A286042 - OEIS

%I #10 Sep 24 2019 07:53:53

%S 13,17,19,23,31,37,41,43,47,53,59,61,67,73,79,83,89,97,101,103,107,

%T 109,113,127,131,137,149,151,157,163,167,173,179,181,191,193,197,199,

%U 211,223,227,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353

%N Largest prime factor of A285993(n), the largest odd abundant number (A005231) equal to the product of n consecutive primes.

%C The smallest term is a(5), there is no odd abundant number (A005231) equal to the product of less than 5 consecutive primes.

%C The corresponding abundant numbers are A285993(n) = prime(k-n+1)*...*prime(k), with prime(k) = a(n).

%H Amiram Eldar, <a href="/A286042/b286042.txt">Table of n, a(n) for n = 5..10000</a>

%F a(n) = A006530(A285993(n)) >= A151800(a(n-1)) = nextprime(a(n-1)), with strict inequality for n = 9, 18, 31, 46, 67, ..., in which case a(n) = nextprime(nextprime(a(n-1))). This is the case if A285993(n) is in A007741.

%e For n < 5, there is no odd abundant number equal to the product of n distinct primes.

%e For 5 <= n <= 8, the largest odd abundant number equal to the product of n consecutive primes is 3*...*a(n) with a(n) = prime(n+1).

%e For 9 <= n <= 17, the largest odd abundant number equal to the product of n consecutive primes is 5*...*a(n) with a(n) = prime(n+2).

%e For 18 <= n <= 30, the largest odd abundant number equal to the product of n consecutive primes is 7*...*a(n) with a(n) = prime(n+3).

%e For 31 <= n <= 45, the largest odd abundant number equal to the product of n consecutive primes is 11*...*a(n) with a(n) = prime(n+4).

%e For 46 <= n <= 66, the largest odd abundant number equal to the product of n consecutive primes is 13*...*a(n) with a(n) = prime(n+5).

%o (PARI) a(r,f=vector(r,i,prime(i+1)),o)={ while(sigma(factorback(f),-1)>2, o=f; f=concat(f[^1],nextprime(f[r]+1)));o[#o]} \\ Intentionally throws an error when n < 5.

%Y Cf. A285993, A005231, A006038, A007707, A007708, A007741.

%K nonn

%O 5,1

%A _M. F. Hasler_, May 01 2017

%E a(66) corrected by _Amiram Eldar_, Sep 24 2019