A098962 Smallest sequence such that over all terms each prime p occurs exactly p times as prime factor; a(1)=1.

1, 2, 3, 5, 6, 7, 11, 13, 15, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209, 211

Offset: 1

Author

Reinhard Zumkeller, Oct 22 2004

Keywords

• nonn

Comments

A000040 is a subsequence.

A099619 gives k such that a(k) = n-th prime; A099620 gives the greatest term divisible by the n-th prime.

A001222(a(n)) <= 2.

Examples

			p=2: a(2)=2, a(5)=6=2*3: 4=2*2 is missing, otherwise 2 would occur more than 2 times, there are no more even terms greater than 6;
p=3: a(3)=3, a(5)=6=2*3, a(9)=15=3*5: 9=3*3 is missing, otherwise 3 would occur more than 3 times, there are no more multiples of 3 greater than 15;
p=5: a(4)=5, a(9)=15=3*5, a(13)=25=5*5, a(16)=35=5*7: no more multiples of 5 greater than 35;
p=7: a(6)=7, a(16)=35=5*7, a(21)=49=7*7, a(28)=77=7*11, a(32)=91=7*13, a(39)=119=7*17: no more multiples of 7 greater than 119.
- _Reinhard Zumkeller_, Feb 17 2013
		

Links

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

Crossrefs

Cf. A020639, A008578, A005145, subsequence of A037143.

Programs

Haskell

import Data.List (delete)
a098962 n = a098962_list !! (n-1)
a098962_list = 1 : f [2..] (tail a175944_list) where
   f xs'@(x:xs) ps'@(p:ps)
     | a010051 x == 1    = x : f xs (delete x ps')
     | u == q && v == q' = x : f xs' zs
     | otherwise         = f xs ps'
     where q = a020639 x; q' = div x q
           (us, u:us') = span (< q) ps'
           (vs, v:vs') = span (< q') us'
           zs@(z:_) = us ++ vs ++ vs'
           xs' = if z == p then xs else filter ((> 0) . (`mod` p)) xs
-- Reinhard Zumkeller, Feb 17 2013

Formula

#{(n,k): A027746(a(n),k)=p, 1<=k<=A001222(a(n))} = p for all primes p. - Reinhard Zumkeller, Feb 17 2013