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polite and impolite numbers
A number is said polite if it can be expressed as the sum of at least two consecutive natural numbers.

For example,  $12=3+4+5$,  $666=1+2+\cdot+36$  and all the triangular numbers greater than 1 are polite numbers (I do not allow  $0+1=1$.)

It turns out that almost all the numbers are polite, and the only impolite numbers are the powers of 2.

Indeed, a number with  $k$  odd divisors greater than 1 can be represented in  $k$  ways as a nontrivial sum of consecutive naturals.

For example,  $451$  has 3 odd divisors greater than 1, namely  $11$,  $41$  and  $451$  itself, and 3 representations

\[451 = 225+226 = 10+\cdots+31=36+\cdots+46.\]

Since polite numbers are so common, I prefer to list impolite numbers.

The first impolite numbers are 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of impolite numbers
Useful links Wikipedia, Polite number
OEIS, Sequence A138591
Impolite numbers can also be... (you may click on names or numbers and on + to get more values)

ABA 32 64 128 512 1024 2048 8192 16384 32768 131072 + 549755813888 2199023255552 4398046511104 8796093022208 aban 16 32 64 128 256 512 alternating 16 32 256 amenable 16 32 64 128 256 512 1024 2048 4096 8192 + 67108864 134217728 268435456 536870912 apocalyptic 8192 16384 c.pentagonal 16 c.triangular 64 cake 64 2048 constructible 16 32 64 128 256 512 1024 2048 4096 8192 + 70368744177664 140737488355328 281474976710656 562949953421312 cube 64 512 4096 32768 262144 2097152 16777216 134217728 1073741824 8589934592 + 549755813888 4398046511104 35184372088832 281474976710656 d-powerful 2048 deficient 16 32 64 128 256 512 1024 2048 4096 8192 + 1048576 2097152 4194304 8388608 Duffinian 16 32 64 128 256 512 1024 2048 4096 8192 + 1048576 2097152 4194304 8388608 eban 32 64 economical 16 32 64 128 256 512 1024 2048 4096 8192 + 2097152 4194304 8388608 16777216 enlightened 256 2048 262144 2097152 268435456 2147483648 274877906944 equidigital 16 32 64 esthetic 32 Friedman 128 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 frugal 128 256 512 1024 2048 4096 8192 16384 32768 65536 + 67108864 134217728 268435456 536870912 gapful 1048576 16777216 33554432 67108864 happy 32 4096 4194304 8388608 Harshad 512 house 32 iban 1024 idoneal 16 inconsummate 8192 524288 interprime 64 4096 Jordan-Polya 16 32 64 128 256 512 1024 2048 4096 8192 + 70368744177664 140737488355328 281474976710656 562949953421312 junction 1024 33554432 katadrome 32 64 Leyland 32 512 33554432 Lynch-Bell 128 magnanimous 16 32 512 metadrome 16 128 256 nialpdrome 32 64 nude 128 oban 16 512 odious 16 32 64 128 256 512 1024 2048 4096 8192 + 67108864 134217728 268435456 536870912 pancake 16 4096 plaindrome 16 128 256 power 16 32 64 128 256 512 1024 2048 4096 8192 + 4398046511104 8796093022208 17592186044416 35184372088832 powerful 16 32 64 128 256 512 1024 2048 4096 8192 + 70368744177664 140737488355328 281474976710656 562949953421312 practical 16 32 64 128 256 512 1024 2048 4096 8192 + 1048576 2097152 4194304 8388608 Ruth-Aaron 16 self 64 512 65536 524288 536870912 square 16 64 256 1024 4096 16384 65536 262144 1048576 4194304 + 4398046511104 17592186044416 70368744177664 281474976710656 super-d 16384 2097152 tau 128 32768 uban 16 32 Ulam 16 8192 unprimeable 512 2048 32768 2097152 8388608 untouchable 2048 65536 upside-down 64 8192 Zuckerman 128