A190136 Largest prime factor of n*(n+1)*(n+2)*(n+3).

3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 23, 23, 23, 13, 13, 29, 29, 31, 31, 31, 31, 17, 17, 37, 37, 37, 37, 41, 41, 43, 43, 43, 43, 47, 47, 47, 47, 17, 17, 53, 53, 53, 53, 19, 29, 59, 59, 61, 61, 61, 61, 31, 13, 67, 67, 67

Offset: 1

Author

Reinhard Zumkeller, May 07 2011

Keywords

• nonn

Comments

a(n) > 11 for n > 9;

a(A086801(n)) = A000040(n) for n > 2.

It follows from Størmer's theorem that lim inf a(n) = infinity, and in fact a(n) >> log log n. - Charles R Greathouse IV, Feb 19 2013

Examples

			Numbers m <= 10^6 such that a(m) = p:
p=13: 10, 11, 12, 13, 24, 25, 63;
p=17: 14, 15, 32, 33, 48, 49;
p=19: 16, 17, 18, 19, 54, 75, 168;
p=23: 20, 21, 22, 23, 207, 322;
p=29: 26, 27, 55, 114;
p=31: 28, 29, 30, 31, 62, 90, 152, 153, 340, 493, 1518;
p=37: 34, 35, 36, 37, 74, 184, 405;
p=41: 38, 39, 123, 245, 285, 286, 287, 492, 1023, 1517, 1680;
p=43: 40, 41, 42, 43, 84, 85, 169, 258, 341, 342, 558, 1330, 1331, 2106, 5289, 10878;
p=47: 44, 45, 46, 47, 91, 92, 93, 185, 186, 187, 374, 375, 702, 986, 987, 17575;
p=53: 50, 51, 52, 53, 159, 368, 369, 527, 845, 899, 900, 1375;
p=59: 56, 57, 115, 116, 117, 118, 174, 294, 528, 529, 530, 648, 943, 1885, 6783;
p=61: 58, 59, 60, 61, 119, 120, 121, 122, 182, 183, 242, 243, 244, 549, 608, 609, 1034, 1218, 1219, 1767, 1768, 2013, 2254, 2622;
p=67: 64, 65, 66, 67, 132, 133, 735, 1271, 1272, 1273, 2208, 2277, 3885, 4958, 5828, 5829;
p=71: 68, 69, 140, 141, 142, 284, 423, 424, 494, 636, 637, 779, 780, 781, 3477, 3478, 3549, 3550, 4899;
p=73: 70, 71, 72, 73, 143, 144, 145, 219, 363, 510, 728, 729, 803, 1022, 1239, 1679, 2772, 70224;
p=79: 76, 77, 78, 79, 158, 234, 235, 472, 473, 474, 550, 867, 868, 1024, 1104, 1419, 2209, 2448, 2923, 3476, 3869, 4898, 5290, 7502, 46136, 70150;
p=83: 80, 81, 82, 83, 246, 247, 413, 495, 663, 664, 1078, 1159, 1824, 2736, 3483, 4232, 4896, 4897, 7137, 8214, 12614, 36517, 97524;
p=89: 86, 87, 88, 89, 175, 264, 265, 354, 531, 710, 711, 712, 798, 1245, 1332, 2847, 4895, 5073, 6318, 18423, 28302, 29279;
p=97: 94, 95, 96, 97, 288, 289, 483, 580, 581, 582, 774, 873, 1064, 1065, 1455, 2132, 2133, 3007, 3975, 4556, 4557, 6496, 6497, 6887, 7564, 7565, 7566, 13869, 17457.
		

References

• Paulo Ribenboim, Galimatias Arithmeticae (Chap 11), in 'My Numbers, My Friends', Springer-Verlag 2000 NY, page 345.
• J. J. Sylvester, "On arithmetical series", Messenger of Mathematics 21 (1892), pp. 1-19 and 87-120.
• M. Faulkner, "On a theorem of Sylvester and Schur", J. London Math. Soc. 41:1 (1966), pp. 107-110.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Greatest Prime Factor

Crossrefs

Cf. A006530, A074399, A093074, A193945.

Programs

Haskell

a190136 n = maximum $ map a006530 [n..n+3]

Mathematica

Table[FactorInteger[Times@@(n+Range[0,3])][[-1,1]],{n,70}] (* Harvey P. Dale, Mar 19 2018 *)

PARI

gpf(n)=vecmax(factor(n)[,1])
a(n)=my(p=precprime(n+3));if(pCharles R Greathouse IV, Feb 19 2013

Formula

a(n) = max{gpf(n), gpf(n+1), gpf(n+2), gpf(n+3)} = gpf(A052762(n+3)) with gpf = A006530, greatest prime factor.

a(n) > 47 for n > 17575. - Charles R Greathouse IV, Feb 19 2013