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A128701
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Highly abundant numbers that are not products of consecutive primes with non-increasing 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.
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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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).
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REFERENCES
| Alaoglu, L., Erdos, P.; On Highly Composite and Similar Numbers, Transactions of the American Mathematical Society, Vol. 56, No. 3, (November 1944), pp. 448-469.
Lagarias, J. C.; An elementary problem equivalent to the Riemann hypothesis, American Mathematical Monthly 109 (2002), pp. 534-543.
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LINKS
| Lagarias, Jeffrey C., An Elementary Problem Equivalent to the Riemann Hypothesis.
Wikepedia, Highly Abundant Numbers.
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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)
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EXAMPLE
| As 10 is the third highly abundant number that cannot be expressed as a product of consecutive primes with non-increasing exponents, then a(3)=10.
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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[ # ]]]&]
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CROSSREFS
| Cf. A002093, A004394, A000203, A004490, A002182, A002201, A128699, A128700, A128702.
Sequence in context: A003615 A043293 A178644 * A030390 A063220 A063234
Adjacent sequences: A128698 A128699 A128700 * A128702 A128703 A128704
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KEYWORD
| nonn
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AUTHOR
| Ant King (mathstutoring(AT)ntlworld.com), Mar 28 2007
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