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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. 2
1729, 172081, 294409, 1773289, 4463641, 56052361, 118901521, 172947529, 216821881, 228842209, 295643089, 798770161, 1150270849, 1299963601, 1504651681, 1976295241, 2301745249, 9624742921, 11346205609, 13079177569 (list; graph; refs; listen; history; text; internal format)
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

E. W. Weisstein, MathWorld: 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: A154728 A194263 A212920 * A033502 A050794 A138130

Adjacent sequences:  A182084 A182085 A182086 * A182088 A182089 A182090

KEYWORD

nonn

AUTHOR

Marius Coman, Apr 11 2012

STATUS

approved

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Last modified December 18 08:26 EST 2014. Contains 252114 sequences.