PrimePi

The prime-counting function: how many primes are ≤ n.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaPrime-counting function↗MathWorldPrimeCountingFunction↗WikidataQ251922↗DLMF27.12↗27.2.E2↗FungrimPrimePi✓↗Fungrim entry04427b↗4fa169✓↗OEISA000720↗Wolfram LanguagePrimePi!↗

PrimePi(n)count of primes $\le n$.

SageMathprime_pi(n)✓↗SymPyprimepi(n)↗mpmathprimepi(n)Rustprime_pi(n)

Typed signatures

Type	Package	Applies
(real) -> integer	compute-engine	always
(real) -> integer	number-theoryreplaces compute-engine	always

Domain: Number theory

Details

• Inverse of NthPrime: $\pi(p_n)=n$.
• Asymptotically $\pi(x)\sim x/\ln x$, the prime number theorem.
• Counts exactly up to $10^{15}$ (the Deléglise–Rivat combinatorial count, seconds at the top); past that the call stays unevaluated.
• n need not be an integer or prime itself -- PrimePi(n) counts primes up to whatever real value is given.
• Threads element-wise over a list, as Wolfram's Listable heads do.

Plot

Examples

15 more kept as data, checked against the oracles

Basic #

No primes are ≤ 1

live ↗

In$\mathrm{PrimePi}(1)$Out$0$

live ↗

In$\mathrm{PrimePi}(2)$Out$1$

live ↗

In$\mathrm{PrimePi}(15)$Out$6$

live ↗

In$\mathrm{PrimePi}(100)$Out$25$

Scope #

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

live ↗

In$\mathrm{PrimePi}(\bigl\lbrack10, 100\bigr\rbrack)$Out$\bigl\lbrack4, 25\bigr\rbrack$

Rational arguments: the primes up to $7/2$ are 2 and 3

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In$\mathrm{PrimePi}(\frac{7}{2})$Out$2$

Real arguments

live ↗

In$\mathrm{PrimePi}(15.25)$Out$6.0$

The primes below a million

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In$\mathrm{PrimePi}(1\,000\,000)$Out$78\,498$

Listable: threads over a list

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In$\mathrm{PrimePi}(\bigl\lbrack3, 5, 17, 25\bigr\rbrack)$Out$\bigl\lbrack2, 3, 7, 9\bigr\rbrack$

Exact at $10^{11}$

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In$\mathrm{PrimePi}(100\,000\,000\,000)$Out$4\,118\,054\,813$

Exact at $10^{13}$, by the combinatorial count

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In$\mathrm{PrimePi}(10\,000\,000\,000\,000)$Out$346\,065\,536\,839$

Known value · OEIS A006880

Exact at $10^{14}$, a couple of seconds by the combinatorial count

live ↗

In$\mathrm{PrimePi}(100\,000\,000\,000\,000)$Out$3\,204\,941\,750\,802$

Known value · OEIS A006880

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In$\mathrm{PrimePi}(1+10^{17})$Out$\mathrm{PrimePi}(100\,000\,000\,000\,000\,001)$

Applications #

The 25 primes below 100, via the count

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In$\mathrm{NthPrime}(1..\mathrm{PrimePi}(100))$Out$\bigl\lbrack2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97\bigr\rbrack$

669 primes up to 5000

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In$\mathrm{PrimePi}(5\,000)$Out$669$

Properties #

Same inverse relation viewed from PrimePi's side. See NthPrime

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In$\mathrm{PrimePi}(\mathrm{NthPrime}(50))=50$Out$\mathrm{True}$

Counting up to the 100th prime returns 100

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In$\mathrm{PrimePi}(\mathrm{NthPrime}(100))$Out$100$

168 primes below a thousand

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In$\mathrm{PrimePi}(1\,000)$Out$168$

Still 4: the primes up to 10 are 2, 3, 5, 7

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In$\mathrm{PrimePi}(10)$Out$4$

Known value · Wolfram MathematicalFunctionData, PrimePi

Possible issues #

No primes are ≤ a negative number

live ↗

In$\mathrm{PrimePi}(-1)$Out$0$

Past $10^{15}$ the exact tiers decline, so the call stays unevaluated at once instead of counting for minutes

live ↗

In$\mathrm{PrimePi}(10\,000\,000\,000\,000\,000.0)$Out$\mathrm{PrimePi}(10\,000\,000\,000\,000\,000.0)$

About

Examples	36
Checked against	Wolfram Language 34 agree2 differSageMath 33 agree2 unevaluatedSymPy 31 agree4 errormpmath 27 agree6 errormathlib4 22 agree11 errorRust 26 agree9 error
Crosswalk	WikipediaMathWorldWikidataDLMF ×2FungrimFungrim entry ×2OEISWolfram LanguageSageMathSymPympmathRust

Implementation

mappedwolframpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

See also: NthPrime, NextPrime