IntegerDigits

The digits of $n$, most significant first — base 10 by default. The base slot takes a whole numeral SYSTEM, not only an integer — factorial, Zeckendorf, Ostrowski, balanced, negative, bijective, mixed-radix, primorial, combinatorial or residue.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaNumerical digit↗MathWorldDigit↗WikidataQ82990↗Wolfram LanguageIntegerDigits!↗Rosetta CodeSum digits of an integer↗

IntegerDigits(n)digits of $n$ in base 10, most significant first.

SymPydigits(n)[1:]Oscarreverse(digits(BigInt(n)))

IntegerDigits(n, base)digits of $n$ in the given base.

SymPydigits(n, base)[1:]Oscarreverse(digits(BigInt(n), base=base))

IntegerDigits(n, base, len)the $len$ least-significant digits, zero-padded if needed.

IntegerDigits(n, system)digits in any of the systems below

SymPydigits(n, system)[1:]Oscarreverse(digits(BigInt(n), base=system))

IntegerDigits(n, system, width)left-padded with zeros to width digits

Typed signatures

Type	Package	Applies
(integer, integer?, integer?) -> list<integer>	compute-engine	always
(integer, any?, integer?) -> list<integer>	numeralsreplaces compute-engine	always

Domain: Numeral systems

Details

• Digits come out most significant first, matching ordinary positional notation.
• In a fixed base the sign of n is discarded, so negative integers give the same digits as their absolute value.
• IntegerDigits(0) is $\{0\}$ -- there's always at least one digit.
• The 3-argument form keeps only the len least-significant digits, truncating or zero-padding as needed.
• Systems: PositionalNumerals(b), MixedRadixNumerals([…]), FactorialNumerals, PrimorialNumerals, BalancedNumerals(b), NegativeNumerals(b), BijectiveNumerals(k), ZeckendorfNumerals, OstrowskiNumerals([…]), CombinatorialNumerals(k), ResidueNumerals([…]), AdicNumerals(b, prec?) — also written Radix, MixedRadix, Factoradic, PrimorialRadix, BalancedRadix, NegativeRadix, BijectiveRadix, Zeckendorf, Ostrowski, CombinatorialSystem, ResidueSystem
• PositionalNumerals(b) is ordinary base-$b$ notation, $b\ge2$ — the same digits an integer base already gives, wrapped as a system value so it can stand wherever the others do (NumeralSystemShape, a constant-radix MixedRadixNumerals comparison)
• OstrowskiNumerals([a₁, …]) is the numeral system a CONTINUED FRACTION defines: place values are the convergents' denominators, and a digit at its ceiling forbids a non-zero digit below it. All quotients 1 is $\varphi$, and that case IS Zeckendorf
• BalancedNumerals and NegativeNumerals represent NEGATIVE integers with no sign at all; fixed radix drops the sign instead
• The factoradic digits of $n$ are the Lehmer code of the $n$-th permutation, so padding to the permutation's size makes the two line up
• An integer with no numeral in a system — anything past $\prod m_i$ in a residue system, say — leaves the call standing rather than answering
• Bijective bases have no zero DIGIT, but zero itself is the empty numeral: that is what makes the correspondence with strings a bijection
• Inverted by FromDigits with the same system

Examples

19 more kept as data, checked against the oracles

Basic #

live ↗

In$\mathrm{IntegerDigits}(1\,234)$Out$\bigl\lbrack1, 2, 3, 4\bigr\rbrack$

live ↗

In$\mathrm{IntegerDigits}(2\,147, 2)$Out$\bigl\lbrack1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1\bigr\rbrack$

live ↗

In$\mathrm{IntegerDigits}(58\,127)$Out$\bigl\lbrack5, 8, 1, 2, 7\bigr\rbrack$

live ↗

In$\mathrm{IntegerDigits}(58\,127, 2)$Out$\bigl\lbrack1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1\bigr\rbrack$

93 784 seconds as days, hours, minutes, seconds

live ↗

In$\mathrm{IntegerDigits}(93\,784, \mathrm{MixedRadixNumerals}(\bigl\lbrack24, 60, 60\bigr\rbrack))$Out$\bigl\lbrack1, 2, 3, 4\bigr\rbrack$

$100 = 89 + 8 + 3$, with no two adjacent ones

live ↗

In$\mathrm{IntegerDigits}(100, \mathrm{ZeckendorfNumerals})$Out$\bigl\lbrack1, 0, 0, 0, 0, 1, 0, 1, 0, 0\bigr\rbrack$

102341 seconds split into days, hours, minutes, seconds

live ↗

In$\mathrm{IntegerDigits}(102\,341, \mathrm{MixedRadixNumerals}(\bigl\lbrack24, 60, 60\bigr\rbrack))$Out$\bigl\lbrack1, 4, 25, 41\bigr\rbrack$

Scope #

Digits past 9 are plain integers: $58127 = \mathrm{e30f}_{16}$

live ↗

In$\mathrm{IntegerDigits}(58\,127, 16)$Out$\bigl\lbrack14, 3, 0, 15\bigr\rbrack$

a mixed radix, spelled MixedRadix: 102 341 seconds is 1d 4h 25m 41s

live ↗

In$\mathrm{IntegerDigits}(102\,341, \mathrm{MixedRadix}(\bigl\lbrack24, 60, 60\bigr\rbrack))$Out$\bigl\lbrack1, 4, 25, 41\bigr\rbrack$

the leading digit is unbounded: $137 = (3 \cdot 3 + 2) \cdot 12 + 5$

live ↗

In$\mathrm{IntegerDigits}(137, \mathrm{MixedRadix}(\bigl\lbrack3, 12\bigr\rbrack))$Out$\bigl\lbrack3, 2, 5\bigr\rbrack$

a width longer than the numeral pads with leading zeros

live ↗

In$\mathrm{IntegerDigits}(5, 2, 8)$Out$\bigl\lbrack0, 0, 0, 0, 0, 1, 0, 1\bigr\rbrack$

Listable in the base: one digit list per base

live ↗

In$\mathrm{IntegerDigits}(7, \bigl\lbrack2, 3, 4\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, 1, 1\bigr\rbrack, \bigl\lbrack2, 1\bigr\rbrack, \bigl\lbrack1, 3\bigr\rbrack\bigr\rbrack$

Listable: threads over a range of integers

live ↗

In$\mathrm{IntegerDigits}(0..7, 2)$Out$\bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack1\bigr\rbrack, \bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack1, 1\bigr\rbrack, \bigl\lbrack1, 0, 0\bigr\rbrack, \bigl\lbrack1, 0, 1\bigr\rbrack, \bigl\lbrack1, 1, 0\bigr\rbrack, \bigl\lbrack1, 1, 1\bigr\rbrack\bigr\rbrack$

Threads element-wise over a list, as Wolfram's Listable heads do

live ↗

In$\mathrm{IntegerDigits}(\bigl\lbrack6, 7, 2\bigr\rbrack, 2)$Out$\bigl\lbrack\bigl\lbrack1, 1, 0\bigr\rbrack, \bigl\lbrack1, 1, 1\bigr\rbrack, \bigl\lbrack1, 0\bigr\rbrack\bigr\rbrack$

ordinary base 2, as a system value — same digits as the native 2-argument form

live ↗

In$\mathrm{IntegerDigits}(2\,147, \mathrm{PositionalNumerals}(2))$Out$\bigl\lbrack1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1\bigr\rbrack$

the same $20 = 13 + 5 + 2$, over a continued fraction's convergents

live ↗

In$\mathrm{IntegerDigits}(20, \mathrm{OstrowskiNumerals}(\bigl\lbrack1, 1, 1, 1, 1, 1, 1, 1\bigr\rbrack))$Out$\bigl\lbrack0, 1, 0, 1, 0, 1, 0, 0\bigr\rbrack$

a negative integer, with no sign

live ↗

In$\mathrm{IntegerDigits}(-5, \mathrm{BalancedNumerals}(3))$Out$\bigl\lbrack-1, 1, 1\bigr\rbrack$

spreadsheet column AAA

live ↗

In$\mathrm{IntegerDigits}(703, \mathrm{BijectiveNumerals}(26))$Out$\bigl\lbrack1, 1, 1\bigr\rbrack$

independent residues — no place values

live ↗

In$\mathrm{IntegerDigits}(23, \mathrm{ResidueNumerals}(\bigl\lbrack3, 5, 7\bigr\rbrack))$Out$\bigl\lbrack2, 3, 2\bigr\rbrack$

the 10-adic truncation: a negative is spelled by its digits, and the width is part of the numeral

live ↗

In$\mathrm{IntegerDigits}(-3, \mathrm{AdicNumerals}(10, 6))$Out$\bigl\lbrack9, 9, 9, 9, 9, 7\bigr\rbrack$

A mixed radix with radices $[3,12]$ expands 137 in three places

live ↗

In$\mathrm{IntegerDigits}(137, \mathrm{MixedRadixNumerals}(\bigl\lbrack3, 12\bigr\rbrack))$Out$\bigl\lbrack3, 2, 5\bigr\rbrack$

live ↗

In$\mathrm{IntegerDigits}(1\,084, \mathrm{MixedRadixNumerals}(\bigl\lbrack17, 15, 11\bigr\rbrack), 2)$Out$\bigl\lbrack8, 6\bigr\rbrack$

Applications #

The 3-argument form keeps only the 4 least-significant digits

live ↗

In$\mathrm{IntegerDigits}(6\,345\,354, 10, 4)$Out$\bigl\lbrack5, 3, 5, 4\bigr\rbrack$

all 3-bit words, by threading a padded form over a range

live ↗

In$\mathrm{IntegerDigits}(0..7, 2, 3)$Out$\bigl\lbrack\bigl\lbrack0, 0, 0\bigr\rbrack, \bigl\lbrack0, 0, 1\bigr\rbrack, \bigl\lbrack0, 1, 0\bigr\rbrack, \bigl\lbrack0, 1, 1\bigr\rbrack, \bigl\lbrack1, 0, 0\bigr\rbrack, \bigl\lbrack1, 0, 1\bigr\rbrack, \bigl\lbrack1, 1, 0\bigr\rbrack, \bigl\lbrack1, 1, 1\bigr\rbrack\bigr\rbrack$

the Lehmer code of the 6th permutation of four things

live ↗

In$\mathrm{IntegerDigits}(5, \mathrm{FactorialNumerals}, 4)$Out$\bigl\lbrack0, 2, 1, 0\bigr\rbrack$

Properties #

an integer base is still the native handler

live ↗

In$\mathrm{IntegerDigits}(255, 16)$Out$\bigl\lbrack15, 15\bigr\rbrack$

Another time-of-day style split, inverse of FromDigits

live ↗

In$\mathrm{IntegerDigits}(98\,534, \mathrm{MixedRadixNumerals}(\bigl\lbrack24, 60, 60\bigr\rbrack))$Out$\bigl\lbrack1, 3, 22, 14\bigr\rbrack$

Possible issues #

Unlike IntegerString there's always at least one digit, even for 0

live ↗

In$\mathrm{IntegerDigits}(0)$Out$\bigl\lbrack0\bigr\rbrack$

The sign is discarded

live ↗

In$\mathrm{IntegerDigits}(-3\,134)$Out$\bigl\lbrack3, 1, 3, 4\bigr\rbrack$

no numeral for a negative in a system that spells only $n \ge 0$; the message says which integers it does spell

live ↗

In$\mathrm{IntegerDigits}(-3, \mathrm{FactorialNumerals})$Out$\mathrm{IntegerDigits}(-3, \mathrm{FactorialNumerals})$

Neat examples #

$100!$ has 158 digits

live ↗

In$\mathrm{Length}(\mathrm{IntegerDigits}(100!))$Out$158$

About

Examples	50
Checked against	Wolfram Language 25 agree1 differSymPy 12 agree1 errorOscar 8 agree
Crosswalk	WikipediaMathWorldWikidataWolfram LanguageSymPy ×2Oscar ×2Rosetta Code

Implementation

mappedoscarpackages/oracle/src/mappings.ts

mappedoscarpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

SymPy's digits returns the base first, then the digits, so the template drops that leading base; for a negative n the base is negated and the digits are those of |n|. A numeral system in the base slot has no counterpart.

mappedsympypackages/oracle/src/mappings.ts

SymPy's digits returns the base first, then the digits, so the template drops that leading base; for a negative n the base is negated and the digits are those of |n|. A numeral system in the base slot has no counterpart.

See also: FromDigits, NumeralSystemShape, DigitCount, IntegerString