cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

1729, 172081, 294409, 1773289, 4463641, 56052361, 118901521, 172947529, 216821881, 228842209, 295643089, 798770161, 1150270849, 1299963601, 1504651681, 1976295241, 2301745249, 9624742921, 11346205609, 13079177569
Offset: 1

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Author

Marius Coman, Apr 11 2012

Keywords

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).

Crossrefs

Programs

  • 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
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