OFFSET
1,2
COMMENTS
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..8404
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.
Jeffrey C. Lagarias, An Elementary Problem Equivalent to the Riemann Hypothesis, arXiv:math/0008177 [math.NT], 2000-2001.
Jeffrey C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, American Mathematical Monthly 109 (2002), pp. 534-543.
Wikipedia, Highly Abundant Numbers.
FORMULA
EXAMPLE
As 10 is the third highly abundant number that cannot be expressed as a product of consecutive primes with nonincreasing exponents, then a(3)=10.
MATHEMATICA
hadata1=FoldList[Max, 1, Table[DivisorSigma[1, n], {n, 2, 10000}]]; data1=Flatten[Position[hadata1, #, 1, 1]&/@Union[hadata1]]; primefactorlist[1]={1}; primefactorlist[k_]:=First[Transpose[FactorInteger[k]]]; exponentlist[1]={1}; exponentlist[k_]:=Last[Transpose[FactorInteger[k]]]; g[k_List]:=If[MemberQ[Table[k[[i]]<= k[[i-1]], {i, 1, Length[k]}], False], False, True]; h[k_]:=If[primefactorlist[k]==(Prime[ # ]&/@Range[Length[primefactorlist[k]]]), True, False]; Select[data1, Or[ ! h[ # ], !g[exponentlist[ # ]]]&]
seq = {1}; sm = 0; Do[f = FactorInteger[n]; p = f[[;; , 1]]; e = f[[;; , 2]]; s = Times @@ ((p^(e + 1) - 1)/(p - 1)); If[s > sm, sm = s; m = Length[p]; If[p[[-1]] != Prime[m] || (m > 1 && ! AllTrue[Differences[e], # <= 0 &]), AppendTo[seq, n]]], {n, 2, 10^4}]; seq (* Amiram Eldar, Jun 18 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Ant King, Mar 28 2007
STATUS
approved