A115343 Products of 9 distinct primes.
223092870, 281291010, 300690390, 340510170, 358888530, 363993630, 380570190, 397687290, 406816410, 417086670, 434444010, 455885430, 458948490, 481410930, 485555070, 497668710, 504894390, 512942430, 514083570, 531990690, 538047510, 547777230, 551861310
Offset: 1
Examples
514083570 is in the sequence as it is equal to 2*3*5*7*11*13*17*19*53.
Links
- David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1045 terms from Vincenzo Librandi and Chai Wah Wu)
Crossrefs
Programs
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Maple
N:= 10^9: # to get all terms < N n0:= mul(ithprime(i),i=1..8): Primes:= select(isprime,[$1..floor(N/n0)]): nPrimes:= nops(Primes): for i from 1 to 9 do for j from 1 to nPrimes do M[i,j]:= convert(Primes[1..min(j,i)],`*`); od od: A:= {}: for i9 from 9 to nPrimes do m9:= Primes[i9]; for i8 in select(t -> M[7,t-1]*Primes[t]*m9 <= N, [$8..i9-1]) do m8:= m9*Primes[i8]; for i7 in select(t -> M[6,t-1]*Primes[t]*m8 <= N, [$7..i8-1]) do m7:= m8*Primes[i7]; for i6 in select(t -> M[5,t-1]*Primes[t]*m7 <= N, [$6..i7-1]) do m6:= m7*Primes[i6]; for i5 in select(t -> M[4,t-1]*Primes[t]*m6 <= N, [$5..i6-1]) do m5:= m6*Primes[i5]; for i4 in select(t -> M[3,t-1]*Primes[t]*m5 <= N, [$4..i5-1]) do m4:= m5*Primes[i4]; for i3 in select(t -> M[2,t-1]*Primes[t]*m4 <= N, [$3..i4-1]) do m3:= m4*Primes[i3]; for i2 in select(t -> M[1,t-1]*Primes[t]*m3 <= N, [$2..i3-1]) do m2:= m3*Primes[i2]; for i1 in select(t -> Primes[t]*m2 <= N, [$1..i2-1]) do A:= A union {m2*Primes[i1]}; od od od od od od od od od: A; # Robert Israel, Sep 02 2014 -
Mathematica
Module[{n=6*10^8,k},k=PrimePi[n/Times@@Prime[Range[8]]];Select[ Union[ Times@@@ Subsets[Prime[Range[k]],{9}]],#<=n&]](* Harvey P. Dale with suggestions from Jean-François Alcover, Sep 03 2014 *) n = 10^9; n0 = Times @@ Prime[Range[8]]; primes = Select[Range[Floor[n/n0]], PrimeQ]; nPrimes = Length[primes]; Do[M[i, j] = Times @@ primes[[1 ;; Min[j, i]]], {i, 1, 9}, {j, 1, nPrimes}]; A = {}; Do[m9 = primes[[i9]]; Do[m8 = m9*primes[[i8]]; Do[m7 = m8*primes[[i7]]; Do[m6 = m7*primes[[i6]]; Do[m5 = m6*primes[[i5]]; Do[m4 = m5*primes[[i4]]; Do[m3 = m4*primes[[i3]]; Do[m2 = m3*primes[[i2]]; Do[A = A ~Union~ {m2*primes[[i1]]}, {i1, Select[Range[1, i2-1], primes[[#]]*m2 <= n &]}], {i2, Select[Range[2, i3-1], M[1, #-1]*primes[[#]]*m3 <= n &]}], {i3, Select[Range[3, i4-1], M[2, #-1]*primes[[#]]*m4 <= n &]}], {i4, Select[Range[4, i5-1], M[3, #-1]*primes[[#]]*m5 <= n &]}], {i5, Select[Range[5, i6-1], M[4, #-1]*primes[[#]]*m6 <= n &]}], {i6, Select[Range[6, i7-1], M[5, #-1]*primes[[#]]*m7 <= n &]}], {i7, Select[Range[7, i8-1], M[6, #-1]*primes[[#]]*m8 <= n &]}], {i8, Select[Range[8, i9-1], M[7, #-1]*primes[[#]]*m9 <= n &]}], {i9, 9, nPrimes}]; A (* Jean-François Alcover, Sep 03 2014, translated and adapted from Robert Israel's Maple program *) -
PARI
is(n)=omega(n)==9 && bigomega(n)==9 \\ Hugo Pfoertner, Dec 18 2018
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Python
from operator import mul from functools import reduce from sympy import nextprime, sieve from itertools import combinations n = 190 m = 9699690*nextprime(n-1) A115343 = [] for x in combinations(sieve.primerange(1,n),9): y = reduce(mul,(d for d in x)) if y < m: A115343.append(y) A115343 = sorted(A115343) # Chai Wah Wu, Sep 02 2014
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Python
from math import prod, isqrt from sympy import primerange, integer_nthroot, primepi def A115343(n): def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1))) def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,9))) def bisection(f,kmin=0,kmax=1): while f(kmax) > kmax: kmax <<= 1 while kmax-kmin > 1: kmid = kmax+kmin>>1 if f(kmid) <= kmid: kmax = kmid else: kmin = kmid return kmax return bisection(f) # Chai Wah Wu, Aug 31 2024
Extensions
Corrected and extended by Don Reble, Mar 09 2006
More terms and corrected b-file from Chai Wah Wu, Sep 02 2014