A076978
Product of the distinct primes dividing the product of composite numbers between consecutive primes.
Original entry on oeis.org
1, 2, 6, 30, 6, 210, 6, 2310, 2730, 30, 39270, 7410, 42, 7590, 46410, 1272810, 30, 930930, 82110, 6, 21111090, 1230, 48969690, 1738215570, 2310, 102, 144690, 6, 85470, 29594505363092670, 16770, 49990710, 138, 7849357706190, 30, 300690390, 20223210, 1122990, 37916970
Offset: 1
a(4) = product of prime divisors of the product of composite numbers between 7 and 11 = 2 * 3 * 5 = 30.
a(5)=6 because 12 is the only composite number between the 5th and the 6th primes (11 and 13) and largest squarefree divisor of 12 is 6.
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with(numtheory): b:=proc(j) if issqrfree(j) then j else fi end: a:=proc(n) local B,BB: B:=divisors(product(i,i=ithprime(n)+1..ithprime(n+1)-1)): BB:=(seq(b(B[j]),j=1..nops(B))): max(BB); end: seq(a(n),n=1..33); # Emeric Deutsch, Jul 28 2006
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f[n_] := Times @@ (First@# & /@ FactorInteger[Times @@ Range[Prime[n] + 1, Prime[n + 1] - 1]]); Array[f, 50] (* Robert G. Wilson v, Dec 02 2020 *)
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a(n) = my(p=1); forcomposite(c=prime(n), prime(n+1), p*=c); factorback(factorint(p)[, 1]); \\ Michel Marcus, May 29 2022
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from sympy import sieve as p, primefactors
def A076978(n):
result = 1
for composites in range(p[n]+1, p[n+1]):
for primefactor in primefactors(composites):
if result % primefactor != 0: result *= primefactor
return result # Karl-Heinz Hofmann, May 30 2022
Original entry on oeis.org
1, 3, 15, 105, 1155, 1365, 19635, 23205, 636405, 10555545, 24484845, 869107785, 14797252681546335, 92442344345566215, 40334203530676690635, 1451417351374223318085, 6087255082617244520985, 469256253416014832182075245585, 1519012498286389398934397206158552189345
Offset: 1
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1/2*Union@ FoldList[Max, Array[Times @@ FactorInteger[Times @@ Range[#1 + 1, #2 - 1]][[All, 1]] & @@ Map[Prime, # + {0, 1}] &, 180, 2]] (* Michael De Vlieger, May 20 2022 *)
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a354218(maxterm) = {my(mp=0,pp=3); forprime(p=5,oo, my(L=List());for(j=pp+1,p-1, my(f=factor(j), nf=#f~); for(k=1,nf, listput(~L,f~[1,k]))); listsort(L,1); my(mpp=prod(k=1,#L,L[k])); if(mpp>mp, if(mpp<2*maxterm, print1(mpp/2,", "); mp=mpp, return)); pp=p)};
a354218(10^75)
-
from sympy import sieve as p, primefactors
def A076978_halfen(n): # for all integers n > 1
result = 1
for composites in range(p[n]+1, p[n+1]):
for primefactor in primefactors(composites):
if result % primefactor != 0: result *= primefactor
return result//2
A354218 = [1]
for n in range(2,180):
if A076978_halfen(n) > A354218[-1]: A354218.append(A076978_halfen(n))
print(A354218) # Karl-Heinz Hofmann, May 20 2022
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.
Original entry on oeis.org
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
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.
A354220 provides the corresponding values of omega.
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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 *)
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