In pi

Browse

Well known sequences

  1. A000045 Fibonacci numbers7 terms at 4,253,057
  2. A000040 Prime numbers6 terms at 8,157,777
  3. A000290 Squares6 terms at 16,685,566
  4. A000079 Powers of 26 terms at 20,467,663
  5. A000108 Catalan numbers6 terms at 75,744,697
  6. A000217 Triangular numbers6 terms at 104,349,019
  7. A000142 Factorials5 terms at 1,380,572
  8. A000796 Digits of pi8 terms at 50,366,472
  9. A005132 Recamán’s sequence7 terms at 376,501,687
  10. A005150 Look and say4 terms at 802,177,755

Deepest

Sequences with the most leading terms found in the first 1,000,000,000 digits.

sequencetermsdigitsposition
A277535 Decimal expansion of Pi*(10^761) - floor(Pi*(10^761)).1212762
A243908 Decimal expansion of Johannes Kepler's polyhedron inscribing constant.12124,026,518
A399279 Decimal expansion of Sum_{k>=1} H(4*k)/k^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.121227,143,463
A334686 Start with n, and successively apply phi, phi, sigma', phi, phi, sigma', phi, ... until reaching either 0 or 1; a(n) is the number of steps needed (phi = A000010, sigma' = A001065); or a(n) = -1 if 0 or 1 is never reached.121246,821,206
A197585 Decimal expansion of least x > 0 having cos(2*Pi*x) = cos(x)^2.121266,733,046
A376761 Number of primes between the n-th composite number c(n) and 2*c(n).1212148,259,824
A293416 Decimal expansion of the minimum ripple factor for a ninth-order, reflectionless, Chebyshev filter.1212167,560,773
A244463 Number of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 9.1212172,330,849
A082597 Number of sets of consecutive primes whose arithmetic mean is an integer, the largest prime of a set is n-th prime.1212175,427,157
A019803 Decimal expansion of sqrt(2*e)/11.1212176,320,929
A074070 Number of Paley classes of Hadamard matrices of order 4n.1212184,662,853
A179835 Smallest k > 0 such that prime(n-k) + prime(n+k) = 2*prime(n), or 0 if there is no such k.1212185,259,080
A199471 Decimal expansion of x>0 satisfying 2*x^2-x*sin(x)=cos(x).1212198,033,753
A366789 Fully multiplicative with a(p) = oddpart(primepi(p)).1212207,686,791
A201615 Decimal expansion of Sum_{n>=1} 1/F(n)^n, where F=A000045 (Fibonacci numbers).1212267,314,535
A201892 Decimal expansion of the number x satisfying x^2+2x+3=e^x.1212268,271,280
A079530 a(n) = phi(n) - ceiling(sqrt(n)).1212293,103,575
A131333 A131332 * A000012.1212299,407,085
A356582 T(n,k) is the number of degree n polynomials in GF_2[x] that have exactly k linear factors in their prime factorization when the factors are counted with multiplicity, n >= 0, 0 <= k <= n. Triangular array read by rows.1212299,607,826
A147602 A nonsense sequence.1212349,294,029
A355035 Consider the least base b >= 2 where the sum of digits of n is a prime number; a(n) corresponds to this prime number.1212352,613,889
A088467 Decimal expansion of 6/(Pi^2 A086724).1212379,875,985
A093626 Decimal expansion of e^(-4*e).1212388,406,980
A337981 Decimal expansion of Pi*BesselY(0,2)/(2*BesselJ(0,2)) - gamma, where BesselJ and BesselY are the Bessel functions of the first and second kind, respectively, and gamma is Euler's constant (A001620).1212445,771,920
A212647 a(n) = product of exponents in canonical prime factorization of A181800(n) (n-th powerful number that is the first integer of its prime signature); a(1) = 1 by convention.1212447,446,626

Earliest

Sequences whose first eight terms appear soonest.

sequenceposition of 8 terms
A277535 Decimal expansion of Pi*(10^761) - floor(Pi*(10^761)).762
A334194 a(n) = n - d1*d2, where d1, d2 are the distances from n to the previous and the next prime number respectively.1,096
A337967 Triangle read by rows, application of the transformation A337966 to Euler's triangle A173018. T(n, k) for 0 <= k <= n.3,844
A367631 Triangle read by rows: T(n,k) is the number of permutations of length n avoiding simultaneously the patterns 123 and 132 with the maximum number of non-overlapping descents equal k.3,844
A031057 Write 2n-1 in base 8 and juxtapose.6,112
A066899 a(n) = card({k : phi(k) == n (mod k)}).8,920
A332955 a(0) = 1 and a(n) = A309807(n) - A309807(n-1) for n > 0.16,015
A252371 a(n) = A243055(A251727(n)).16,490
A275817 Least positive integer s such that an integer square k^2 lies between s^2*n and s^2*(n+1), with s^2*n < k^2 < s^2*(n+1).16,490
A088523 a(1) = 2; for n > 1, a(n) = (a(n-1) + prime(n)) mod n.19,392
A080883 Distance of n to next square.20,023
A021628 Decimal expansion of 1/624.27,548
A390746 Number of strict integer partitions of n > 0 such that the least part plus the greatest part is odd.28,181
A395409 a(n) = A033677(n) - A135034(n); excess of the smallest divisor of n >= sqrt(n) over ceiling(sqrt(n)).28,181
A010604 Decimal expansion of cube root of 33.28,337
A094965 A continued fraction transformation of e.33,386
A381485 Decimal expansion of sqrt(13)/6.34,551
A019770 Decimal expansion of 2*e/17.35,718
A021318 Decimal expansion of 1/314.38,428
A107853 Expansion of g.f. x*(x-1)*(x+1)^3/((2*x^3+x^2-1)*(x^4+1)).40,836
A010470 Decimal expansion of square root of 13.41,857
A002193 Decimal expansion of square root of 2.52,638
A020807 Decimal expansion of 1/sqrt(50).52,638
A364711 Decimal expansion of (negative of) the real part of (-sqrt(2))^^10, where ^^ indicates tetration or hyper-4 (e.g., 2^^4 = 2^(2^(2^2))).52,638
A245474 a(n) = smallest positive integer s such that s*n - floor(sqrt(s*n))^2 is a square.53,099

Rarest

Sequences whose first three terms never appear, shortest strings first.

sequencedigits in 3 terms
A275218 Numbers in 2-cycles of RATS sequences.8
A000318 Generalized tangent numbers d(4,n).9
A000320 Generalized tangent numbers d(5,n).9
A000487 Number of permutations of length n with exactly two valleys.9
A000488 Generalized tangent numbers d_(n,3).9
A000492 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-6 places.9
A000525 Number of partially labeled rooted trees with n nodes (4 of which are labeled).9
A000536 Number of 3-line Latin rectangles.9
A000555 Number of labeled trees of diameter 4 with n nodes.9
A000597 Central factorial numbers: A008955(n,3).9
A001135 Primes p such that the multiplicative order of 2 modulo p is (p-1)/5.9
A001239 Numbers that are the sum of 3 nonnegative cubes in more than 1 way.9
A001293 Leech triangle: k-th number (0 <= k <= n) in n-th row (0 <= n) is number of octads in S(5,8,24) containing k given points and missing n-k given points.9
A001374 Number of relational systems on n nodes. Also number of directed 3-multigraphs with loops on n nodes.9
A001520 a(n) = (6*n+1)*(6*n+3)*(6*n+5).9
A001583 Artiads: the primes p == 1 (mod 5) for which Fibonacci((p-1)/5) is divisible by p.9
A002303 Generalized tangent numbers.9
A002847 Number of ways of getting a straight flush, 4 of a kind, full house, flush, straight, 3 of a kind, 2 pair, a pair, no pair in poker.9
A002975 Primitive weird numbers: weird numbers with no proper weird divisors.9
A003031 Denominators of expansion of Fresnel integral S(z).9
A003294 Numbers k such that k^4 can be written as a sum of four positive 4th powers.9
A003399 Sum of 10 positive 9th powers.9
A003435 Number of directed Hamiltonian circuits on n-octahedron with a marked starting node.9
A003503 The larger of a betrothed pair.9
A003552 Divisors of 2^47 - 1.9

Pi approximations

Write a sequence's first terms out as one number, then reach pi with a small constant, a power or root, a logarithm or an exponential, a shift of the decimal point, and at most one added integer. Ranked by digits matched less a penalty for each of those moves, so plain forms win. Sequences whose names mention pi, and decimal expansions of constants, are left out.

sequenceexpressionvaluedigits
A091473 Integral_{x>=0} (cos(2x) * Product_{n>=1} cos(x/n)) dx.8 * 392699081698 / 10^123.14159311.7
A137506 a(2*n+1) = 141 + 124*n, a(2*n+2) = |a(2*n) - 24| with a(2)=59, thus a(4,6,8,...) = 35,11,13,11,13...3 + 1415926535 / 10^103.14159310.5
A250224 Number of length n+1 0..3 arrays with the sum of the cubes of adjacent differences multiplied by some arrangement of +-1 equal to zero.1 / 42052208704^2 / 18 * 10^233.1415939.8
A216939 Number of side-3 hexagonal 0..n arrays with values nondecreasing E, SW and SE.3 + 521211^3 / 10^183.1415939.6
A062876 Numbers of lattice points corresponding to incrementally largest circle radii in A062875.3 + 412202844^2 / 12 / 10^173.14159310.1
A081770 Numbers twice their squarefree kernel (A007947).3 + 412202844^2 / 12 / 10^173.14159310.1
A083697 a(n) = 2^(2^n - 1) * Fibonacci(2^n).1 / 12242688 / 26 * 10^93.1415938.2
A178794 These are the x coordinates of the isolated visible lattice points in the plane.2199115 / 7 / 10^53.1415937.2
A396784 Least positive integer k such that A001414(k+1) - A001414(k) = n.3 + 5212110^3 / 10^213.1415939.6
A089086 Greatest common divisor of n^2-5 and n^2+5.3 + 5212110^3 / 10^213.1415939.6
A171337 Number of 0..31 integer arrays v[1..n] of length n with all autocorrelation values sum(i){v[i]*v[i-k]} distinct for k in 0..n-1.3 + 1 / 32102332635 / 22 * 10^113.1415939.1
A223657 Number of 7Xn 0..1 arrays with all rows having a nonnegative second derivative, and all and columns having a positive second derivative in a quadratic least squares fit, with one and two element arrays taken as having a zero second derivative.60360010425^3 / 7 / 10^313.1415937.9
A234889 Number of (n+1) X (7+1) 0..5 arrays with every 2 X 2 subblock having its diagonal sum differing from its antidiagonal sum by 1, with no adjacent elements equal (constant-stress tilted 1 X 1 tilings).sqrt(98696) / 10^23.1415926.7
A322748 Primes p such that q=p^2+p+1 is prime and (q^2+q+1)/3 is prime.3 + 1 / 2354171 / 3 * 10^63.1415938.7
A381825 Odd cubefull exponentially odd numbers: numbers whose prime factorization has only odd primes and odd exponents that are larger than 1 (except for 1 whose prime factorization is empty).3 + 18 / 127125243 * 10^63.1415938.9
A346767 a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(6*k,k) / (5*k + 1).3 + 1 / 11770855 / 6 * 10^73.1415938.7
A132606 Numbers m such that A132601(m) = A132601(m-1).203941^3 / 27 / 10^143.1415938.1
A380901 Integers k such that k = Sum k/(p_i + j), where p_i are the prime factors of k (with multiplicity). Case j = 2.3 + 23 / 1624378125 * 10^73.1415938.8
A251340 Number of (n+1)X(5+1) 0..3 arrays with every 2X2 subblock summing to a nonzero multiple of 3.3 + sqrt(72375) / 19 / 10^23.1415938.5
A063164 Dimension of the space of weight 2n cusp forms for Gamma_0( 96 ).3 + 16 * 94072^2 / 10^123.1415938.8
A087054 Primes of the form pq + qr + rp where p, q and r are distinct primes.314159 / 10^53.1415906.1
A245650 Primes in the sequence 12*n - prime(n), (A245071).314159 / 10^53.1415906.1
A363187 Prime numbers that are the average of three consecutive odd semiprimes.314159 / 10^53.1415906.1
A238397 Numbers of the form pq + qr + rp where p, q and r are distinct primes (sorted sequence without duplicates).314159 / 10^53.1415906.1
A361402 a(1) = 5; a(n+1) is the smallest prime p > a(n) such that digsum(p) = a(n).6 * 523599 / 10^63.1415946.4

OEIS snapshot 2026-09-04.