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The number 19 is not a harshad number in base 10, because the sum of the digits 1 and 9 is 10, and 19 is not divisible by 10. In base 10, every natural number expressible in the form 9R n a n , where the number R n consists of n copies of the single digit 1, n > 0, and a n is a positive integer less than 10 n and multiple of n , is a harshad ...
264 is an even composite number [1] with three distinct prime factors (2 3 × 3 × 11). [2] 264 is a Harshad number in base ten, [3] also divisible by each of its digits.264 is the sum of all even composite numbers that are not the sum of two abundant numbers (not necessarily distinct): 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 26 + 28 + 34 + 46. [4]
[a] Furthermore, it is the first in the family of absolute Euler pseudoprimes, a subset of Carmichael numbers. [7] 1729 is divisible by 19, the sum of its digits, making it a harshad number in base 10. [8] 1729 is the dimension of the Fourier transform on which the fastest known algorithm for multiplying two numbers is based. [9]
The number of iterations needed for () to reach a fixed point is the sum-product function's persistence of , and undefined if it never reaches a fixed point. Any integer shown to be a sum-product number in a given base must, by definition, also be a Harshad number in that base.
The sum of the base 10 digits of the integers 0, 1, 2, ... is given by OEIS: A007953 in the On-Line Encyclopedia of Integer Sequences. Borwein & Borwein (1992) use the generating function of this integer sequence (and of the analogous sequence for binary digit sums) to derive several rapidly converging series with rational and transcendental sums.
666 is a Smith number and Harshad number in base ten. [13] [14] The 27th indexed unique prime in decimal features a "666" in the middle of its sequence of digits. [15] [c] The Roman numeral for 666, DCLXVI, has exactly one occurrence of all symbols whose value is less than 1000 in decreasing order (D = 500, C = 100, L = 50, X = 10, V = 5, I = 1 ...
102 is the first three-digit base 10 polydivisible number, since 1 is divisible by 1, 10 is divisible by 2 and 102 is divisible by 3. This also shows that 102 is a Harshad number. 102 is the first 3-digit number divisible by the numbers 3, 6, 17, 34 and 51. 102 64 + 1 is a prime number. There are 102 vertices in the Biggs–Smith graph.
The article says: H.G. Grundman proved in 1994 that in base 10 there are no sequences with more than 20 consecutive Harshad numbers, but there is a missing word here, since clearly there is a sequence with more than 20 consecutive Harshad numbers, namely the sequence of Harshad numbers, whose initial segment is cited at the top of the article.