A138620 -id:A138620 - cp's OEIS frontend

A167057 Numbers k such that 12*k + 11 is prime.

0, 1, 3, 4, 5, 6, 8, 10, 13, 14, 15, 18, 19, 20, 21, 25, 28, 29, 31, 34, 35, 36, 38, 39, 40, 41, 46, 48, 49, 53, 54, 56, 59, 61, 68, 69, 71, 73, 75, 78, 80, 81, 84, 85, 90, 91, 95, 96, 98, 101, 104, 106, 108, 109, 113, 118, 119, 120, 123, 124, 125, 126, 129, 130, 131, 133

Offset: 1

Michael B. Porter, Oct 27 2009

Corresponds to even numbers in A024898.

			3 is in the sequence since 12*3+11 = 47 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A110801, A167055, A167056, A024898, primes are in A068231.

[n: n in [0..200] |IsPrime(12*n+11)]; // Vincenzo Librandi, Mar 25 2010

Select[Range[0, 200], PrimeQ[12 # + 11] &] (* Vincenzo Librandi, May 20 2014 *)

PARI
```
isA167057(n) = isprime(12*n+11)
```

a(n) = A138620(n)-1. [From R. J. Mathar, Oct 29 2009]

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

Original entry on oeis.org

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).

A294614 Sum of the divisors of 12*n - 1, divided by 12, minus n.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 2, 3, 0, 0, 0, 3, 4, 0, 0, 0, 0, 8, 4, 3, 0, 3, 6, 0, 0, 5, 0, 7, 4, 0, 0, 0, 18, 0, 0, 0, 0, 9, 4, 12, 4, 0, 14, 0, 0, 5, 8, 11, 0, 0, 6, 0, 12, 9, 0, 5, 0, 13, 6, 5, 10, 7, 14, 0, 0, 5, 0, 31, 0, 5, 0, 7, 30, 0, 12, 0, 0, 17, 6, 0, 0, 13, 18, 9, 8

Offset: 1

Omar E. Pol and Robert G. Wilson v, Nov 04 2017

a(n) = 0 iff n is in A138620.

First occurrence of k > -1: 1, 3, 8, 13, 18, 31, 28, 33, 23, 43, 66, 53, 45, 63, 48, 101, 166, etc.

			a(13) = 3 since d(12*13-1)/12 - 13 = 192/12 - 13 = 16 - 13 = 3.

Inspired by A291900.

Cf. A000203, A017653, A068231, A086463, A138620.

a[n_] := DivisorSigma[1, 12 n - 1]/12 - n; Array[a, 90]

PARI
```
a(n) = sigma(12*n-1)/12 - n;
```

a(n) = sigma(12*n-1)/12 - n = A000203(A017653(n-1))/12 - n.

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/18 - 1/2 = 0.048311... . - Amiram Eldar, Mar 28 2024

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A167057 Numbers k such that 12*k + 11 is prime.

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

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A294614 Sum of the divisors of 12*n - 1, divided by 12, minus n.

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