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Least prime number that produces the highest abundancy number when multiplied by the product of all previous n-1 terms.
2

%I #9 Apr 03 2023 10:36:11

%S 2,3,5,2,7,11,3,13,2,17,19,23,29,2,5,31,37,3,41,43,47,53,7,59,61,2,67,

%T 71,73,79,83,89,97,101,103,107,109,113,3,2,127,131,11,137,139,149,151,

%U 5,157,163,167,173,179,181,13,191,193,197,199,211,223,227,229,233,239,241

%N Least prime number that produces the highest abundancy number when multiplied by the product of all previous n-1 terms.

%C "Least" is required in the definition, otherwise a(14) could be either 2 or 5 because 2*77636318760 and 5*77636318760 have the same abundancy. It appears that only a(14) has this property. - _T. D. Noe_, Jan 24 2010

%H Jon E. Schoenfield, <a href="/A137826/b137826.txt">Table of n, a(n) for n = 1..10000</a>

%H The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=AbundantNumber">Abundant Numbers</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Abundancy.html">Abundancy</a>.

%e a(4)=2 since the product a(1)*a(2)*a(3) is 2*3*5=30, and

%e 30*2 = 60 has abundancy 2.8, whereas

%e 30*3 = 90 has abundancy 2.6,

%e 30*5 = 150 has abundancy 2.48,

%e 30*7 = 210 has abundancy 2.7428571..., etc.

%t Abundancy[k_Integer] := DivisorSigma[1, k]/k; SetAttributes[Abundancy,Listable]; nn=100; lastPrime=1; n=1; Table[a=Abundancy[n*Prime[Range[lastPrime+1]]]; pos=Position[a,Max[a]]; p=Prime[pos[[1,1]]]; If[pos[[1,1]>lastPrime, lastPrime++ ]; n=n*p; p, {nn}] (* _T. D. Noe_, Jan 24 2010 *)

%Y Cf. A005101, A017665, A017666, A137825 (product of terms).

%K easy,nonn

%O 1,1

%A _Sergio Pimentel_, Feb 11 2008

%E Edited by _T. D. Noe_, Jan 24 2010

%E Extended by _T. D. Noe_, Jan 24 2010

%E Edited by _Jon E. Schoenfield_, Mar 02 2019