Missing-digit sum

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In number theory, a -missing-digit sum in a given number base is a natural number that is equal to the sum of numbers created by deleting digits from the original number. For example, the OEIS lists these two integers as 1-missing-digit sums in base ten:

Number theory Branch of pure mathematics

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers.

Natural number A kind of number, used for counting

In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers". The natural numbers can, at times, appear as a convenient set of codes ; that is, as what linguists call nominal numbers, forgoing many or all of the properties of being a number in a mathematical sense.

In mathematics, summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total. Besides numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any types of mathematical objects on which an operation denoted "+" is defined.

Contents

1,729,404 = 729404 (missing 1) + 129404 (missing 7) + 179404 (missing 2) + 172404 + 172904 + 172944 + 172940
1,800,000 = 800000 (missing 1) + 100000 (missing 8) + 180000 (missing first 0) + 180000 + 180000 + 180000 + 180000 [1]

Definition

Let be a natural number. We recursively define the -missing-digit sum function for base to be the following:

Recursion Process of repeating items in a self-similar way

Recursion occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances, it is often done in such a way that no loop or infinite chain of references can occur.

where is the number of digits in the number in base .

A natural number is a -missing-digit sum if it is a fixed point for , which occurs if .

Fixed point (mathematics) Point preserved by an endomorphism

In mathematics, a fixed point of a function is an element of the function's domain that is mapped to itself by the function. That is to say, c is a fixed point of the function f(x) if f(c) = c. This means f(f ) = fn(c) = c, an important terminating consideration when recursively computing f. A set of fixed points is sometimes called a fixed set.

For example, the number 121 in base is a missing-digit sum with , because .

A natural number is a sociable missing-digit sum if it is a periodic point for , where for a positive integer , and forms a cycle of period . A missing-digit sum is a sociable missing-digit sum with , and a amicable missing-digit sum is a missing-digit sum with .

In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.

In mathematics, a periodic sequence is a sequence for which the same terms are repeated over and over:

The number of iterations needed for to reach a fixed point is the missing-digit sum function's persistence of , and undefined if it never reaches a fixed point.

In mathematics, the persistence of a number is the number of times one must apply a given operation to an integer before reaching a fixed point at which the operation no longer alters the number.

As a result of the recursive definition of the -missing-digit sum function, when digits are deleted from a -digit natural number, the number of natural numbers in the sum is equal to the binomial coefficient

Binomial coefficient family of positive integers that occur as coefficients in the binomial theorem

In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers nk ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n, and it is given by the formula

.

For a given base , for every number , or n is a trivial missing-digit sum, defined below. This means that non-trivial missing-digit sums and cycles of only exist for .

Missing-digit sums for

b = k (trivial)

Let be a positive integer and the number base . Then for all integers :

b = 2k

Let be a positive integer and the number base . Then:

Searching for missing-digit sums

Searching for 1-missing-digit sums is simplified when one notes that the final two digits of n determine the final digit of its missing-digit sum. One can therefore test simply the final two digits of a given n to determine whether or not it is a potential missing-digit sum. In this way, the search space is considerably reduced. For example, consider the set of seven-digit base-ten numbers ending in ...01. For these numbers, the final digit of the sum is equal to (digit-0 x 1 + digit-1 x 6) modulo 10 = (0 + 6) mod 10 = 6 mod 10 = 6. Therefore, no seven-digit number ending in ...01 is equal to its own missing-digit-sum in base ten.

Now consider the set of seven-digit base-ten numbers ending in ...04. For these numbers, the final digit of the sum is equal to (0 x 1 + 4 x 6) modulo 10 = (0 + 24) mod 10 = 24 mod 10 = 4. This set may therefore contain one or more missing-digit sums. Next consider seven-digit numbers ending ...404. The penultimate (last-but-one) digit of the sum is equal to (2 + 4 x 2 + 0 x 4) modulo 10 = (2 + 8 + 0) mod 10 = 10 mod 10 = 0 (where the 2 is the tens digit of 24 from the sum for the final digit). This set of numbers ending ...404 may therefore contain one or more missing-digit sums. Similar reasoning can be applied to sums in which two, three and more digits are deleted from the original number.

Missing-digit sums and cycles of Fm,b for specific m and b

All numbers are represented in base . This table is currently incomplete.

Missing-digit sumsCycles
1 2 100
1 3 112, 121, 1000, 2000
1 4 1200, 2223, 10000, 20000, 300001332→2121→2031→1332
1 5 33334, 100000, 200000, 300000, 400000
1 6 14000, 444445, 1000000, 2000000, 3000000, 4000000, 5000000
175555556, 10000000, 20000000, 30000000, 40000000, 50000000, 60000000
1 8 160000, 66666667, 100000000, 200000000, 300000000, 400000000, 500000000, 600000000, 700000000
1 9 777777778, 1000000000, 2000000000, 3000000000, 4000000000, 5000000000, 6000000000, 7000000000, 8000000000
1 10 1800000, 14358846, 14400000, 15000000, 28758846, 28800000, 29358846, 29400000, 1107488889, 1107489042, 1111088889, 1111089042, 3277800000, 3281400000, 4388888889, 4388889042, 4392488889, 4392489042, 4500000000, [2] 5607488889, 5607489042, 5611088889, 5611089042, 7777800000, 7781400000, 8888888889, 8888889042, 8892488889, 8892489042, 10000000000, 20000000000, 30000000000, 40000000000, 50000000000, 60000000000, 70000000000, 80000000000, 90000000000
1119999999999A, 100000000000, 200000000000, 300000000000, 400000000000, 500000000000, 600000000000, 700000000000, 800000000000, 900000000000, A00000000000
1 12 1A000000, AAAAAAAAAAAB, 1000000000000, 2000000000000, 3000000000000, 4000000000000, 5000000000000, 6000000000000, 7000000000000, 8000000000000, 9000000000000, A000000000000, B000000000000
2 10 167564622641, 174977122641, 175543159858, 175543162247, 183477122641, 183518142444, 191500000000, 2779888721787, 2784986175699, 212148288981849, 212148288982006, 315131893491390, 321400000000000, 417586822240846, 417586822241003, 418112649991390, 424299754499265, 424341665637682, 526796569137682, 527322398999265, 533548288981849, 533548288982006, 636493411120423, 636531893491390, 642800000000000, 650000000000000, 738986822240846, 738986822241003, 739474144481849, 739474144482006, 739474144500000, 739512649991390, 745699754499265, 745741665637682, 746186822240846, 746186822241003, 751967555620423, 848722398999265, 849167555620423, 854948288981849, 854948288982006, 855396569137682, 862148288981849, 862148288982006, 957893411120423, 957931893491390, 965131893491390, 971400000000000
4 10 1523163197662495253514, 47989422298181591480943, 423579919359414921365511, 737978887988727574986738

Extension to negative integers

Missing-digit sums numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.

See also

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References

  1. Jon Ayres. "Sequence A131639". Neil Sloane. Retrieved 10 March 2014.
  2. Jon Ayres. "Sequence A131639". Neil Sloane. Retrieved 10 March 2014.