Excluding the single-digit perfect number 6, the smallest anti-perfect number (244)
An anti-perfect number is a number such that the sum of the decimal reverses of the proper divisors of the number equals the number. In this case, the proper divisors are 1,2,4,61,122 and 1+2+4+16+221=244.
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Bundesstraße 245
A German federal highway
The number of distinct two colored necklaces with 7 beads per color. (246)
Necklaces are defined as continuous (no clasp), and necklaces which can be rotated to the same state are not considered distinct.
See in particular the square array of the tabular form, third column eighth row.
Prooflet
Allowing rotations as distinct (or equivalently introducing a clasp), there are clearly 14 choose 7 = 3432 distinct necklace orientations with 14 beads, seven of each of 2 colors.
Because half of the beads are each color, no rotations of a number not a divisor of 14 can return to the same state; repeated rotations will require a single-color necklace.
Further, no rotations of 7 can produce the same result, because each half of 7 beads will necessarily not have the same number of beads of each color as the other half.
This means that the only clasped patterns which do not have 14 distinct rotations are the two which are strictly alternating black and white.
So we have 2 clasped orientations which are the same necklace in groups of 2, and 3430 which are the same necklace in groups of 14.
So the number of unclasped orientations are 2/2 + 3430/14 = 1 + 245 = 246.
The smallest positive number which is the difference between two numbers whose decimal expressions contain all ten decimal digits exactly once between them. (50123-49876=247)
Also, the concatenation of the number of hours in a day and the number of days in a week, used as an expression to mean “all day every day”.
Year of the Consulship of Philippus and Severus, or 1001 since the founding of Rome, which is 248
(Also a leap year)
A371491 evaluated at 2.
\sum_{i=1}^n\sum_{j=1}^n\sum_{k=1}^n\sum_{l=1}^n\sum_{m=1}^n(\frac{n}{gcd(i,j,k,l,m,n)})^3|_{n=2}
\hspace{3em}=8+8+8+8+8+8+8+8+8+8+8+8+8+8+8+8+8
\hspace{4em}+8+8+8+8+8+8+8+8+8+8+8+8+8+8+1=249
The country code of Rwanda
Also about 3200 times less than the higher side of the death toll in the Rwandan Genocide
The number of nonempty square submatrices of a 5x5 matrix (251).
OEIS does not specify nonempty, but that’s the only way the numbers fit.
{5\choose1}^2+{5\choose2}^2+{5\choose3}^2+{5\choose4}^2+{5\choose5}^2 =5^2+10^2+10^2+5^2+1^2 =25+100+100+25+1 =251
The 16th Mian-Chowla number (252)
https://oeis.org/A005282
The number of ways to populate two rows of a chess board with any number of pieces such that every piece is a kings move from exactly two other pieces. (253)
Ref: A183450
Essentially, all pieces must be in “island groups” of 3 pieces in 2 columns, each group separated by at least one column. How many groups?
0: 1 solution (no pieces at all)
1: 7*4=28 solutions (7 column pairs, 4 “missing” in each pair)
2: 10*4^2=160 solutions (10 pairs of column pairs, 4 "missing in each pair)
3: 1*4^3=64 solutions (column pairs must be [12] [45] [78], 4 “missing” in each pair)
Total = 1 + 28 + 160 + 64=253
One of the most well known South Park refrences in FRC (254)
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The maximum number which can be expressed in an unsigned byte of 8 bits. (255)
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The total number of values in 8 bytes, counting all 0 to 255 is 256
Edit: the “8 bytes” should be either 8 bits, or 1 byte/octet if I am understanding Wikipedia correctly
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The largest known prime number of the form k^k+1. (257)
It is conjectured and considered likely that 1^1+1=2, 2^2+1=5, and 4^4+1=257 are the only three.
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-335 of the Bengali calendar, a calendar system which, modified, still is used in modern day Bangladesh.
Also, congrats to @GeeTwo for hitting 100 posts on the thread
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The smallest composite number of the form \sum_{k=0}^n6^n. (259)
Almost equivalently, the smallest composite number of the form (6^n-1)/5.
About 9.29 times the number of days till Kickoff (28), equals roughly 260
The number of prime numbers in the interstellar message in Carl Sagan’s Contact (261)