Top 100 important LODA programs
The LODA programs repository is a directed acyclic graph (DAG). This top 100 list is the most important programs. Several programs in the entire graph may depend on one or more of the top 100 programs, but the top 100 programs do not depend on any programs outside the top 100.
There is a relationship between the top 100 programs and the OEIS number of references. Although there are some exceptions. Several programs in top 100 have few references in OEIS. Are these integer sequences underappreciated by humans?
| Rank | LODA (callers) | OEIS (refs) | Name |
|---|---|---|---|
| 1 | A080339 (876) | A080339 (20) | Characteristic function of {1} union {primes}: 1 if n is 1 or a prime, else 0. |
| 2 | A010051 (2506) | A010051 (1040) | Characteristic function of primes: 1 if n is prime, else 0. |
| 3 | A159477 (146) | A159477 (2) | a(n) = smallest prime >= n, if 1 is counted as a prime. |
| 4 | A000040 (1784) | A000040 (9520) | The prime numbers. |
| 5 | A000203 (643) | A000203 (4060) | a(n) = sigma(n), the sum of the divisors of n. Also called sigma_1(n). |
| 6 | A000005 (641) | A000005 (3852) | d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n. |
| 7 | A064911 (437) | A064911 (73) | If n is semiprime (or 2-almost prime) then 1 else 0. |
| 8 | A001222 (686) | A001222 (2010) | Number of prime divisors of n counted with multiplicity (also called big omega of n, bigomega(n) or Omega(n)). |
| 9 | A151800 (73) | A151800 (155) | Least prime > n (version 2 of the "next prime" function). |
| 10 | A000010 (357) | A000010 (3307) | Euler totient function phi(n): count numbers <= n and prime to n. |
| 11 | A000041 (252) | A000041 (3224) | a(n) is the number of partitions of n (the partition numbers). |
| 12 | A000142 (444) | A000142 (2319) | Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters). |
| 13 | A004086 (245) | A004086 (442) | Read n backwards (referred to as R(n) in many sequences). |
| 14 | A006005 (298) | A006005 (46) | The odd prime numbers together with 1. |
| 15 | A000593 (163) | A000593 (255) | Sum of odd divisors of n. |
| 16 | A006530 (199) | A006530 (877) | Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1. |
| 17 | A000045 (297) | A000045 (5066) | Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. |
| 18 | A000110 (171) | A000110 (1205) | Bell or exponential numbers: number of ways to partition a set of n labeled elements. |
| 19 | A020639 (126) | A020639 (829) | Lpf(n): least prime dividing n (when n > 1); a(1) = 1. Or, smallest prime factor of n, or smallest prime divisor of n. |
| 20 | A008683 (133) | A008683 (1157) | M\öbius (or Moebius) function mu(n). mu(1) = 1; mu(n) = (-1)^k if n is the product of k different primes; otherwise mu(n) = 0. |
| 21 | A122825 (34) | A122825 (1) | a(n) = n + number of previous prime terms, a(1) = 1. |
| 22 | A003415 (204) | A003415 (478) | a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(mn) = m*a(n) + n*a(m). |
| 23 | A007953 (185) | A007953 (956) | Digital sum (i.e., sum of digits) of n; also called digsum(n). |
| 24 | A010052 (81) | A010052 (347) | Characteristic function of squares: a(n) = 1 if n is a square, otherwise 0. |
| 25 | A000120 (114) | A000120 (1479) | 1's-counting sequence: number of 1's in binary expansion of n (or the binary weight of n). |
| 26 | A046660 (19) | A046660 (86) | Excess of n = number of prime divisors (with multiplicity) - number of prime divisors (without multiplicity). |
| 27 | A000108 (184) | A000108 (3500) | Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). |
| 28 | A010816 (12) | A010816 (17) | Expansion of Product_{k>=1} (1 - x^k)^3. |
| 29 | A116916 (16) | A116916 (7) | Expansion of q^(-1/8) * (eta(q)^3 + 3 * eta(q^9)^3) in powers of q^3. |
| 30 | A046897 (92) | A046897 (47) | Sum of divisors of n that are not divisible by 4. |
| 31 | A007088 (269) | A007088 (697) | The binary numbers (or binary words, or binary vectors, or binary expansion of n): numbers written in base 2. |
| 32 | A001221 (137) | A001221 (1606) | Number of distinct primes dividing n (also called omega(n)). |
| 33 | A109606 (55) | A109606 (7) | Number of numbers k with 1 < k < n which are relatively prime to n. |
| 34 | A010815 (111) | A010815 (1522) | From Euler's Pentagonal Theorem: coefficient of q^n in Product_{m>=1} (1 - q^m). |
| 35 | A087172 (26) | A087172 (7) | Greatest Fibonacci number that does not exceed n. |
| 36 | A008966 (70) | A008966 (245) | a(n) = 1 if n is squarefree, otherwise 0. |
| 37 | A064989 (56) | A064989 (300) | Multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p. |
| 38 | A001951 (85) | A001951 (106) | A Beatty sequence: a(n) = floor(n*sqrt(2)). |
| 39 | A002131 (15) | A002131 (45) | Sum of divisors d of n such that n/d is odd. |
| 40 | A013632 (58) | A013632 (42) | Difference between n and the next prime greater than n. |
| 41 | A252736 (33) | A252736 (7) | a(1) = a(2) = 0; for n > 2: a(2n) = 1 + a(n), a(2n+1) = a(A064989(2n+1)). |
| 42 | A090406 (81) | A090406 (3) | a(n) = PrimePi(n+3) - PrimePi(n). |
| 43 | A353463 (47) | A353463 (2) | Characteristic function of lesser twin primes: a(n) = 1 if both n and n+2 are primes, otherwise 0. |
| 44 | A099267 (44) | A099267 (12) | Numbers generated by the golden sieve. |
| 45 | A171688 (41) | A171688 (2) | Twin primes > 3. |
| 46 | A071960 (85) | A071960 (1) | Largest k >= 0 such that Product_{i=0..k} (n+i) divides n!. |
| 47 | A173919 (44) | A173919 (4) | Numbers that are prime or one less than a prime. |
| 48 | A007318 (111) | A007318 (1854) | Pascal's triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!), 0 <= k <= n. |
| 49 | A181819 (42) | A181819 (354) | Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)). |
| 50 | A036234 (61) | A036234 (28) | Number of primes <= n, if 1 is counted as a prime. |
| 51 | A086436 (82) | A086436 (9) | Maximum number of parts possible in a factorization of n; a(1) = 1, and for n > 1, a(n) = A001222(n) = bigomega(n). |
| 52 | A276086 (82) | A276086 (354) | Primorial base exp-function: digits in primorial base representation of n become the exponents of successive prime factors whose product a(n) is. |
| 53 | A199685 (93) | A199685 (6) | a(n) = 5*10^n+1. |
| 54 | A010163 (34) | A010163 (3) | Continued fraction for sqrt(92). |
| 55 | A139764 (36) | A139764 (8) | Smallest term in Zeckendorf representation of n. |
| 56 | A083811 (90) | A083811 (5) | Numbers n such that 2n+1 is the digit reversal of n+1. |
| 57 | A122045 (42) | A122045 (69) | Euler (or secant) numbers E(n). |
| 58 | A230980 (104) | A230980 (2) | Number of primes <= n, starting at n=0. |
| 59 | A001358 (45) | A001358 (1615) | Semiprimes (or biprimes): products of two primes. |
| 60 | A010158 (27) | A010158 (4) | Continued fraction for sqrt(85). |
| 61 | A106325 (8) | A106325 (4) | Smallest semiprime not less than n. |
| 62 | A033182 (28) | A033182 (3) | Number of pairs (p,q) such that 5*p + 6*q = n. |
| 63 | A001223 (46) | A001223 (598) | Prime gaps: differences between consecutive primes. |
| 64 | A116958 (2) | A116958 (1) | Numbers k such that 2*k + 5 and 2*k + 7 are twin primes. |
| 65 | A054785 (35) | A054785 (23) | a(n) = sigma(2n) - sigma(n), where sigma is the sum of divisors of n, A000203. |
| 66 | A002436 (3) | A002436 (14) | E.g.f.: Sum_{n >= 0} a(n)*x^(2*n)/(2*n)! = sec(2*x). |
| 67 | A264668 (90) | A264668 (3) | a(n) = A264600(n) - A061486(n). |
| 68 | A030101 (53) | A030101 (148) | a(n) is the number produced when n is converted to binary digits, the binary digits are reversed and then converted back into a decimal number. |
| 69 | A066096 (37) | A066096 (7) | Duplicate values in A060143. |
| 70 | A089026 (18) | A089026 (23) | a(n) = n if n is a prime, otherwise a(n) = 1. |
| 71 | A003961 (60) | A003961 (458) | Completely multiplicative with a(prime(k)) = prime(k+1). |
| 72 | A100661 (19) | A100661 (6) | Quet transform of A006519 (see A101387 for definition). Also, least k such that n+k has at most k ones in its binary representation. |
| 73 | A034444 (38) | A034444 (270) | a(n) is the number of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1). |
| 74 | A000111 (41) | A000111 (316) | Euler or up/down numbers: e.g.f. sec(x) + tan(x). Also for n >= 2, half the number of alternating permutations on n letters (A001250). |
| 75 | A002129 (12) | A002129 (76) | Generalized sum of divisors function: excess of sum of odd divisors of n over sum of even divisors of n. |
| 76 | A027760 (24) | A027760 (32) | Denominator of Sum_{p prime, p-1 divides n} 1/p. |
| 77 | A324969 (74) | A324969 (10) | Number of unlabeled rooted identity trees with n vertices whose non-leaf terminal subtrees are all different. |
| 78 | A073184 (24) | A073184 (8) | Number of cubefree divisors of n. |
| 79 | A035363 (20) | A035363 (111) | Number of partitions of n into even parts. |
| 80 | A028296 (3) | A028296 (36) | Expansion of e.g.f. Gudermannian(x) = 2*arctan(exp(x)) - Pi/2. |
| 81 | A007913 (50) | A007913 (242) | Squarefree part of n: a(n) is the smallest positive number m such that n/m is a square. |
| 82 | A008472 (70) | A008472 (335) | Sum of the distinct primes dividing n. |
| 83 | A064722 (32) | A064722 (15) | a(1) = 0; for n >= 2, a(n) = n - (largest prime <= n). |
| 84 | A039653 (41) | A039653 (29) | a(0) = 0; for n > 0, a(n) = sigma(n)-1. |
| 85 | A032742 (57) | A032742 (212) | a(1) = 1; for n > 1, a(n) = largest proper divisor of n. |
| 86 | A018252 (86) | A018252 (352) | The nonprime numbers: 1 together with the composite numbers, A002808. |
| 87 | A122111 (31) | A122111 (265) | Self-inverse permutation of the positive integers induced by partition enumeration in A112798 and partition conjugation. |
| 88 | A000720 (75) | A000720 (1468) | pi(n), the number of primes <= n. Sometimes called PrimePi(n) to distinguish it from the number 3.14159... |
| 89 | A016231 (1) | A016231 (1) | Inverse of 2222nd cyclotomic polynomial. |
| 90 | A078308 (65) | A078308 (13) | a(n) = Sum_{d divides n} d^(n/d + 1). |
| 91 | A002822 (10) | A002822 (90) | Numbers m such that 6m-1, 6m+1 are twin primes. |
| 92 | A300786 (63) | A300786 (1) | L.g.f.: log(Product_{k>=1} (1 + k*x^k)) = Sum_{n>=1} a(n)*x^n/n. |
| 93 | A010125 (13) | A010125 (4) | Continued fraction for sqrt(21). |
| 94 | A010054 (32) | A010054 (1547) | a(n) = 1 if n is a triangular number, otherwise 0. |
| 95 | A010225 (19) | A010225 (0) | Continued fraction for sqrt(183). |
| 96 | A151799 (52) | A151799 (89) | Version 2 of the "previous prime" function: largest prime < n. |
| 97 | A001414 (61) | A001414 (554) | Integer log of n: sum of primes dividing n (with repetition). Also called sopfr(n). |
| 98 | A000196 (77) | A000196 (298) | Integer part of square root of n. Or, number of positive squares <= n. Or, n appears 2n+1 times. |
| 99 | A001157 (51) | A001157 (359) | a(n) = sigma_2(n): sum of squares of divisors of n. |
| 100 | A136522 (45) | A136522 (55) | a(n) = 1 if n is a palindrome, otherwise 0. |