A128701 -id:A128701 - cp's OEIS frontend

A128699 Highly abundant numbers that are not superabundant, i.e., the complement of A004394 w.r.t. A002093.

3, 8, 10, 16, 18, 20, 30, 42, 72, 84, 90, 96, 108, 144, 168, 210, 216, 288, 300, 336, 420, 480, 504, 540, 600, 630, 660, 960, 1008, 1080, 1200, 1440, 1560, 1620, 1800, 1920, 1980, 2100, 2160, 2340, 2400, 2880, 3024, 3120, 3240, 3360, 3600, 3780, 3960, 4200

Offset: 1

Ant King, Mar 28 2007

nonn

In 1944, Alaoglu and Erdős conjectured that this sequence was infinite and this was proved to be true by Nicolas in 1969.

			The sequence of highly abundant numbers begins 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20 and the sequence of superabundant numbers begins 1, 2, 4, 6, 12, 24. Because 10 is the third number which is in the first sequence but not in the second, it follows that a(3)=10.

Amiram Eldar, Table of n, a(n) for n = 1..1000
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.
Jean-Louis Nicolas, Ordre maximal d'un élément du groupe Sn des permutations et "highly composite numbers" Bull. Soc. Math. France 97 (1969), pp. 129-191.
Eric Weisstein's World of Mathematics, Superabundant Number.
Wikipedia, Highly Abundant Numbers.

Cf. A000203, A002093, A004394, A128700, A128701, A128702.

habdata1=FoldList[Max,1,Table[DivisorSigma[1,n],{n,2,10000}]]; data1=Flatten[Position[habdata1,#,1,1]&/@Union[habdata1]];sabdata2=FoldList[Max,1,Table[DivisorSigma[1,n]/n,{n,2,10000}]]; data2=Flatten[Position[sabdata2,#,1,1]&/@Union[sabdata2]];sabdata2=FoldList[Max,1,Table[DivisorSigma[1,n]/n,{n,2,10000}]]; Complement[data1,data2]

The highly abundant numbers are those integers for which sigma(n) > sigma(m) for all m < n (A002093) and the superabundant numbers are those integers for which sigma(n)/n > sigma(m)/m for all m < n (A004394).

A128702 Highly abundant numbers (A002093) that are not Harshad numbers (A005349).

Original entry on oeis.org

16, 96, 168, 47880, 85680, 95760, 388080, 458640, 526680, 609840, 637560, 776160, 887040, 917280, 942480, 1219680, 1244880, 1607760, 1774080, 2439360, 3880800, 5266800, 5569200, 6098400, 7761600, 9424800, 12196800, 17907120, 20900880

Offset: 1

Ant King, Mar 28 2007

All superabundant numbers (A004394), colossally abundant numbers (A004490), highly composite numbers (A002182) and superior highly composite numbers (A002201) are Harshad numbers. However, this is not true of the highly abundant numbers (A002093) and there are 32 exceptions in the 394 highly abundant numbers less than 50 million.

The previous comment is erroneous. The first superabundant number that is not a Harshad number is A004394(105) = 149602080797769600. The first highly composite number that is not a Harshad number is A002182(61) = 245044800. For all exceptions I found, the sum of digits is a power of 3. Although the first 60000 terms of the colossally abundant numbers and the superior highly composite numbers are Harshad numbers, I am not aware of a proof that all terms are Harshad numbers. There may be large counterexamples. [T. D. Noe, Oct 27 2009]

			The third highly abundant number that is not a Harshad number is 168. So a(3)=168.

Giovanni Resta, Table of n, a(n) for n = 1..400
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.
Wikipedia, Highly Abundant Numbers.
Wikipedia, Harshad Number.

Cf. A002093, A005349, A000203, A004394, A004490, A002182, A002201, A128699, A128700, A128701.

hadata1=FoldList[Max,1,Table[DivisorSigma[1,n],{n,2,10^6}]]; data1=Flatten[Position[hadata1,#,1,1]&/@Union[hadata1]];HarshadQ[k_]:=If[IntegerQ[ k/(Plus @@ IntegerDigits[ k ])],True,False];Select[data1,!HarshadQ[ # ] &]

The highly abundant numbers (A002093) are those values of n for which sigma(n)>sigma(m) for all mA000203(n). Harshad numbers (A005349) are divisible by the sum of their digits.

a(16)-a(29) from Donovan Johnson, May 09 2009

cp's OEIS Frontend

A128699 Highly abundant numbers that are not superabundant, i.e., the complement of A004394 w.r.t. A002093.

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