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Smallest prime divisor of 10*T(n)+1 = 5*n*(n+1)+1, where T(n) = 1 + 2 + ... + n.
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%I #14 Feb 11 2024 12:50:27

%S 11,31,61,101,151,211,281,19,11,19,661,11,911,1051,1201,1361,1531,29,

%T 1901,11,2311,2531,11,3001,3251,3511,19,31,19,4651,11,5281,31,11,6301,

%U 6661,79,7411,29,59,79,11,9461,9901,11,19,29,19,12251,41,89,13781,11

%N Smallest prime divisor of 10*T(n)+1 = 5*n*(n+1)+1, where T(n) = 1 + 2 + ... + n.

%C All divisors of 10*T(n)+1 are congruent to 1 or -1 modulo 10; that is, they end in the decimal digit 1 or 9.

%H Harvey P. Dale, <a href="/A125239/b125239.txt">Table of n, a(n) for n = 1..1000</a>

%H Nick Hobson, <a href="https://web.archive.org/web/20111019022839/http://www.qbyte.org/puzzles/p149s.html#triangular">Triangular Numbers</a>.

%e 10*T(9) + 1 = 5*9*10 + 1 = 451 = 11*41, so a(9) = 11.

%t FactorInteger[#][[1,1]]&/@(10*Accumulate[Range[60]]+1) (* _Harvey P. Dale_, Dec 12 2011 *)

%o (PARI) a(n) = if(n<1, 0, factor(5*n*(n+1)+1)[1,1])

%Y Cf. A000217, A062786, A090562, A124989.

%K easy,nonn

%O 1,1

%A _Nick Hobson_, Nov 25 2006