Answer: FORTUNE
Each clue points to an entry in the Online Encyclopedia of Integer Sequences, and each answer is one of the values in the corresponding sequence (with the correct number of digits). As illustrated by the filled-in examples, you should also keep track of the index of the chosen entry (between 1 and 26) and the corresponding letter. The flavortext and examples indicate that this index should be computed as if the first number in the entry has index 1, instead of paying attention to OEIS's "offset" field.
The four longer answers are larger than the values given in their OEIS entries. Because they come from well-known sequences (Fibonacci numbers, partition numbers, sixth powers, and 3-smooth numbers), these values may be discovered by other means, such as querying Wolfram Alpha, looking at extended tables from the "LINKS" section of their OEIS entries, or computing from scratch.
We tried to provide an entertaining tour of some of the best—and worst—that OEIS has to offer. Full answers and descriptions are below.
| 21 | 82 | 43 | 44 | 15 | 46 | 17 | |||
| 98 | 3 | 19 | 8 | 7 | 610 | 9 | 5 | 911 | |
| 412 | 1 | 0 | 213 | 1 | 7 | 3 | 5 | ||
| 114 | 3 | 2 | 7 | 715 | 1 | 0 | 0 | 7 | 6 |
| 816 | 5 | 517 | 1 | 618 | 3 | ||||
| 119 | 2 | 920 | 1 | 621 | 522 | ||||
| 223 | 124 | 7 | 6 | 725 | 8 | 2 | 326 | 3 | 6 |
| 127 | 4 | 8 | 9 | 1 | 728 | 3 | 9 | ||
| 329 | 7 | 3 | 7 | 630 | 3 | 5 | 3 | 631 | |
| 732 | 5 | 6 | 933 | 6 | 634 | 7 |
| Across | Clue | Answer | Index | Letter | Description |
|---|---|---|---|---|---|
| 1 | A000053 | 28 | 4 | d | Local stops on New York City Broadway line (IRT #1) subway. |
| 3 | A169769 | 44 | 9 | i | Number of geometrically distinct closed knight's tours of a 3 X n chessboard. |
| 5 | A201553 | 141 | 1 | a | Number of arrays of 6 integers in -n..n with sum zero. |
| 8 | A125197 | 93187 | 7 | g | Numbers such that pre- and post-concatenating 7, 13 and 31 results in six primes. |
| 10 | A003214 | 6959 | 15 | o | Number of binary forests with n nodes. |
| 12 | A135850 | 410 | 14 | n | Numbers n such that there are precisely 6 groups of order n. |
| 13 | A274807 | 21735 | 1 | a | Numbers n such that n and n+1 both have 32 divisors. |
| 14 | A000041 | 1327710076 | a(n) = number of partitions of n (the partition numbers). | ||
| 16 | A180326 | 85 | 12 | l | The non-common part of the smaller number of an amicable pair. |
| 17 | A033873 | 51 | 20 | t | Difference between first prime > 10^n (A003617) and 10^n. |
| 18 | A173933 | 63 | 23 | w | The number of numbers m < k/2 such that m/k is in the Cantor set, where k= A173931(n). |
| 19 | A081621 | 12 | 15 | o | Number of n-node triangulations of the sphere with minimal degree 5. |
| 20 | A006359 | 91 | 4 | d | Number of distributive lattices; also number of paths with n turns when light is reflected from 6 glass plates. |
| 21 | A240277 | 65 | 9 | i | Minimal number of people such that exactly n days are required to spread gossip. |
| 23 | A001014 | 2176782336 | 6th powers: a(n) = n^6. | ||
| 27 | A056146 | 14891 | 7 | g | Palindromic primes in bases 4 and 8. |
| 28 | A180499 | 739 | 9 | i | n^3 + n-th cube-free number. |
| 29 | A006827 | 3737 | 20 | t | Number of partitions of 2n with all subsums different from n. |
| 30 | A179403 | 63536 | 19 | s | Number of ways to place 2 nonattacking kings on an n X n toroidal board. |
| 32 | A256089 | 756 | 1 | a | Non-palindromic balanced numbers in base 9. |
| 33 | A067996 | 96 | 20 | t | Number of ways of making change for n cents using coins of 1, 2, 3, 5, 10, 20, 25, 50, 100 cents (all historical U.S.A. coinage from 1 to 100 cents). |
| 34 | A087536 | 67 | 1 | a | Primes consisting only of digits 6 and 7 occurring with equal frequency. |
| Down | Clue | Answer | Index | Letter | Description |
|---|---|---|---|---|---|
| 1 | A245660 | 29 | 20 | t | Number of unitary polyominoes without holes with n cells. A unitary polyomino is a polyomino whose edges all have length 1. |
| 2 | A004577 | 83435 | 9 | i | Erroneous version of A000940. |
| 3 | A000045 | Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. | |||
| 4 | A109837 | 47 | 13 | m | Smallest prime factor of the reverse concatenation of the first n odd numbers. |
| 5 | A003586 | 3-smooth numbers: numbers of the form 2^i*3^j with i, j >= 0. | |||
| 6 | A068987 | 49703 | 5 | e | a(n) = first position in the digit sequence 3,1,4,1,5,9,.... of Pi where the pattern "1,2,...,n" occurs. |
| 7 | A062306 | 1537 | 12 | l | Number of ways writing 2^n as a sum of two nonprime numbers. |
| 9 | A095615 | 112 | 1 | a | a(n) = 112 written in base 10 - n. |
| 11 | A251742 | 956 | 19 | s | 8-step Fibonacci sequence starting with 0,0,0,1,0,0,0,0. |
| 13 | A118874 | 21 | 20 | t | A halting sequence: let f_n be the n-th recursive function, relative to the Godel numbering given in Cutland, then a(n) is f_n(n)+1 if the corresponding program halts on input n, 0 otherwise. |
| 14 | A075297 | 18 | 14 | n | Duplicate of A057597. |
| 15 | A126648 | 71 | 1 | a | A 3 x 3 magic square. |
| 19 | A034960 | 17835 | 13 | m | Divide odd numbers into groups with prime(n) elements and add together. |
| 20 | A001893 | 98 | 5 | e | Number of permutations of (1,...,n) having n-3 inversions (n>=3). |
| 21 | A038808 | 63936 | 15 | o | Palindromic numbers which are the difference of two positive cubes. |
| 22 | A063083 | 56 | 6 | f | Number of permutations of n elements with an odd number of fixed points. |
| 23 | A018564 | 213 | 5 | e | Divisors of 639. |
| 24 | A198588 | 1477 | 16 | p | Odd numbers producing 5 odd numbers in the Collatz iteration. |
| 25 | A119015 | 71 | 15 | o | Denominators of "Farey fraction" approximations to e. |
| 26 | A135367 | 335 | 14 | n | Not the sum of three perfect powers (assuming 1, but not 0, is a perfect power). |
| 30 | A039923 | 69 | 25 | y | Continued fraction for 2^(1/3) + sqrt(3). |
| 31 | A211395 | 67 | 13 | m | Number of Sophie Germain primes between 2^n and 2^(n+1). |
The message in the Letter column reads "diagonal two digits at a time last name of eponym". The main diagonal thus produces the sequence 23,17,19,23,37, which appears in just one OEIS Sequence, A005235: "Fortunate numbers: least m>1 such that m+prime(n)# is prime, where p# denotes the product of all primes <= p." This sequence was named after Reo FORTUNE.