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4 số armstrong đầu tiên

1, 2, 3, 4

The sequence is finite — exactly 88 exist in base 10, the largest being the 39-digit 115,132,219,018,763,992,565,095,597,973,971,522,401, which is beyond what this in-browser search can enumerate.

Thiết lập

Thiết lập nhanh

At most 41 of the 88 base-10 Armstrong numbers are short enough to search for here.

Each extra digit roughly doubles the work. 10 is instant; 14 takes a moment and is the furthest exact double-precision arithmetic reaches.

Every one-digit number equals itself to the first power, so 1 through 9 qualify for free.

Prints 153 = 1³ + 5³ + 3³ instead of just 153.

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Kết quả

4 giá trị

1, 2, 3, 4

The sequence is finite — exactly 88 exist in base 10, the largest being the 39-digit 115,132,219,018,763,992,565,095,597,973,971,522,401, which is beyond what this in-browser search can enumerate.

Tạo ảnh

Hãy bật JavaScript để tạo kiểu cho những số này và tải về dưới dạng ảnh. Bản thân các giá trị đã được liệt kê ở trên.

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4 số armstrong đầu tiên là những số nào?

4 số armstrong đầu tiên là:

1, 2, 3, 4

Bài viết nền bên dưới chưa được dịch và đang hiển thị bằng tiếng Anh.

Về số armstrong

Three of the four interesting base-10 examples were already a curiosity in the 1930s. G. H. Hardy, in A Mathematician's Apology (1940), used 153, 370, 371 and 407 — the only numbers above 1 that equal the sum of the cubes of their digits — as his specimen of a mathematical fact that goes nowhere: a coincidence of decimal notation with no generalisation behind it, fit for a puzzle column rather than a research paper. He meant it as a put-down, and he was right about the mathematics, which is exactly why the family has thrived in recreational and teaching contexts ever since.

The name "narcissistic number" is generally traced to Joseph S. Madachy's Mathematics on Vacation (1966), which collected the pattern alongside other self-referential digit games. "Pluperfect digital invariant" and "plus perfect number" are the terms used in the more systematic literature on digital invariants. "Armstrong number" — overwhelmingly the name used in programming courses — is attributed to one Michael F. Armstrong, said to have set the problem as a class exercise, but the attribution is repeated far more often than it is documented, and no primary source for it is easy to find. Treat it as folklore with a plausible name attached.

The one genuinely satisfying result about them is that there are finitely many. A k-digit number is at least 10^(k−1), while the sum of the k-th powers of its digits is at most k · 9^k. For k = 61 the second quantity is already smaller than the first, so no narcissistic number can have 61 or more digits. That bounds the search, and the full base-10 enumeration comes to 88 numbers, ending at a 39-digit term.

Tính chất chính

  • There are exactly 88 narcissistic numbers in base 10; OEIS A005188 lists 89 terms because it includes 0.
  • The largest is 115,132,219,018,763,992,565,095,597,973,971,522,401, which has 39 digits.
  • No narcissistic number has 61 or more digits, because k · 9^k < 10^(k−1) for every k ≥ 61 — so the whole sequence sits below 10^60.
  • The three-digit members are exactly 153, 370, 371 and 407; the four-digit members are exactly 1634, 8208 and 9474.
  • There are no two-digit narcissistic numbers, and none with 12 or 13 digits either.
  • Each of 1 through 9 qualifies trivially, being a one-digit number equal to itself raised to the first power.
  • 153 is also the 17th triangular number and equals 1! + 2! + 3! + 4! + 5!.

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