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A128701
Highly abundant numbers that are not products of consecutive primes with nonincreasing exponents, i.e., that are not of the form n=2^{e_2} * 3^{e_3} * ...* p^{e_p}, with e_2>=e_3>=...>=e_p.
3
1, 3, 10, 18, 20, 42, 84, 90, 108, 168, 300, 336, 504, 540, 600, 630, 660, 1008, 1200, 1560, 1620, 1980, 2100, 2340, 2400, 3024, 3120, 3240, 3780, 3960, 4200, 4680, 5880, 6120, 6240, 7920, 8400, 8820
OFFSET
1,2
COMMENTS
This is the subsequence of those highly abundant numbers (A002093) that have a different canonical structure to the superabundant numbers (A004394), the colossally abundant numbers (A004490), the highly composite numbers (A002182) and the superior highly composite numbers (A002201).
LINKS
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.
FORMULA
The highly abundant numbers (A002093) are those values of n for which sigma(n)>sigma(m) for all m<n, where sigma(n)= A000203(n).
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 *)
KEYWORD
nonn
AUTHOR
Ant King, Mar 28 2007
STATUS
approved