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

%I A098962 #13 Jul 12 2024 22:19:21
%S A098962 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,
%T A098962 71,73,77,79,83,89,91,97,101,103,107,109,113,119,121,127,131,137,139,
%U A098962 143,149,151,157,163,167,169,173,179,181,187,191,193,197,199,209,211
%N A098962 Smallest sequence such that over all terms each prime p occurs exactly p times as prime factor; a(1)=1.
%C A098962 A000040 is a subsequence.
%C A098962 A099619 gives k such that a(k) = n-th prime; A099620 gives the greatest term divisible by the n-th prime.
%C A098962 A001222(a(n)) <= 2.
%H A098962 Reinhard Zumkeller, <a href="/A098962/b098962.txt">Table of n, a(n) for n = 1..10000</a>
%F A098962 #{(n,k): A027746(a(n),k)=p, 1<=k<=A001222(a(n))} = p for all primes p. - _Reinhard Zumkeller_, Feb 17 2013
%e A098962 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;
%e A098962 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;
%e A098962 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;
%e A098962 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.
%e A098962 - _Reinhard Zumkeller_, Feb 17 2013
%o A098962 (Haskell)
%o A098962 import Data.List (delete)
%o A098962 a098962 n = a098962_list !! (n-1)
%o A098962 a098962_list = 1 : f [2..] (tail a175944_list) where
%o A098962    f xs'@(x:xs) ps'@(p:ps)
%o A098962      | a010051 x == 1    = x : f xs (delete x ps')
%o A098962      | u == q && v == q' = x : f xs' zs
%o A098962      | otherwise         = f xs ps'
%o A098962      where q = a020639 x; q' = div x q
%o A098962            (us, u:us') = span (< q) ps'
%o A098962            (vs, v:vs') = span (< q') us'
%o A098962            zs@(z:_) = us ++ vs ++ vs'
%o A098962            xs' = if z == p then xs else filter ((> 0) . (`mod` p)) xs
%o A098962 -- _Reinhard Zumkeller_, Feb 17 2013
%Y A098962 Cf. A020639, A008578, A005145, subsequence of A037143.
%K A098962 nonn
%O A098962 1,2
%A A098962 _Reinhard Zumkeller_, Oct 22 2004