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A128702 Highly abundant numbers (A002093) that are not Harshad numbers (A005349). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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]
LINKS
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.
Wikipedia, Harshad Number.
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). Harshad numbers (A005349) are divisible by the sum of their digits.
EXAMPLE
The third highly abundant number that is not a Harshad number is 168. So a(3)=168.
MATHEMATICA
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[ # ] &]
CROSSREFS
Sequence in context: A320406 A241936 A321338 * A322639 A277044 A192037
KEYWORD
nonn,base
AUTHOR
Ant King, Mar 28 2007
EXTENSIONS
a(16)-a(29) from Donovan Johnson, May 09 2009
STATUS
approved

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Last modified April 21 14:48 EDT 2024. Contains 371874 sequences. (Running on oeis4.)