A165713 -id:A165713 - cp's OEIS frontend

A165714 Let the prime factorization of m be m = product p(m,k)^b(m,k), where p(m,j)A165713(n), k)^b(n,k)}.

3, 2, 25, 7, 10, 2, 27, 121, 6, 13, 28, 2, 15, 6, 83521, 19, 50, 23, 63, 22, 6, 5, 104, 9, 14, 24389, 99, 31, 42, 2, 69343957, 34, 35, 6, 1444, 41, 39, 10, 88, 43, 30, 47, 45, 92, 6, 7, 80, 2809, 867, 26, 12, 59, 6655, 14, 513, 58, 62, 61, 132, 2, 21, 325, 90458382169, 34

Offset: 2

Leroy Quet, Sep 24 2009

nonn

A165713(n) = the smallest integer > n that is divisible by exactly the same number of distinct primes as n is.

			12 = 2^2 * 3^1, which is divisible by 2 distinct primes. The next larger integer divisible by exactly 2 distinct primes is 14 = 2^1 * 7^1. Taking the primes from the factorization of 14 and the exponents from the factorization of 12, we have a(12) = 2^2 * 7^1 = 28.

Cf. A165713, A165715.

Extended by Ray Chandler, Mar 12 2010

A165715 Let the prime factorization of m be m = product p(m,k)^b(m,k), where p(m,j) < p(m, j+1) for all j, the p's are the distinct primes dividing m, and each b is a positive integer. Then a(n) = product {p(n,k)^b(A165713(n), k)}.

Original entry on oeis.org

2, 9, 2, 5, 6, 343, 4, 3, 20, 11, 6, 28561, 14, 75, 2, 17, 12, 19, 10, 21, 88, 529, 6, 125, 52, 3, 14, 29, 30, 28629151, 2, 33, 34, 1225, 6, 37, 38, 351, 20, 41, 84, 43, 44, 15, 368, 2209, 18, 7, 10, 153, 4394, 53, 6, 1375, 14, 57, 58, 59, 30, 51520374361, 124, 21, 2

Offset: 2

Leroy Quet, Sep 24 2009

nonn

A165713(n) = the smallest integer > n that is divisible by exactly the same number of distinct primes as n is.

			12 = 2^2 * 3^1, which is divisible by 2 distinct primes. The next larger integer divisible by exactly 2 distinct primes is 14 = 2^1 * 7^1. Taking the primes from the factorization of 12 and the exponents from the factorization of 14, we have a(12) = 2^1 * 3^1 = 6.

Cf. A165713, A165714.

Extended by Ray Chandler, Mar 12 2010

A079892 Least number > n having one more distinct prime factor than n.

Original entry on oeis.org

2, 6, 6, 6, 6, 30, 10, 10, 10, 30, 12, 30, 14, 30, 30, 18, 18, 30, 20, 30, 30, 30, 24, 30, 26, 30, 28, 30, 33, 210, 33, 33, 42, 42, 42, 42, 38, 42, 42, 42, 44, 210, 44, 60, 60, 60, 48, 60, 50, 60, 60, 60, 54, 60, 60, 60, 60, 60, 62, 210, 62, 66, 66, 65, 66, 210, 68, 70, 70, 210

Offset: 1

Reinhard Zumkeller, Jan 14 2003

nonn

A001221(a(n)) = A001221(n) + 1;

a(A002110(k)) = A002110(k+1).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A079893, A079890.

Cf. A165713.

a079892 n = head [x | x <- [n + 1 ..], a001221 x == 1 + a001221 n]
-- Reinhard Zumkeller, Aug 29 2013

A165712 a(n) = the smallest integer > n that is divisible by exactly the same number of primes (counted with multiplicity) as n is.

Original entry on oeis.org

3, 5, 6, 7, 9, 11, 12, 10, 14, 13, 18, 17, 15, 21, 24, 19, 20, 23, 27, 22, 25, 29, 36, 26, 33, 28, 30, 31, 42, 37, 48, 34, 35, 38, 40, 41, 39, 46, 54, 43, 44, 47, 45, 50, 49, 53, 72, 51, 52, 55, 63, 59, 56, 57, 60, 58, 62, 61, 81, 67, 65, 66, 96, 69, 68, 71, 70, 74, 75, 73, 80

Offset: 2

Leroy Quet, Sep 24 2009

nonn

			8 = 2^3, and so is divisible by exactly 3 primes counted with multiplicity. The next larger number divisible by exactly 3 primes counted with multiplicity is 12, which is 2^2 *3. So a(8) = 12.

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A165713.

Cf. A001222, A079890.

a165712 n = head [x | x <- [n + 1 ..], a001222 x == a001222 n]
-- Reinhard Zumkeller, Aug 29 2013

a[n_] := For[Om = PrimeOmega[n]; k = n+1, True, k++, If[PrimeOmega[k] == Om, Return[k]]]; Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Jul 21 2017 *)
Module[{nn = 10^2, s, t}, s = PositionIndex@ Array[PrimeOmega, {nn}]; t = ConstantArray[0, nn]; TakeWhile[#, # > 0 &] &@ Rest@ ReplacePart[t, Flatten@ Map[#1 -> #2 & @@ # &, Map[Partition[Lookup[s, #], 2, 1] &, Keys@ s], {2}]]] (* Michael De Vlieger, Jul 21 2017 *)

a(n) = {my(bon = bigomega(n)); my(k = n+1); while (bigomega(k) != bon, k++); k;} \\ Michel Marcus, Jul 21 2017

Extended by Ray Chandler, Mar 12 2010

cp's OEIS Frontend

A165714 Let the prime factorization of m be m = product p(m,k)^b(m,k), where p(m,j)A165713(n), k)^b(n,k)}.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A165715 Let the prime factorization of m be m = product p(m,k)^b(m,k), where p(m,j) < p(m, j+1) for all j, the p's are the distinct primes dividing m, and each b is a positive integer. Then a(n) = product {p(n,k)^b(A165713(n), k)}.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A079892 Least number > n having one more distinct prime factor than n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

A165712 a(n) = the smallest integer > n that is divisible by exactly the same number of primes (counted with multiplicity) as n is.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions