A165712 -id:A165712 - cp's OEIS frontend

A079890 Least number > n having one more prime factor than n, not necessarily distinct.

2, 4, 4, 8, 6, 8, 9, 16, 12, 12, 14, 16, 14, 18, 18, 32, 21, 24, 21, 24, 27, 27, 25, 32, 27, 27, 36, 36, 33, 36, 33, 64, 42, 42, 42, 48, 38, 42, 42, 48, 46, 54, 46, 54, 54, 50, 49, 64, 50, 54, 52, 54, 55, 72, 63, 72, 63, 63, 62, 72, 62, 63, 81, 128, 66, 81, 69, 81, 70, 81, 74, 96

Offset: 1

Reinhard Zumkeller, Jan 14 2003

A001222(a(n)) = A001222(n) + 1;
a(2^k) = 2^(k+1).
a(A076156(n)) = A076156(n)+1. - Reinhard Zumkeller, Feb 01 2008

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

Cf. A079891, A079892.

Cf. A165712.

a079890 n = head [x | x <- [n + 1 ..], a001222 x == 1 + a001222 n]
-- Reinhard Zumkeller, Aug 29 2013

lng[n_]:=Module[{x=n+1,pon=PrimeOmega[n]},While[PrimeOmega[x]-pon!=1, x++]; x]; Array[lng,80] (* Harvey P. Dale, Nov 09 2011 *)

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

Original entry on oeis.org

3, 4, 5, 7, 10, 8, 9, 11, 12, 13, 14, 16, 15, 18, 17, 19, 20, 23, 21, 22, 24, 25, 26, 27, 28, 29, 33, 31, 42, 32, 37, 34, 35, 36, 38, 41, 39, 40, 44, 43, 60, 47, 45, 46, 48, 49, 50, 53, 51, 52, 54, 59, 55, 56, 57, 58, 62, 61, 66, 64, 63, 65, 67, 68, 70, 71, 69, 72, 78, 73, 74

Offset: 2

Leroy Quet, Sep 24 2009

nonn

			12 = 2^2 *3, and so is divisible by exactly 2 distinct primes. The next larger number divisible by exactly 2 distinct primes is 14, which is 2*7. So a(12) = 14.

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

Cf. A165712, A137929.

Cf. A001221, A079892.

a165713 n = head [x | x <- [n + 1 ..], a001221 x == a001221 n]
-- Reinhard Zumkeller, Aug 29 2013

a[n_] := For[nu = PrimeNu[n]; k = n+1, True, k++, If[PrimeNu[k] == nu, Return[k]]]; Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Nov 18 2013 *)

More terms from Sean A. Irvine, Feb 10 2010

A364052 a(n) is the least k such that no number with distinct base-n digits is the product of k (not necessarily distinct) primes.

Original entry on oeis.org

2, 3, 7, 9, 12, 14, 14, 21, 26, 28, 33, 36, 40, 45, 36, 50, 59, 61, 65, 70, 75, 77, 85, 89, 94, 97, 104, 107, 113, 118, 84

Offset: 2

Robert Israel, Jul 04 2023

A364049(n) <= a(n) <= 1 + floor(log_2(A062813(n))).

			a(4) = 7 because 2 = 2^1 = 2_4, 4 = 2^2 = 10_4, 8 = 2^3 = 20_4, 24 = 2^3 * 3 = 120_4, 108 = 2^2 * 3^3 = 1230_4 and 216 = 2^3 * 3^3 = 3120_4 have distinct base-4 digits and are products of 1 to 6 primes respectively, but there is no product of 7 primes that has distinct base-4 digits.

Cf. A001222, A062813, A165712, A364049.

f:= proc(n) local d,S,V,k;
  V:= {};
  for d from 1 to n do
    S:= select(t -> t[-1] <> 0, combinat:-permute([$0..n-1],d));
    S:= map(proc(t) local i;  numtheory:-bigomega(add(t[i]*n^(i-1),i=1..d)) end proc, S);
    V:= V union convert(S,set);
  od;
  min({$1..1+max(V)} minus V)
end proc:
map(f, [$2..10]);

a(11) from Jon E. Schoenfield, Jul 05 2023
a(12) from Martin Ehrenstein, Jul 07 2023
a(13)-a(18) from Jon E. Schoenfield, Jul 08 2023
a(19)-a(22) from Pontus von Brömssen, Jul 13 2023
a(23)-a(32) from Bert Dobbelaere, Jul 20 2023

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A079890 Least number > n having one more prime factor than n, not necessarily distinct.

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

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A364052 a(n) is the least k such that no number with distinct base-n digits is the product of k (not necessarily distinct) primes.

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