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.

Showing 1-10 of 48 results. Next

A097212 Numbers n such that A076078(n) > A076078(m) for all m < n, A076078(n) being the number of sets of distinct positive integers with a least common multiple of n.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 30, 36, 48, 60, 120, 180, 240, 360, 420, 720, 840, 1260, 1680, 2520, 4620, 5040, 7560, 9240, 10080, 12600, 13860, 15120, 18480, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 138600, 166320, 221760, 277200, 332640
Offset: 1

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Author

Matthew Vandermast, Aug 07 2004

Keywords

Comments

RECORDS transform of A076078. All highly composite numbers (A002182) are members. All members belong to A025487 and A067128.

Links

Crossrefs

Cf. A140999 (intersection of A025487 and A067128). - Matthew Vandermast, Oct 11 2008

Programs

  • Mathematica
    f[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu[n/d](2^DivisorSigma[0, d] - 1))]; b = 0; l = {}; Do[c = a[n]; If[c > b, b = c; AppendTo[l, n]], {n, 10^6}]; l (* Robert G. Wilson v, Aug 13 2004 *)

Extensions

More terms from Robert G. Wilson v, Aug 13 2004

A097214 Numbers m such that A076078(m) = m, where A076078(m) equals the number of sets of distinct positive integers with a least common multiple of m.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 32, 44, 64, 128, 184, 256, 512, 752, 1024, 2048, 4096, 8192, 12224, 16384, 32768, 49024, 61064, 65536, 131072, 262144, 524288, 981520, 1048576, 2097152, 4194304, 8388608, 12580864, 16777216, 33554432, 67108864, 134217728
Offset: 1

Views

Author

Matthew Vandermast, Aug 12 2004

Keywords

Comments

Contains all powers of 2 (A000079). Union of A000079 and A097215.
If 3*2^k - 1 is prime then 2^k*(3*2^k-1) is in the sequence. So 2^A002235*(3*2^A002235-1) is a subsequence of this sequence. - Farideh Firoozbakht, Aug 06 2005

Examples

			A total of 10 sets of distinct positive integers have a least common multiple of 10: {1,2,5}, {1,2,5,10}, {1,2,10}, {1,5,10}, {1,10}, {2,5}, {2,5,10}, {2,10}, {5,10} and {10}. Hence 10 is in the sequence.

Links

Crossrefs

Cf. A076078, A097215.
Cf. A002235.

Extensions

a(26) corrected by Jinyuan Wang, Feb 11 2020

A097215 Numbers m such that A076078(m) = m and bigomega(m) >= 2; or in other words, A097214, excluding powers of 2.

Original entry on oeis.org

10, 44, 184, 752, 12224, 49024, 61064, 981520, 12580864, 206158168064, 16492668126208, 1080863908958322688, 18374686467592175488, 885443715520878608384, 4703919738602662723328, 226673591177468092350464, 232113757366000005450563584, 3894222643901120685369075227951104
Offset: 1

Views

Author

Matthew Vandermast, Aug 12 2004

Keywords

Comments

A076078(m) equals the number of sets of distinct positive integers with a least common multiple of m.
If 3*2^k - 1 is an odd prime then 2^k*(3*2^k-1) is in the sequence. - Farideh Firoozbakht, May 03 2009
For what seems to be an appearance of this sequence in a different context, see Harborth (2013). - N. J. A. Sloane, Jun 08 2013

Examples

			For example, there are 184 sets of distinct positive integers with a least common multiple of 184.

Links

Crossrefs

Cf. A097214, A097416, A281993.
Cf. A002235. - Farideh Firoozbakht, May 03 2009

Programs

  • Mathematica
    f[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu[n/d](2^DivisorSigma[0, d] - 1))]; t = Union[ Table[ f[n], {n, 28000000}]]; Select[t, f[ # ] == # && !IntegerQ[ Log[2, # ]] &] (* Robert G. Wilson v, Aug 17 2004 *)
  • PARI
    A076078(n) = {local(f, l, s, t, q); f = factor(n); l = matsize(f)[1]; s = 0; forvec(v = vector(l, i, [0, 1]), q = sum(i = 1, l, v[i]); t = (-1)^(l - q)*2^prod(i = 1, l, f[i, 2] + v[i]); s += t); s; }
lista(nn) = {my(w=List([]), m=1, q=2, g); for(k=1, logint(nn, 2)-1, q=nextprime(q+1); m=m*q; for(r=1, nn\2^k-1, g=factor(A076078(m*2^r))[, 2]; if(#g==k+1&&g[2]==1, listput(w, A076078(m*2^r))))); Set(w); } \\ Jinyuan Wang, Feb 11 2020

Extensions

More terms from Robert G. Wilson v, Aug 18 2004
More terms from Jinyuan Wang, Feb 11 2020

A097218 Numbers n such that A076078(n) < n, where A076078(n) equals the number of sets of distinct positive integers with a least common multiple of n.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101, 103, 106
Offset: 1

Views

Author

Matthew Vandermast, Aug 12 2004

Keywords

A097416 Numbers n such that A076078(m)=n for some m, excluding powers of 2.

Original entry on oeis.org

10, 44, 184, 218, 400, 752, 3040, 3392, 3748, 12224, 27904, 49024, 57856, 61064, 64594, 196352, 226304, 253808, 785920, 954368, 981520, 1822720, 3144704, 12580864, 14630912, 15499264, 15722528, 16450240, 16700300, 31522816, 50327552
Offset: 1

Views

Author

Robert G. Wilson v, Aug 17 2004

Keywords

Comments

A076078(n) equals the number of sets of distinct positive integers with a least common multiple of n.

Links

Crossrefs

Cf. A076078, A097215.

Programs

  • Mathematica
    f[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu[n/d](2^DivisorSigma[0, d] - 1))]; t = Union[ Table[ f[n], {n, 2^20}]]; Complement[ Take[t, 52], Table[2^i, {i, 0, 20}]]

A097210 Numbers that appear in A076078.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 32, 44, 64, 128, 184, 218, 256, 400, 512, 752, 1024, 2048, 3040, 3392, 3748, 4096, 8192, 12224, 16384, 27904, 32768, 49024, 57856, 61064, 64594, 65536, 131072, 196352, 226304, 253808, 262144, 524288, 785920, 954368, 981520, 1048576
Offset: 1

Views

Author

Matthew Vandermast, Aug 01 2004

Keywords

Comments

A076078(n) = the number of sets of distinct positive integers with a least common multiple of n.
All powers of 2 are in the sequence. - Robert G. Wilson v, Aug 14 2004

Crossrefs

Cf. A076078, A097211.

Programs

  • Mathematica
    f[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu[n/d](2^DivisorSigma[0, d] - 1))]; Take[ Union[ Table[ f[n], {n, 2^20}]], 42] (* Robert G. Wilson v, Aug 14 2004 *)

A097211 a(n) = the number of sets of distinct positive integers with a least common multiple of A025487(n), i.e., A076078(A025487(n)).

Original entry on oeis.org

1, 2, 4, 10, 8, 44, 16, 184, 218, 32, 400, 752, 3748, 64, 3392, 3040, 61064, 128, 27904, 253808, 12224, 64594, 57856, 981520, 256, 226304, 16450240, 49024, 16700300, 954368, 15722528, 512, 1822720, 1055953664, 196352, 4278006328, 15499264
Offset: 1

Views

Author

Matthew Vandermast, Aug 09 2004

Keywords

Comments

The sequence A025487 contains the least number of each prime signature.
Sequence is a rearrangement of A097210 unless two or more members of A025487 are LCMs of an identical number of sets of distinct positive integers.

Links

Crossrefs

Cf. A025487, A076078, A097210.

Programs

  • Mathematica
    f[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu[n/d](2^DivisorSigma[0, d] - 1))]; PrimeExponents[n_] := Flatten[ Table[ # [[2]], {1}] & /@ FactorInteger[n]]; lpe = {}; ln = {1}; Do[pe = Sort[PrimeExponents[n]]; If[ Position[lpe, pe] == {}, AppendTo[lpe, pe]; AppendTo[ln, f[n]]], {n, 1000}]; ln (* Robert G. Wilson v, Aug 14 2004 *)

Extensions

Second comment edited by Matthew Vandermast, Oct 21 2008

A097216 Numbers n such that A076078(n) > n, where A076078(n) equals the number of sets of distinct positive integers with a least common multiple of n.

Original entry on oeis.org

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 130, 132, 135, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 165, 168, 170, 174, 176, 180, 182, 186, 190, 192, 195
Offset: 1

Views

Author

Matthew Vandermast, Aug 12 2004

Keywords

A097217 Odd numbers n such that A076078(n) > n, where A076078(n) equals the number of sets of distinct positive integers with a least common multiple of n.

Original entry on oeis.org

105, 135, 165, 195, 225, 315, 405, 495, 525, 567, 585, 675, 693, 765, 819, 825, 855, 945, 975, 1035, 1071, 1125, 1155, 1197, 1215, 1275, 1287, 1305, 1323, 1365, 1395, 1425, 1449, 1485, 1575, 1665, 1683, 1701, 1725, 1755, 1785, 1827, 1845, 1881, 1925
Offset: 1

Views

Author

Matthew Vandermast, Aug 13 2004

Keywords

Comments

Odd members of A097216.

Crossrefs

Cf. A097214, A097218.

A285572 Number of finite sets of pairwise indivisible positive integers with least common multiple n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 9, 1, 1, 2, 2, 2, 6, 1, 2, 2, 4, 1, 9, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 23, 1, 2, 3, 1, 2, 9, 1, 3, 2, 9, 1, 10, 1, 2, 3, 3, 2, 9, 1, 5, 1, 2, 1, 23, 2, 2, 2, 4, 1, 23, 2, 3, 2, 2, 2, 6, 1, 3, 3, 6
Offset: 1

Views

Author

Gus Wiseman, Apr 21 2017

Keywords

Examples

			The a(72)=10 sets are {72}, {8,9}, {8,18}, {8,36}, {9,24}, {18,24}, {24,36}, {6,8,9}, {8,9,12}, {8,12,18}.

Links

Crossrefs

Cf. A000666, A006126, A048143, A054921, A076078, A076413, A285573.

Programs

  • Mathematica
    nn=50;
stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[Rest[stableSets[Divisors[n],Divisible]],LCM@@#===n&]],{n,1,nn}]
Showing 1-10 of 48 results. Next

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