A089016 -id:A089016 - cp's OEIS frontend

A383081 a(n) = A003557(A089016(n)).

1, 1, 6, 2, 4, 2, 4, 1, 1, 4, 15, 1, 4, 10, 15, 1, 5, 24, 52, 1, 2, 5, 1, 3, 5, 31, 45, 2, 4, 14, 41, 1, 2, 5, 7, 2, 4, 5, 11, 30, 43, 2, 6, 18, 26, 71, 139, 3, 36, 69, 96, 5, 14, 69, 95, 1, 4, 11, 30, 57, 1, 2, 8, 39, 54, 255, 2, 3, 9, 32, 60, 82, 2, 7, 13, 45

Offset: 0

Michael De Vlieger, Apr 18 2025

nonn

Largest n-round number A089016(n) = k*P. Then a(n) = k = rad(k*P), where P is in A002110 and rad = A007947.

Michael De Vlieger, Table of n, a(n) for n = 0..5000

Cf. A002110, A003557, A007947, A048597, A089016, A122936, A122937.

Table[k = P = 1;
  While[P *= Prime[k]; P < Prime[k + 1]^(n + 1), k++];
  P /= Prime[k]; Floor[Prime[k]^(n + 1)/P], {n, 0, 75}]

A122937 3-Round numbers: numbers n such that every number less than n and relatively prime to n has at most three prime factors (counting multiplicities).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192

Offset: 1

T. D. Noe, Sep 21 2006

This sequence, for r=3 prime factors, is finite. Maillet proved that such sequences are finite for any fixed r. The case r=1 is A048597; case r=2 is A122936.

Dickson, History of the Theory of Numbers, Vol. I, Chelsea, New York, 1952, p. 134.

T. D. Noe, Table of n, a(n) for n=1..265 (complete sequence)

Cf. A048597 (very round numbers), A051250, A089016 (largest n-round number).

Omega[n_] := If[n==1, 0, Plus@@(Transpose[FactorInteger[n]][[2]])]; nn=60060; r=3; moreThanR=Select[Range[nn], Omega[ # ]>r&]; lst={1}; Do[s=Select[Range[n],GCD[n,# ]==1&]; If[Intersection[s,moreThanR]=={}, AppendTo[lst,n]], {n,2,nn}]; lst

A122936 2-Round numbers: numbers n such that every number less than n and relatively prime to n has at most two prime factors (counting multiplicities).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 150, 180, 210, 240, 270, 300, 330, 420, 630, 840, 1050, 1260

Offset: 1

T. D. Noe, Sep 21 2006

This sequence, for r=2 prime factors, is finite. Maillet proved that such sequences are finite for any fixed r. The case r=1 is A048597; case r=3 is A122937.

Dickson, History of the Theory of Numbers, Vol. I, Chelsea, New York, 1952, p. 134.

H. Bonse, Über eine bekannte Eigenshaft der Zahl 30 und ihre Verallgemeinerung, Archiv d. Math. u. Physik (3) vol. 12 (1907) 292-295. See page 295.

Cf. A048597 (very round numbers), A051250, A089016 (largest n-round number).

Omega[n_] := If[n==1, 0, Plus@@(Transpose[FactorInteger[n]][[2]])]; nn=1260; r=2; moreThanR=Select[Range[nn], Omega[ # ]>r&]; lst={1}; Do[s=Select[Range[n],GCD[n,# ]==1&]; If[Intersection[s,moreThanR]=={}, AppendTo[lst,n]], {n,2,nn}]; lst
tpfQ[n_] :=Max[PrimeOmega /@ Select[Range[n - 1], CoprimeQ[#, n] &]] < 3; Select[Range[1300],tpfQ] (* Harvey P. Dale, Mar 16 2016 *)

cp's OEIS Frontend

A383081 a(n) = A003557(A089016(n)).

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A122937 3-Round numbers: numbers n such that every number less than n and relatively prime to n has at most three prime factors (counting multiplicities).

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Author

Keywords

Comments

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Programs

Mathematica

A122936 2-Round numbers: numbers n such that every number less than n and relatively prime to n has at most two prime factors (counting multiplicities).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica