A182087 Carmichael numbers of the form C = (30n-p)*(60n-(2p+1))*(90n-(3p+2)), where n is a natural number and p, 2p+1, 3p+2 are all three prime numbers.

1729, 172081, 294409, 1773289, 4463641, 56052361, 118901521, 172947529, 216821881, 228842209, 295643089, 798770161, 1150270849, 1299963601, 1504651681, 1976295241, 2301745249, 9624742921, 11346205609, 13079177569

Offset

1,1

Comments

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).

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Carmichael Number.

Prog

(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))
}; \\ Charles R Greathouse IV, Oct 02 2012

Crossrefs

Cf. A033502, A206347.

Sequence in context: A212920 A317126 A318646 * A327787 A352970 A033502

Adjacent sequences: A182084 A182085 A182086 * A182088 A182089 A182090

Keyword

nonn

Author

Marius Coman, Apr 11 2012

Status

approved