Old sequence challenge (Joe Shipman)

What is the next number in this sequence: 1 2 3 4 5 6 8 9 10 11 12 14 15 16 17 18 20 21 22 24 ... ?

Answer

The next number is 26. The sequence is all differences of prime numbers. For example, 24 = 29 - 5 and 26 = 29 - 3. 25 is not on the list; if it were a difference of primes, one of them would have to be even and hence 2 (the only even prime), and therefore the other one would have to be 27, but 27 is not prime.

It looks as if every even number will be in this list of differences of primes. Whenever we have tried to write an even number as a difference of primes, we have always succeeded.

Nevertheless, this "obvious" fact is not known to be true for all even numbers. In fact, there is no known sure-fire way to tell whether any given even number is a difference of primes. Of course if it is, you'd find the primes after looking long enough, but if it were not, you could look forever and never know for sure, ever wondering if the next primes you'd try might work.

For this reason, we do not know whether a computer could compute this sequence. Under all known methods, if some even number failed to be a difference of primes, the computer would just get stuck. This is the simplest example I know of a sequence which is not known to be computable.

Computer calculations for primes up to a trillion indicate that the most-common difference of consecutive primes is 6 (2 x 3), but mathematicians suspect that for much huger primes, of perhaps 35 digits or more, 30 (2 x 3 x 5) and later 210 (2 x 3 x 5 x 7) and other differences will become more common. No one has been able to decide for sure.

Ron Douglass has a different interpretation of our original sequence. He points out that the numbers missing are the 4th, 6th, 8th, and 9th primes and notes that 4, 6, 8, 9 are the first four composite numbers.

Since the next composite number is 10, he concludes that the next missing number in the sequence should be the 10th prime, namely, 29.

A fanciful alternative interpretation of the sequence comes from Herb Helbig: "Suppose that Joe plays the violin . You see, after subtracting multiples of 9, the four missing numbers (7, 13, 19, 23) become 7, 4, 1, 5. Assigning these to the alphabet gives G, D, A, E, all four strings of the violin. The last number required is 24, which therefore ends the sequence. There is no next number!"

New hymn numbers challenge (Joe Herman)

"Occasionally in church I will mentally add up the numbers of the hymns for the service as they appear on a sign on the wall. Sometimes I even add them upside down and sideways. On almost all occasions when I do this arithmetic, I note a strange coincidence. Consider, for example, these hymn numbers:

181

158

297

They total to 636 when added normally; 1,824 if totaled upside down (792+851+181); 1,950 from the right side (211+958+781); and 1,158 from the left (187+859+112). In every instance when adding the digits in these separate totals, 6+6+3, 1+8+2+4, 1+9+5+0, and 1+1+5+8, the total is identical for this example: 15.

Is this some unusual situation or will it always be the case; and is there some reason for it?"

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