OFFSET
1,2
COMMENTS
Let b = A319100. Note that:
- if k is an odd number, then b(2*k) = b(k), b(4*k) = 2*b(k), b(2^e*k) = 4*b(k) for e >= 3;
- if k is not divisible by 3, then b(3*k) = 2*b(k), b(3^e*k) = 6*b(k) for e >= 2;
- for all primes p > 3, if k is not divisible by p, then b(p^e*k) = b(p*k).
As a result, it is easy to see that for every n, a(n) is not congruent to 2 modulo 4 and is not divisible by 16 or 27 or p^2 for any prime p > 3.
LINKS
Jianing Song, Table of n, a(n) for n = 1..1000
FORMULA
Let p(j) = A002476(j), q(i) = A007528(i), P(j) = Product_{k=1..j} p(k) = A121940(j) if j > 0, Q(i) = Product_{k=1..i} q(k) = A057130(i) if i > 0. If A025610(n) = 2^i*6^j, then:
(a) if i = 0, then a(n) = 1 if j = 0, 7 if j = 1 and 9*P(j-1) if j >= 2;
(b) if i = 1, then a(n) = 3 if j = 0, 21 if j = 1 and 36*P(j-1) if j >= 2;
(c) if i = 2, then a(n) = 8 if j = 0, 56 if j = 1 and 72*P(j-1) if j >= 2;
(d) if i >= 3, then a(n) = 24*Q(i-3) if j = 0 and P(j-1)*8*Q(i-3)*min{9*q(i-2), 3*p(j)} if j >= 1. [Rewritten by Jianing Song, Jun 04 2019]
PROG
(PARI) isA025610(n) = omega(6*n)==2&&valuation(n, 2)>=valuation(n, 3)
b(n) = if(isA025610(n), i=1; while(A319100(i)!=n, i++); i)
for(n=1, 216, if(isA025610(n), print1(b(n), ", "))) \\ See A319100 for its program
(PARI) p(j) = my(t=0, v=vector(j)); for(k=1, oo, if(prime(k)%6==1, t++; v[t]=prime(k)); if(t==j, return(v)))
q(i) = my(t=0, v=vector(i)); for(k=1, oo, if(prime(k)%6==5, t++; v[t]=prime(k)); if(t==i, return(v)))
b(i, j) = {
if(j<=1 && i<=2, my(M=[1, 3, 8; 7, 21, 56]); return(M[j+1, i+1]));
if(j==0 && i>=3, my(Q=q(i-3)); return(24*prod(k=1, i-3, Q[k])));
if(j>=2 && i<=2, my(P=p(j-1), w=[9, 36, 72]); return(w[i+1]*prod(k=1, j-1, P[k])));
if(j>=1 && i>=3, my(P=p(j), Q=q(i-2)); return(prod(k=1, j-1, P[k])*8*prod(k=1, i-3, Q[k])*min(9*Q[i-2], 3*P[j])));
}
list(lim) = my(v=A025610(lim), u=vector(#v)); for(k=1, #v, my(i=valuation(v[k], 2)-valuation(v[k], 3), j=valuation(v[k], 3)); u[k]=b(i, j)); u \\ Jianing Song, Jun 04 2019, See A025610 for its program
CROSSREFS
KEYWORD
nonn
AUTHOR
Jianing Song, Mar 27 2019
STATUS
approved