OFFSET
1,1
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1045 terms from Vincenzo Librandi and Chai Wah Wu)
EXAMPLE
514083570 is in the sequence as it is equal to 2*3*5*7*11*13*17*19*53.
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 *)
PROG
(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)
(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
(PARI) is(n)=omega(n)==9 && bigomega(n)==9 \\ Hugo Pfoertner, Dec 18 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jonathan Vos Post, Mar 06 2006
EXTENSIONS
Corrected and extended by Don Reble, Mar 09 2006
More terms and corrected b-file from Chai Wah Wu, Sep 02 2014
STATUS
approved