of records in A171919 = number of solutions to n=x*y*z, x+y=z+1.

Original entry on oeis.org

1, 4, 112, 144, 23400, 28224, 247104, 604800, 26812800, 2677752000, 6805814400, 165145780800, 1248124550400, 17996854730400, 388778796252000
Offset: 1

Views

Author

R. J. Mathar and M. F. Hasler, Oct 23 2010

Keywords

Comments

The sequence lists all n such that A171919(n) > A171919(k) for all k < n.
Also the subsequence of terms of A171920 for which A171919 is larger than for all preceding values.
The actual record values are given in A181486.
a(10) > 5*10^7.
It seems highly probable that all terms of this sequence, except for a(1) = 1, are multiples of 4.
a(14) > 4*10^12. - Donovan Johnson, Jun 14 2011
a(16) > 2*10^17. - Robert Gerbicz, Apr 10 2012

Examples

			a(1) = 1 since there is no smaller value possible.
a(2) = 4 is the smallest number for which there are more than 1 = A171919(1) solutions to n = x*y*z, x + y = z + 1.
a(3) = 112 is the smallest number for which there are more than 2 = A171919(4) solutions to n = x*y*z, x + y = z + 1.

Programs

  • PARI
    m=0;for(n=1,1e9,A171919(n)>m | next; m=A171919(n); print1(n", "))

Extensions

a(10)-a(13) from Donovan Johnson, Jun 14 2011
a(14)-a(15) from Robert Gerbicz, Apr 10 2012
Showing 1-2 of 2 results.

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