The On-Line Encyclopedia of Integer Sequences (OEIS)

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A290947

Primes p1 > 3, such that p2 = 3p1-2 and p3 = (p1*p2+1)/2 are also primes, so p1*p2*p3 is a triangular 3-Carmichael number.
(history; published version)

#10 by Charles R Greathouse IV at Mon Aug 14 21:34:37 EDT 2017

STATUS

editing

approved

#9 by Charles R Greathouse IV at Mon Aug 14 21:34:20 EDT 2017

LINKS

Charles R Greathouse IV, <a href="/A290947/b290947.txt">Table of n, a(n) for n = 1..10000</a>

PROG

(PARI) list(lim)=my(v=List()); forprime(p=7, lim, if(isprime(3*p-2) && isprime((p*(3*p-2)+1)/2), listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Aug 14 2017

STATUS

proposed

editing

#8 by Amiram Eldar at Mon Aug 14 17:03:42 EDT 2017

STATUS

editing

proposed

#7 by Amiram Eldar at Mon Aug 14 17:03:09 EDT 2017

CROSSREFS

Cf. A000217, A002997, A087788, A290945.

Discussion

Mon Aug 14

17:03

Amiram Eldar: Done.

#6 by Michel Marcus at Mon Aug 14 16:58:52 EDT 2017

STATUS

proposed

editing

#5 by Amiram Eldar at Mon Aug 14 16:27:10 EDT 2017

STATUS

editing

proposed

Discussion

Mon Aug 14

16:58

Michel Marcus: Please insert some banks in the crossrefs : see edit screen

#4 by Amiram Eldar at Mon Aug 14 16:23:37 EDT 2017

EXAMPLE

p1 = 7 is in the sequence since with p2 = 3*7-2 = 19 and p3 = (7*19+1)/2 = 67 they are both all primes. 7*19*67 = 8911 is both a triangular and 3-Carmichael number.

MATHEMATICA

If[AllTrue[{p1, p2, p3}, PrimeQ], AppendTo[seq, p1]], {k, 1,

#3 by Amiram Eldar at Mon Aug 14 16:20:54 EDT 2017

EXAMPLE

p1 = 7 is in the sequence since p2 = 3*7-2 = 19 and p3 = (7*19+1)/2 = 67 are both primes. 7*19*67 = 8911 is both triangular and Carmichael number.

MATHEMATICA

seq = {}; Do[p1 = 6 k + 1; p2 = 3 p1 - 2; p3 = (p1*p2 + 1)/2;

If[AllTrue[{p1, p2, p3}, PrimeQ], AppendTo[seq, p1]], {k, 1,

2000}]; seq

#2 by Amiram Eldar at Mon Aug 14 16:15:24 EDT 2017

NAME

allocated for Amiram EldarPrimes p1 > 3, such that p2 = 3p1-2 and p3 = (p1*p2+1)/2 are also primes, so p1*p2*p3 is a triangular 3-Carmichael number.

DATA

7, 13, 37, 43, 61, 193, 211, 271, 307, 331, 601, 673, 727, 757, 823, 1063, 1297, 1447, 1597, 1621, 1657, 1693, 2113, 2281, 2347, 2437, 2503, 3001, 3067, 3271, 3733, 4093, 4201, 4957, 5581, 6073, 6607, 7321, 7333, 7723, 7867, 8287, 8581, 8647, 9643, 10243

OFFSET

1,1

COMMENTS

The primes are of the form p1=(6k+1), p2=(18k+1), and p3=(54k^2+12k+1), with k = 1, 2, 6, 7, 10, 32, 35, 45, 51, 55, 100, ...

The generated triangular 3-Carmichael numbers are: 8911, 115921, 8134561, 14913991, 60957361, 6200691841, 8863329511, 24151953871, 39799655911, 53799052231, 585796503601, ...

CROSSREFS

Cf. A000217,A002997,A087788,A290945.

KEYWORD

allocated

nonn

AUTHOR

Amiram Eldar, Aug 14 2017

STATUS

approved

editing

#1 by Amiram Eldar at Mon Aug 14 16:15:24 EDT 2017

NAME

allocated for Amiram Eldar

KEYWORD

allocated

STATUS

approved