OFFSET
1,2
COMMENTS
The Heinz numbers of the partitions into parts 2,3, and 4 (including the number 1, the Heinz number of the empty partition). We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [2,3,3,4] the Heinz number is 3*5*5*7 = 525; it is in the sequence. - Emeric Deutsch , May 21 2015
Numbers m | 105^e with integer e >= 0. - Michael De Vlieger, Aug 22 2019
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
FORMULA
Sum_{n>=1} 1/a(n) = (3*5*7)/((3-1)*(5-1)*(7-1)) = 35/16. - Amiram Eldar, Sep 22 2020
a(n) ~ exp((6*log(3)*log(5)*log(7)*n)^(1/3)) / sqrt(105). - Vaclav Kotesovec, Sep 23 2020
MAPLE
with(numtheory): S := {}: for j to 3100 do if `subset`(factorset(j), {3, 5, 7}) then S := `union`(S, {j}) else end if end do: S; # Emeric Deutsch, May 21 2015
# alternative
isA108347 := proc(n)
if n = 1 then
true;
else
return (numtheory[factorset](n) minus {3, 5, 7} = {} );
end if;
end proc:
A108347 := proc(n)
option remember;
if n = 1 then
1;
else
for a from procname(n-1)+1 do
if isA108347(a) then
return a;
end if;
end do:
end if;
end proc:
seq(A108347(n), n=1..80); # R. J. Mathar, Jun 06 2024
MATHEMATICA
With[{n = 3087}, Sort@ Flatten@ Table[3^i * 5^j * 7^k, {i, 0, Log[3, n]}, {j, 0, Log[5, n/2^i]}, {k, 0, Log[7, n/(3^i*5^j)]}]] (* Michael De Vlieger, Aug 22 2019 *)
PROG
(Magma) [n: n in [1..4000] | PrimeDivisors(n) subset [3, 5, 7]]; // Bruno Berselli, Sep 24 2012
(Python)
from sympy import integer_log
def A108347(n):
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
def f(x):
c = n+x
for i in range(integer_log(x, 7)[0]+1):
for j in range(integer_log(m:=x//7**i, 5)[0]+1):
c -= integer_log(m//5**j, 3)[0]+1
return c
return bisection(f, n, n) # Chai Wah Wu, Sep 16 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Douglas Winston (douglas.winston(AT)srupc.com), Jul 01 2005
STATUS
approved