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%I #9 May 21 2022 08:29:37
%S 3,5,7,13,19,31,53,73,89,113,211,293,523,887,1129,1327,4297,4831,5351,
%T 5591,8467,12853,15683,19609,25471,31397,134513,155921,338033,360653,
%U 370261,492113,1349533,1357201,1561919,2010733,4652353,8421251,11113933,15203977,17051707
%N Primes p such that the number of distinct prime factors omega of the product of the composite numbers between p and the next prime after p sets a new record.
%e a(6) = 31, because the first product of consecutive composites with 6 primes in its squarefree kernel is P = 32*33*34*35*36 with rad(P) = 2*3*5*7*11*17 = 39270, whereas the interval starting after A354217(6) = 23 leads only to 5 distinct factors, i.e., rad(24*25*26*27*28) = 2*3*5*7*13, not sufficient to beat the record set by the composites after a(5) = A354217(5) = 19 with rad(20*21*22) = 2*3*5*7*11.
%t s = Array[PrimeNu[Times @@ FactorInteger[Times @@ Range[#1 + 1, #2 - 1]][[All, 1]] & @@ Map[Prime, # + {0, 1}]] &, 10^4]; Prime@ Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* _Michael De Vlieger_, May 20 2022 *)
%Y A354220 provides the corresponding values of omega.
%Y Cf. A001221, A076978, A354217.
%K nonn
%O 1,1
%A _Hugo Pfoertner_, May 20 2022