A354219 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.

3, 5, 7, 13, 19, 31, 53, 73, 89, 113, 211, 293, 523, 887, 1129, 1327, 4297, 4831, 5351, 5591, 8467, 12853, 15683, 19609, 25471, 31397, 134513, 155921, 338033, 360653, 370261, 492113, 1349533, 1357201, 1561919, 2010733, 4652353, 8421251, 11113933, 15203977, 17051707

Offset: 1

Author

Hugo Pfoertner, May 20 2022

Keywords

• nonn

Examples

			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.
		

Crossrefs

A354220 provides the corresponding values of omega.

Cf. A001221, A076978, A354217.

Programs

Mathematica

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 *)
DeleteDuplicates[Table[{p,Length[Union[Flatten[FactorInteger[#][[;;,1]]&/@Range[p+1,NextPrime[p]-1]]]]},{p,Prime[Range[ 2,10^6]]}], GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* The program generates the first 40 terms of the sequence, i.e., every term up to the 1 millionth prime. *) (* Harvey P. Dale, Feb 01 2025 *)