A193869 -id:A193869 - cp's OEIS frontend

A077316 Triangle read by rows: T(n,k) is the k-th prime = 1 (mod n).

2, 3, 5, 7, 13, 19, 5, 13, 17, 29, 11, 31, 41, 61, 71, 7, 13, 19, 31, 37, 43, 29, 43, 71, 113, 127, 197, 211, 17, 41, 73, 89, 97, 113, 137, 193, 19, 37, 73, 109, 127, 163, 181, 199, 271, 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 23, 67, 89, 199, 331, 353

Offset: 1

Amarnath Murthy, Nov 04 2002

			Triangle begins:
   2;
   3,  5;
   7, 13, 19;
   5, 13, 17, 29;
  11, 31, 41, 61, 71;
  ...

Nathaniel Johnston, Rows 1..100, flattened

Cf. A034694 (first column), A077317 (main diagonal), A077318 (row sums), A077319, A093870, A193869 (row products).

Tj := proc(n,k) option remember: local j,p: if(k=0)then return 0:fi: for j from procname(n,k-1)+1 do if(isprime(n*j+1))then return j: fi: od: end: A077316 := proc(n,k) return n*Tj(n,k)+1: end: seq(seq(A077316(n,k),k=1..n),n=1..15); # Nathaniel Johnston, Sep 02 2011

Tj[n_, k_] := Tj[n, k] = Module[{j}, If[k == 0, Return[0]];
   For[j = Tj[n, k-1]+1, True, j++, If[PrimeQ[n*j+1], Return[j]]]];
T[n_, k_] := n*Tj[n, k]+1;
Table[Table[T[n, k], {k, 1, n}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Aug 02 2022, after Nathaniel Johnston *)

Edited and extended by Franklin T. Adams-Watters, Aug 29 2006

A193873 Smallest product of three distinct primes of the form n*k+1.

Original entry on oeis.org

30, 105, 1729, 1105, 13981, 1729, 88537, 50881, 51319, 13981, 137149, 29341, 548497, 88537, 285541, 186337, 3372529, 51319, 18326641, 252601, 1152271, 137149, 1809641, 1366633, 3828001, 548497, 4814857, 645569, 4797703, 285541, 79230049, 4811297

Offset: 1

Omar E. Pol, Sep 02 2011

Note that the Hardy-Ramanujan number is the first and the smallest repeated number: a(3) = a(6) = 1729.

			a(1) =  2*3*5 = 30
a(2) =  3*5*7 = 105
a(3) =  7*13*19 = 1729
a(4) =  5*13*17 = 1105
a(5) = 11*31*41 = 13981

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001235, A077316, A092609, A121940, A154729, A193869.

a[n_] := Module[{s = {}, c = 0, m = n + 1}, While[c < 3, While[!PrimeQ[m], m += n]; c++; AppendTo[s, m]; m += n]; Times @@ s]; Array[a, 100] (* Amiram Eldar, Jan 17 2025 *)

a(n)=my(p,q,k=1);while(!isprime(k+=n),);p=k;while(!isprime(k+=n),);q=k;while(!isprime(k+=n),);p*q*k \\ Charles R Greathouse IV, Sep 03 2011

A194263 Numbers that occur more than once in A193873, in order of appearance.

Original entry on oeis.org

1729, 13981, 88537, 51319, 137149, 548497, 285541, 3372529, 18326641, 1152271, 1809641, 3828001, 4814857, 4797703, 79230049, 4413223, 4209661, 19703611, 3882139, 50357677, 70611161, 26536591, 175493677, 85409941, 12932989, 84350561, 192754871, 59756863

Offset: 1

Omar E. Pol, Sep 02 2011

nonn

Note that a(1) = 1729 is known as the Hardy-Ramanujan number (see A001235).
Apparently these numbers occur only twice in A193873.
Many of these numbers occur more than twice. For example, 812405017 occurs 3 times, a(67), a(134), and a(268). Robert Price, Mar 17 2020

			1729 is in the sequence because the Hardy-Ramanujan number is also the smallest product of three distinct primes of the form 3*k+1 and also of the form 6*k+1, so 1729 occurs more than once in A193873.

Robert Price, Table of n, a(n) for n = 1..2554

Cf. A001235, A154729, A193869, A193873.

It appears that a(n) = A193873(2*n+1).
This conjecture only holds for n<62: a(62)=A193873(124)=A193873(2n)=620378449
, a(63)=A193873(125)=A193873(2n-1)=424315751
. Robert Price, Mar 17 2020

A194265 Smallest product of two distinct primes of the form n*k+1.

Original entry on oeis.org

6, 15, 91, 65, 341, 91, 1247, 697, 703, 341, 1541, 481, 4187, 1247, 1891, 1649, 14111, 703, 43739, 2501, 5461, 1541, 6533, 7081, 15251, 4187, 17767, 3277, 13747, 1891, 116003, 18721, 13333, 14111, 14981, 2701, 33227, 43739, 12403, 9881, 61337, 5461, 74563

Offset: 1

Omar E. Pol, Sep 03 2011

nonn

			a(1) = 2*3 = 6
a(2) = 3*5 = 15
a(3) = 7*13 = 91
a(4) = 5*13 = 65
a(5) = 11*31 = 341

Cf. A193869, A193879.

Table[ps = Select[Table[n*k + 1, {k, 100}], PrimeQ, 2]; If[Length[ps] == 2, ps[[1]]*ps[[2]], 0], {n, 100}] (* T. D. Noe, Oct 21 2011 *)

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A077316 Triangle read by rows: T(n,k) is the k-th prime = 1 (mod n).

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A193873 Smallest product of three distinct primes of the form n*k+1.

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A194263 Numbers that occur more than once in A193873, in order of appearance.

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A194265 Smallest product of two distinct primes of the form n*k+1.

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