A216925 -id:A216925 - cp's OEIS frontend

A290810 Numbers k such that 6k-1, 12k-1 and 18k-1 are all primes.

1, 4, 5, 14, 15, 29, 39, 40, 49, 70, 110, 159, 169, 204, 235, 260, 264, 315, 334, 355, 390, 425, 449, 490, 560, 565, 599, 634, 725, 729, 735, 820, 824, 889, 1019, 1029, 1349, 1379, 1419, 1510, 1580, 1590, 1694, 1719, 1765, 1925, 1930, 1950, 1985, 2044, 2150

Offset: 1

Amiram Eldar, Aug 11 2017

nonn

If k is in the sequence then (6k-1)(12k-1)(18k-1) = 36k * (36k^2 - 11k + 1) - 1 is a Lucas-Carmichael number (A006972).

Analogous to A046025 as A006972 (Lucas-Carmichael numbers) is analogous to A002997 (Carmichael numbers).

			1 is in the sequence since 6*1 - 1 = 5, 12*1 - 1 = 11 and 18*1 - 1 = 17 are all primes, and 5*11*17 = 935 is a Lucas-Carmichael number.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A002997, A006972, A046025, A067256, A087788, A216925.

Intersection of A024898, A138620 and A138918.

seq = {}; Do[ If[ AllTrue[{6 m - 1, 12 m - 1, 18 m - 1}, PrimeQ ], AppendTo[seq, m] ], {m, 1, 10^5} ]; seq

isok(n) = isprime(6*n-1) && isprime(12*n-1) && isprime(18*n-1); \\ Michel Marcus, Aug 11 2017

6*a(n) - 1 = A067256(n+1).

A292539 Primes p1 such that p2 = 2p1 + 1 and p3 = p1p2 - 2 are also primes, so p1p2*p3 is a Lucas-Carmichael number of the form k^2 - 1.

Original entry on oeis.org

3, 5, 11, 29, 53, 83, 173, 239, 281, 359, 431, 719, 761, 809, 911, 1031, 1103, 1223, 1289, 1451, 1481, 1511, 1559, 1931, 2069, 2339, 2351, 2393, 2693, 2699, 2819, 2969, 3359, 3491, 3539, 3851, 4019, 4211, 4409, 5039, 6113, 6269, 6329, 6491, 6521, 6551, 6581

Offset: 1

Amiram Eldar, Sep 18 2017

nonn

All the primes, except the first, are of the form p1 = 6k - 1, p2 = 12k - 1, p3 = 72k^2 - 18k - 1, with k = 1, 2, 5, 9, 14, 29, 40, 47, 60, 72, 120, 127, 135, 152, 172, 184, ...
The generated Lucas-Carmichael numbers are 399, 2915, 63503, 2924099, 32148899, 192099599, 3603600899, 13105670399, 25027872803, ...
Subsequence of A005384 (Sophie Germain primes).

			p1 = 3 is in the sequence since with p2 = 2*3 + 1 = 7 and p3 = 3*7 - 2 = 19 they are all primes. 3*7*19 = 399 is a Lucas-Carmichael number.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006972, A216925, A290947, A292538.

aQ[n_] := AllTrue[{n, 2n+1, 2 n^2+n-2}, PrimeQ]; Select[Range[10^3], aQ]
Select[Prime[Range[1000]],AllTrue[{2#+1,#(2#+1)-2},PrimeQ]&] (* Harvey P. Dale, Aug 16 2024 *)

is(n) = if(!ispseudoprime(n), return(0), my(p=2*n+1); if(!ispseudoprime(p), return(0), if(ispseudoprime(n*p-2), return(1)))); 0 \\ Felix Fröhlich, Sep 18 2017

A300959 Number of prime factors of the n-th Lucas-Carmichael number.

Original entry on oeis.org

3, 3, 3, 3, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 4, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 4, 4, 4, 3, 3, 4, 3, 4, 4, 3, 4, 3, 5, 4, 3, 3, 3, 3, 4, 3, 3, 4, 3, 4, 4, 3, 3, 3, 5, 4, 4, 3, 3, 4, 3, 4, 3, 3, 4, 4, 4, 3, 3, 4, 4, 3, 4, 4, 4, 4, 3, 5, 3, 4, 3

Offset: 1

Tim Johannes Ohrtmann, Mar 17 2018

nonn

The number of prime factors is always >= 3.

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..10000

Cf. A006972 (Lucas-Carmichael numbers).
Cf. A216925, A216926, A216927, A217002, A217003, A217091 (Lucas-Carmichael numbers with 3 to 8 prime factors).
Cf. A216928 (Least Lucas-Carmichael number with n prime factors).

islc(n)=my(f=factor(n)); for(i=1, #f[, 1], if((n+1)%(f[i, 1]+1) || f[i, 2]>1, return(0))); #f[, 1]>1; \\ from A006972
lista(nn) = for (n=1, nn, if (islc(n), print1(omega(n), ", "))); \\ Michel Marcus, Mar 17 2018

a(n) = A001221(A006972(n)).

cp's OEIS Frontend

A290810 Numbers k such that 6k-1, 12k-1 and 18k-1 are all primes.

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A292539 Primes p1 such that p2 = 2p1 + 1 and p3 = p1p2 - 2 are also primes, so p1p2*p3 is a Lucas-Carmichael number of the form k^2 - 1.

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PARI

A300959 Number of prime factors of the n-th Lucas-Carmichael number.

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Author

Keywords

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PARI

Formula

cp's OEIS Frontend

A290810 Numbers k such that 6k-1, 12k-1 and 18k-1 are all primes.

Views

Author

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Comments

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Mathematica

PARI

Formula

A292539 Primes p1 such that p2 = 2p1 + 1 and p3 = p1*p2 - 2 are also primes, so p1*p2*p3 is a Lucas-Carmichael number of the form k^2 - 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A300959 Number of prime factors of the n-th Lucas-Carmichael number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A292539 Primes p1 such that p2 = 2p1 + 1 and p3 = p1p2 - 2 are also primes, so p1p2*p3 is a Lucas-Carmichael number of the form k^2 - 1.