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COMMENTS
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These numbers can be reduced to only two possible forms: C =(30n-23)*(60n-47)*(90n-71) or C = (30n-29)*(60n-59)*(90n-89). In the first form, for the particular case when 30n-23,60n-47 and 90n-71 are all three prime numbers, we obtain the Chernick numbers of the form 10m+1 (for k = 5n-4 we have C = (6k+1)*(12k+1)*(18k+1)). In the second form, for the particular case when 30n-29,60n-59 and 90n-89 are all three prime numbers, we obtain the Chernick numbers of the form 10m+9 (for k = 5n-5 we have C = (6k+1)*(12k+1)*(18k+1)).
So the Chernick numbers can be divided into two categories: Chernick numbers of the form (30n+7)*(60n+13)*(90n+19) and Chernick numbers of the form (30n+1)*(60n+1)*(90n+1).
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PROG
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(PARI) list(lim)={
my(v=List(), f);
for(k=1, round(solve(x=(lim/162000)^(1/3), lim^(1/3), (30*x-23)*(60*x-47)*(90*x-71)-lim)),
n=(30*k-23)*(60*k-47)*(90*k-71)-1;
f=factor(30*k-23);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
f=factor(60*k-47);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
f=factor(90*k-71);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
listput(v, n+1)
);
for(k=2, round(solve(x=(lim/162000)^(1/3), lim^(1/3), (30*x-29)*(60*x-59)*(90*x-89)-lim)),
n=(30*k-29)*(60*k-59)*(90*k-89)-1;
f=factor(30*k-29);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
f=factor(60*k-59);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
f=factor(90*k-89);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
listput(v, n+1)
);
vecsort(Vec(v))
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