A122937 -id:A122937 - cp's OEIS frontend

A089016 Largest n-round number.

2, 30, 1260, 60060, 2042040, 446185740, 25878772920, 7420738134810, 304250263527210, 52331045326680120, 9223346738827371150, 1922760350154212639070, 469153525437627883933080

Offset: 0

Paul Boddington, Nov 04 2003

A positive integer m is said to be n-round if it is divisible by all primes p satisfying p^(n+1) < m, or equivalently if all positive integers t < m satisfying GCD(t,m)=1 are divisible by at most n primes (counting multiplicities). Using the fact that p_(t+1)<2*p_t (p_t the (t)th prime) it is easy to prove that there are only finitely many n-round numbers for each n. 1-round numbers are usually called very round (A048597).

			a(4)=2042040 as follows. Certainly it is 4-round since it is <= 19^5 and divisible by all primes < 19. Also it is > 17^5, hence the largest 4-round number must be a multiple of 510510 = 2.3.5.7.11.13.17. But no 4-round number can be > 19^5 (since it is easy to prove that if p is a prime >= 19 and q is the next prime after p then 2.3.5....p > q^5 ). Thus 2042040, being the largest multiple of 510510 which is <= 19^5, must be the largest 4-round number.

T. D. Noe, Table of n, a(n) for n=0..100

Cf. A048597, A122936 (2-round numbers), A122937 (3-round numbers).

Table[k=1; While[prod=Times@@Prime[Range[k]]; prodT. D. Noe, Sep 21 2006 *)

More terms from T. D. Noe, Sep 21 2006

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 *)

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

Original entry on oeis.org

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}]

cp's OEIS Frontend

A089016 Largest n-round number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

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

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica