A307162 a(n) is the smallest k such that A319100(k) = A025610(n).

1, 3, 8, 7, 24, 21, 120, 56, 1320, 63, 168, 22440, 252, 840, 516120, 504, 9240, 819, 14967480, 2184, 157080, 3276, 613666680, 10920, 3612840, 6552, 28842333960, 120120, 15561, 104772360, 32760, 1528643699880, 2042040, 62244, 4295666760, 207480, 90189978292920, 46966920, 124488

Offset: 1

Author

Jianing Song, Mar 27 2019

Keywords

• nonn

Comments

A025610 is the range of A319100.

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

Crossrefs

Cf. A025610, A319100, A002476, A007528, A121940, A057130.

Programs

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

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]