OEIS featured in The New York Times

[Valentin Albillo]

      
Hi again,

Thanks a lot to Fernando del Rey and jonmoore for their interest and very kind words, much appreciated. Re the OEIS (On-Line Encyclopedia of Integer Sequences), I heartily recommend reading these PDF documents to all people new to it:

On-Line Encyclopedia of Integer Sequences (25-page PDF)
The On-Line Encyclopedia of Integer Sequences - An illustrated guide with many unsolved problems (37-page PDF)
A Handbook of Integer Sequences - Fifty Years Later (23-page PDF)

Now for a little self-quoting:

I Wrote:[...] Generating the sequence is trivial [...]

1 3 6 3 8 4 2 1 8 4 2 1 12 6 3 16 8 4 2 1 18 9 28 14 7 30 15 44 22 ...

and there's a number of interesting things to explore in the sequence.

Indeed there is, and presently I'll mention a couple. First of all, the sequence is conjectured to include all positive integers 1, 2, 3, ..., as elements, and we can investigate this alleged fact by using this 5-liner for the HP-71B, which takes as input K, the maximum number of elements to generate, and will output the pairs (number, index) for up to the first 60 numbers (1..60), where index is the lowest one where number first appears in the sequence.

It will also output how many numbers weren't found within the first K elements (plus timing) and finally it will explicitly list all numbers not found:

10 DESTROY ALL @ INPUT K @ INTEGER D(K),M(50) @ N=1 @ P=1 @ SETTIME 0
20 FOR C=1 TO K @ IF N>K THEN STD ELSE IF NOT D(N) THEN D(N)=C
30 IF MOD(N,2) THEN P=FPRIM(P+1) @ N=N+P ELSE N=N DIV 2
40 NEXT C @ C=0 @ FOR I=1 TO MIN(K,60) @ IF D(I) THEN DISP (I,D(I)); ELSE C=C+1 @ M(C)=I
50 NEXT I @ DISP @ DISP "Not found:";C;TIME$ @ FOR I=1 TO C @ DISP M(I); @ NEXT I @ DISP

Let's make a sample run specifying a maximum of 100 elements:

>RUN ? 100


(1,1) (2,7) (3,2) (4,6) (5,71) .. (48,42) (51,58) (53,73) (55,88) (58,33)
Not found: 21 (timing)
13 17 19 25 26 27 32 33 34 36 38 41 47 49 50 52 54 56 57 59 60

This means that 1 first appears at index 1, 2 at index 7, 3 at index 2, 4 at index 6, 5 at index 71, and so on until 55 appears at index 88 and 58 at index 33. We also learn that as many as 21 numbers from 1..60 didn't appear among the first 100 elements of the sequence, the first of them being 13, 17, ... and the last ones being ..., 59, 60.

Of course, increasing the maximum number of elements to explore will probably result in finally locating the first occurrence of some or all the numbers not found within the first 100 elements of the sequence. Indeed, running the above program for 100, 200, ...., 2000 elements we get these results:

#Elem  #  Not found
--------------------------------------------------------------------------
 100  21  13 17 19 25 26 27 32 33 34 36 38 41 47 49 50 52 54 56 57 59 60
 200  12  13 25 26 27 32 36 41 49 50 52 54 60
 500   8  25 27 32 36 49 50 54 60
1000   6  25 27 36 50 54 60
2000   4  27 36 54 60

and in fact only 4 numbers from 1..60 failed to make an appearance within the first 2,000 elements, corroborating the conjecture that all positive integers will eventually appear in the sequence.

Now we can try to locate the first occurrence of the 4 missing numbers (27, 36, 54, 60) in a faster way and using minimal memory by running this 4-line HP-71B program, which will accept the number to locate and will search for it among the first 1,000,000 elements (if not using a very fast emulator, you might want to reduce this upper limit to save time; generating a million elements requires primes up to 4,761,697):

10 DESTROY ALL @ STD @ INPUT K @ N=1 @ P=1 @ SETTIME 0
20 FOR C=1 TO 1000000 @ IF N=K THEN DISP N;C;TIME$ @ END
30 IF MOD(N,2) THEN P=FPRIM(P+1) @ N=N+P ELSE N=N DIV 2
40 NEXT C @ DISP "Not found: ";TIME$

Let's try a sample run with an easily found number, 25:

>RUN
? 25 ->  25  1154 (timing)

which quickly tells us that 25 appears in the sequence at index 1,154. Now let's consider the four numbers which weren't located earlier within the first 2,000 elements, namely 27, 36, 54 and 60, to try and locate their first occurrence in the sequence, if possible:

>RUN
? 27 ->  27  161336  (Emu71/Win @972x:  28", go71b @128x:  3'36", Physical: 7h 41')
? 60 ->  60  614667  (Emu71/Win @972x: 3'3", go71b @128x: 23'14", Physical: ~ 50h)

so 27 appears for the first time in the sequence at index 161,336 and 60 at index 614,667. As for 54, being twice 27, we don't need to search for it, its index will be one less than the one for 27, namely 161,335.

Unfortunately, the search up to index 1,000,000 fails for 36 because its first occurrence in the sequence happens to be at index 77,534,485,877 !!. Even worse, if considering numbers up to 100 instead of up to 60, the number 97 first appears at index 17,282,073,747,557 !!!

Additionally, a second conjecture is that every positive integer appears in the sequence not just once but an infinite number of times. We can check it out by finding multiple indexes for any given input number, simply delete the END statement at line 20 and reduce to 100,000 the maximum index to search up to. Line 20 will then look like this:

20 FOR C=1 TO 100000 @ IF N=K THEN DISP N;C;TIME$

Let's try a few selected numbers (you may break execution before reaching index 100,000, to save time):

>RUN
?  1  ->  1  8  12  20  742 ...             (the very next appearance is at index 513,152,128)
?  2  ->     7  11  19  741 ...             (ditto at index 513,152,127)
?  3  ->  2  4  15  46  95  6355 ...
?  5  ->  71  4849 ...                            (no others up to index 1,000,000)
?  7  ->  25  114  123  446  7104 ...
? 13  ->  345  418  4621 ...                  (no others up to index 1,000,000)
? 25  ->  1154  1519  10359  13330 ... (no others up to index 1,000,000)
? 85  ->  140  3161  72349...

so it seems that indeed there might be infinite appearances of any positive integer but the distance from one appearance to the next increases at an extremely fast rate.

Additional Trivia:

● A few other nice appearances are 31416 at indexes 6,768 and 6,923, 11111 at 12,497, 55555 at 56,551, 100000 at 26,488 and last but not least, 2023 at 2,165. On the other hand, 1992 does not appear within the first 1,000,000 elements.

● There's a number of solutions of A(n) = n, i.e. numbers whose index in the sequence equals the number itself. They can be found very easily with a trivial modification of my second program above, and the first ones are n = 1, 16, 787, 427447 and no others up to index 1,000,000.

Regards.
V.