A079890 -id:A079890 - cp's OEIS frontend

A079891 a(n) = gcd(n, A079890(n)).

1, 2, 1, 4, 1, 2, 1, 8, 3, 2, 1, 4, 1, 2, 3, 16, 1, 6, 1, 4, 3, 1, 1, 8, 1, 1, 9, 4, 1, 6, 1, 32, 3, 2, 7, 12, 1, 2, 3, 8, 1, 6, 1, 2, 9, 2, 1, 16, 1, 2, 1, 2, 1, 18, 1, 8, 3, 1, 1, 12, 1, 1, 9, 64, 1, 3, 1, 1, 1, 1, 1, 24, 1, 1, 3, 1, 1, 3, 1, 16, 27, 2, 1, 12, 1, 2, 1, 4, 1, 18, 1, 4, 1, 2, 1, 32, 1, 2, 1

Offset: 1

Reinhard Zumkeller, Jan 14 2003

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A079893, A079894.

lng[n_]:=Module[{x=n+1,pon=PrimeOmega[n]},While[PrimeOmega[x]-pon! = 1, x++]; x];Table[GCD[n,lng[n]],{n,100}] (* Harvey P. Dale, Jun 10 2013 *)

A079894 a(n) = gcd(A079890(n), A079892(n)).

Original entry on oeis.org

2, 2, 2, 2, 6, 2, 1, 2, 2, 6, 2, 2, 14, 6, 6, 2, 3, 6, 1, 6, 3, 3, 1, 2, 1, 3, 4, 6, 33, 6, 33, 1, 42, 42, 42, 6, 38, 42, 42, 6, 2, 6, 2, 6, 6, 10, 1, 4, 50, 6, 4, 6, 1, 12, 3, 12, 3, 3, 62, 6, 62, 3, 3, 1, 66, 3, 1, 1, 70, 3, 2, 6, 74, 3, 3, 3, 78, 3, 2, 12, 2, 4, 85, 6, 2, 2, 2, 18, 91, 6, 2, 2, 2, 2

Offset: 1

Reinhard Zumkeller, Jan 14 2003

nonn

Cf. A079891, A079893.

A076156 Numbers n such that Omega(n+1) = Omega(n)+1, where Omega(m) (A001222) denotes the number of prime factors of m, counting multiplicity.

Original entry on oeis.org

1, 3, 5, 13, 26, 37, 49, 51, 61, 62, 65, 69, 73, 74, 77, 91, 99, 115, 123, 125, 129, 146, 157, 169, 185, 187, 188, 193, 194, 195, 206, 221, 231, 235, 237, 254, 265, 267, 274, 275, 277, 278, 289, 291, 309, 313, 321, 343, 355, 362, 363, 365, 374, 386, 397, 398

Offset: 1

Joseph L. Pe, Nov 01 2002

A079890(a(n)) = a(n)+1. - Reinhard Zumkeller, Feb 01 2008

			Omega(26 + 1) = 3 = 1 + 2 = 1 + Omega(26), so 26 is a term of the sequence.

Zak Seidov, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller).

Omega[n_] := Apply[Plus, Transpose[FactorInteger[n]][[2]]]; l = {1}; Do[ If[Omega[i + 1] == Omega[i] + 1, l = Append[l, i]], {i, 2, 10^3}]; l
Position[Partition[PrimeOmega[Range[400]],2,1],?(#[[1]]+1==#[[2]]&), 1, Heads->False]//Flatten (* _Harvey P. Dale, May 15 2018 *)

a(n) seems to be asymptotic to c*n where c=7.6.... - Benoit Cloitre, Jan 15 2003

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

A087913 Greatest number less than n such that the number of prime factors is one less than that of n; a(1)=0.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 6, 7, 7, 1, 10, 1, 13, 13, 12, 1, 15, 1, 15, 19, 19, 1, 20, 23, 23, 26, 26, 1, 26, 1, 24, 31, 31, 31, 30, 1, 37, 37, 30, 1, 39, 1, 39, 39, 43, 1, 40, 47, 49, 47, 51, 1, 52, 53, 52, 53, 53, 1, 52, 1, 61, 62, 48, 61, 65, 1, 65, 67, 69, 1, 60, 1, 73, 74, 74, 73

Offset: 1

Reinhard Zumkeller, Oct 26 2003

nonn

A001222(a(n)) = A001222(n) - 1 for n>1.

Cf. A079890.

cp's OEIS Frontend

A079891 a(n) = gcd(n, A079890(n)).

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A079894 a(n) = gcd(A079890(n), A079892(n)).

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A076156 Numbers n such that Omega(n+1) = Omega(n)+1, where Omega(m) (A001222) denotes the number of prime factors of m, counting multiplicity.

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

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Haskell

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

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A087913 Greatest number less than n such that the number of prime factors is one less than that of n; a(1)=0.

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