This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A094268 #33 Feb 02 2024 06:29:50
%S A094268 12,5775,171078830
%N A094268 Starting term of smallest consecutive n-tuples of abundant numbers.
%C A094268 The triple 171078830, 171078831, 171078832 was apparently found by Laurent Hodges and Michael Reid in 1995.
%C A094268 The starting term of the smallest consecutive 4-tuple of abundant numbers is at most 141363708067871564084949719820472453374 - Bruno Mishutka (bruno.mishutka(AT)googlemail.com), Nov 01 2007
%C A094268 Paul Erdős showed that there are two absolute constants c1, c2 such that for all large n there are at least c1 log log log n but not more than c2 log log log n consecutive abundant numbers less than n. - Bruno Mishutka (bruno.mishutka(AT)googlemail.com), Nov 01 2007
%C A094268 From _Jianing Song_, Apr 10 2021: (Start)
%C A094268 a(n) exists for all n. Proof: since the infinite product Product_{p prime} (1 + 1/p) diverges, we can find a strictly increasing sequence {b(m)} such that b(0) = 0, Product_{k=b(m)+1..b(m+1)} (1 + 1/prime(k)) > 2 for all m. Given n, by Chinese Remainder Theorem, we can find N such that N + m divides Product_{k=b(m)+1..b(m+1)} prime(k) for m = 0..n-1, then sigma(N + m)/(N + m) >= Product_{k=b(m)+1..b(m+1)} (1 + 1/prime(k)) > 2.
%C A094268 For example, if N is divisible by 2*3*5, N+1 is divisible by 7*11*...*73, N+2 is divisible by 79*83*...*7499, N+3 is divisible by 7507*7517*...*57081677, N+4 is divisible by 57081679*57081697*...*(some very large prime), then N through N+4 are consecutive abundant numbers.
%C A094268 Of course, the number N found using this method will be extremely large, since Product_{k=1..K} (1 + 1/prime(k)) ~ log(log(K)). (End)
%D A094268 J.-M. De Koninck and A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 771, pp. 98, 327, Ellipses, Paris, 2004.
%D A094268 S. Kravitz, Three Consecutive Abundant Numbers, Journal of Recreational Mathematics, 26:2 (1994), 149. Solution by L. Hodges and M. Reid, JRM, 27:2 (1995), 156-157.
%H A094268 Paul Erdős, <a href="http://www.renyi.hu/~p_erdos/1935-03.pdf">Note on consecutive abundant numbers</a>, J. London Math. Soc. 10, 128-131 (1935).
%H A094268 Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_878.htm">Puzzle 878. Consecutive abundant integers</a>
%Y A094268 Cf. A005101, A005231, A096399.
%K A094268 hard,bref,more,nonn
%O A094268 1,1
%A A094268 _Lekraj Beedassy_, Jun 02 2004