login
A262081
a(1) = 6; for n > 1, a(n) = the least squarefree composite number whose sum of prime factors is prime and whose greatest prime factor is the sum of prime factors of a(n-1).
1
6, 10, 210, 34, 399, 58, 651, 82, 903, 795, 1281, 142, 1533, 2739, 2037, 214, 6213, 1965, 2919, 298, 5285, 3423, 2595, 3801, 382, 12545, 7385, 4683, 3495, 5061, 8785, 3945, 5691, 562, 5943, 9669, 6447, 10461, 12909, 694, 7329, 5385, 12845, 7959, 5835, 13895
OFFSET
1,1
COMMENTS
a(n) is a composite squarefree integer in order to avoid the trivial sequences 3, 6, 5, 5, 5, ... or 6, 10, 49, 49, 49, ... if a(1)=3 or a(1)=6.
The greatest prime factors of a(n) are 3, 5, 7, 17, 19, 29, 31, 41, 43, 53, 61, 71, 73, 83, 97, 107, 109, 131, 139, 149, 151, 163, 173, 181, 191, 193, 211, 223, 233, 241, ...
The definition expressed with A-numbers: a(1) = 6; for n > 1, a(n) = the least term k of A120944 such that A001414(k) is prime and A006530(k) = A001414(a(n-1)). Also A008472 could be used instead of A001414 because all terms are squarefree, thus all prime factors of any single term are distinct. - Antti Karttunen, Sep 28 2015
LINKS
EXAMPLE
{a(4), a(5), a(6)} = {34, 399, 58} because 34 = 2*17 => 2+17 = 19 is the greatest prime factor of 399 = 3*7*19, and 3+7+19 = 29 is the greatest prime divisor of 58 = 2*29 where 29+2 = 31 is prime.
MAPLE
with(numtheory):p0:=3:nn:=10^7:
for n from 1 to 50 do:
ii:=0:
for k from 4 to nn while(ii=0) do:
x:=factorset(k):n0:=nops(x):
s:=sum('x[i]', 'i'=1..n0):
if isprime(s) and x[n0]=p0 and issqrfree(k)and type(k, prime)=false
then
ii:=1: printf(`%d, `, k):p0:=s:k0:=k:
else
fi:
od:
od:
PROG
(PARI) gpf(k) = my(fk=factor(k)); fk[#fk~, 1];
sc(k) = forcomposite(c = k, , if (issquarefree(c), return (c)));
sdpf(k)= my(fk=factor(k)); vecsum(fk[, 1]);
scok(a) = {my(s = sdpf(a)); forcomposite(c = 1, , if (issquarefree(c) && (gpf(c)==s) && isprime(sdpf(c)), return (c)); ); }
lista(nn) = {print1(a=sc(1), ", "); for (n=2, nn, na = scok(a); print1(na, ", "); a = na; ); } \\ Michel Marcus, Sep 27 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Sep 10 2015
STATUS
approved