It’s obvious when you think about: an even number can never be the sum of two consecutive integers. Conversely, an odd number (except 1) is always the sum of two consecutive integers: 3 = 1 + 2; 5 = 2 + 3; 7 = 3 + 4; 9 = 4 + 5; and so on. The sum of three consecutive integers can be either odd or even: 6 = 1 + 2 + 3; 9 = 2 + 3 + 4. The sum of four consecutive integers must always be even: 1 + 2 + 3 + 4 = 10; 2 + 3 + 4 + 5 = 14. And so on.
But notice that 9 is the sum of consecutive integers in two different ways: 9 = 4 + 5 = 2 + 3 + 4. Having spotted that, I decided to look for numbers that were the sums of consecutive integers in the most different ways. These are the first few:
3 = 1 + 2 (number of sums = 1)
9 = 2 + 3 + 4 = 4 + 5 (s = 2)
15 = 1 + 2 + 3 + 4 + 5 = 4 + 5 + 6 = 8 + 7 = (s = 3)
45 (s = 5)
105 (s = 7)
225 (s = 8)
315 (s = 11)
945 (s = 15)
1575 (s = 17)
2835 (s = 19)
3465 (s = 23)
10395 (s = 31)
It was interesting that the number of different consecutive-integer sums for n was most often a prime number. Next I looked for the sequence at the Online Encyclopedia of Integer Sequences and discovered something that I hadn’t suspected:
A053624 Highly composite odd numbers: where d(n) increases to a record.
1, 3, 9, 15, 45, 105, 225, 315, 945, 1575, 2835, 3465, 10395, 17325, 31185, 45045, 121275, 135135, 225225, 405405, 675675, 1576575, 2027025, 2297295, 3828825, 6891885, 11486475, 26801775, 34459425, 43648605, 72747675, 130945815 — A053624 at OEIS
The notes add that the sequence is “Also least number k such that the number of partitions of k into consecutive integers is a record. For example, 45 = 22+23 = 14+15+16 = 7+8+9+10+11 = 5+6+7+8+9+10 = 1+2+3+4+5+6+7+8+9, six such partitions, but all smaller terms have fewer such partitions (15 has four).” When you don’t count the number n itself as a partition of n, you get 3 partitions for 15, i.e. consecutive integers sum to 15 in 3 different ways, so s = 3. I looked at more values for s and found that the stream of primes continued to flow:
3 → s = 1
9 = 3^2 → s = 2 (prime)
15 = 3 * 5 → s = 3 (prime)
45 = 3^2 * 5 → s = 5 (prime)
105 = 3 * 5 * 7 → s = 7 (prime)
225 = 3^2 * 5^2 → s = 8 = 2^3
315 = 3^2 * 5 * 7 → s = 11 (prime)
945 = 3^3 * 5 * 7 → s = 15 = 3 * 5
1575 = 3^2 * 5^2 * 7 → s = 17 (prime)
2835 = 3^4 * 5 * 7 → s = 19 (prime)
3465 = 3^2 * 5 * 7 * 11 → s = 23 (prime)
10395 = 3^3 * 5 * 7 * 11 → s = 31 (prime)
17325 = 3^2 * 5^2 * 7 * 11 → s = 35 = 5 * 7
31185 = 3^4 * 5 * 7 * 11 → s = 39 = 3 * 13
45045 = 3^2 * 5 * 7 * 11 * 13 → s = 47 (prime)
121275 = 3^2 * 5^2 * 7^2 * 11 → s = 53 (prime)
135135 = 3^3 * 5 * 7 * 11 * 13 → s = 63 = 3^2 * 7
225225 = 3^2 * 5^2 * 7 * 11 * 13 → s = 71 (prime)
405405 = 3^4 * 5 * 7 * 11 * 13 → s = 79 (prime)
675675 = 3^3 * 5^2 * 7 * 11 * 13 → s = 95 = 5 * 19
1576575 = 3^2 * 5^2 * 7^2 * 11 * 13 → s = 107 (prime)
2027025 = 3^4 * 5^2 * 7 * 11 * 13 → s = 119 = 7 * 17
2297295 = 3^3 * 5 * 7 * 11 * 13 * 17 → s = 127 (prime)
3828825 = 3^2 * 5^2 * 7 * 11 * 13 * 17 → s = 143 = 11 * 13
6891885 = 3^4 * 5 * 7 * 11 * 13 * 17 → s = 159 = 3 * 53
11486475 = 3^3 * 5^2 * 7 * 11 * 13 * 17 → s = 191 (prime)
26801775 = 3^2 * 5^2 * 7^2 * 11 * 13 * 17 → s = 215 = 5 * 43
34459425 = 3^4 * 5^2 * 7 * 11 * 13 * 17 → s = 239 (prime)
43648605 = 3^3 * 5 * 7 * 11 * 13 * 17 * 19 → s = 255 = 3 * 5 * 17
72747675 = 3^2 * 5^2 * 7 * 11 * 13 * 17 * 19 → s = 287 = 7 * 41
130945815 = 3^4 * 5 * 7 * 11 * 13 * 17 * 19 → s = 319 = 11 * 29
I can’t spot any way of predicting when n will yield a primal s, but I like the way that a simple question took an unexpected turn. When a number sets a record for the number of different ways it can be the sum of consecutive integers, that number will also be a highly composite odd number.
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