Comma Prime-sums

(based on this idea)

a(n) is the smallest integer not yet present in S such that the leftmost digit of a(n) and the rightmost digit of a(n-1) sum up to a prime – with a(1)=1.

S = 1,2,3,4,7,6,5,8,9,20,21,10,22,11,12,13,23,24,14,15,25,26,16,...

(by hand)

I think S is a derangement of N.

Say a record of the successive "prime-sums" is kept [those sums can only be equal to 2, 3, 5, 7, 11, 13 and 17]; will the frequency of each sum slowly converge to 1/7th?

Best,

É.

Christopher Gribble:

The first 100 terms are:

1, 2, 3, 4, 7, 6, 5, 8, 9, 20, 21, 10, 22, 11, 12, 13, 23, 24, 14, 15, 25, 26, 16, 17, 40, 27, 41, 18, 30, 28, 31, 19, 29, 42, 32, 33, 43, 44, 34, 35, 60, 36, 50, 37, 45, 61, 46, 51, 47, 48, 38, 39, 49, 80, 52, 53, 81, 62, 54, 70, 55, 63, 82, 56, 57, 64, 71, 65, 66, 58, 59, 83, 84, 72, 90, 73, 85, 67, 68, 91, 69, 86, 74, 75, 87, 400, 76, 77, 401, 100, 78, 92, 93, 88, 94, 79, 89, 200, 201, 101, ...

The frequencies of occurrence of the 7 prime sums over the first 9999 terms are:

2 773

3 483

5 1219

7 1835

11 3720

13 1197

17 772

... suggesting non-uniformity of distribution.

(...)

The ways in which prime sums can be formed from the least significant digit (LSD) of a(n-1) and the most significant digit (MSD) of a(n) are:

2 0 + 2, 1 + 1

3 0 + 3, 1 + 2, 2 + 1

5 0 + 5, 1 + 4, 2 + 3, 3 + 2, 4 + 1

7 0 + 7, 1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2, 6 + 1

11 2 + 9, 3 + 8, 4 + 7, 5 + 6, 6 + 5, 7 + 4, 8 + 3, 9 + 2

13 4 + 9, 5 + 8, 6 + 7, 7 + 6, 8 + 5, 9 + 4

17 8 + 9, 9 + 8

These partitions are not uniformly distributed as can be seen from their frequencies in the first 9999 pairs.

MSD 0 1 2 3 4 5 6 7 8 9

LSD

0 0 0 273 219 0 225 0 284 0 0

1 0 500 1 0 198 0 303 0 0 0

2 0 263 0 209 0 227 0 0 0 302

3 0 0 323 0 207 0 0 0 471 0

4 0 264 0 209 0 0 0 511 0 17

5 0 0 325 0 0 0 507 0 168 0

6 0 280 0 0 0 436 0 284 0 0

7 0 0 0 0 698 0 301 0 0 0

8 0 0 0 474 0 223 0 0 0 300

9 0 0 321 0 204 0 0 0 472 0

(...)

My C++ program halts with a stack overflow exception for n = 100000. However, for n = 10000 and 20000 the prime sum frequencies are:

2 773 2762

3 483 1315

5 1219 3051

7 1835 3835

11 3720 6732

13 1197 1362

17 772 942

The frequencies with which the partitions of the prime sums into the least significant digit (LSD) of a(n-1) and the most significant digit (MSD) of a(n) are for n = 20000:

MSD 0 1 2 3 4 5 6 7 8 9

LSD

0 0 0 1262 219 0 225 0 295 0 0

1 0 1500 1 0 198 0 303 0 0 0

2 0 1095 0 209 0 227 0 0 0 469

3 0 0 1322 0 207 0 0 0 471 0

4 0 1097 0 209 0 0 0 522 0 172

5 0 0 1325 0 0 0 507 0 168 0

6 0 1269 0 0 0 436 0 294 0 0

7 0 0 0 0 1698 0 301 0 0 0

8 0 0 0 1306 0 223 0 0 0 470

9 0 0 1323 0 204 0 0 0 472 0

It is interesting to note that some frequencies have not changed.

For n = 10000:

The smallest integer not present in the first 10000 terms = 7968

The largest integer present in the first 10000 terms = 40195

For n = 20000:

The smallest integer not present in the first 20000 terms = 13851

The largest integer present in the first 20000 terms = 41195

__________

Thanks, Chris!

The graphs below were received a couple of days later from Jean-Marc Falcoz. The first one is confirmed by Hans Havermann.

Hans:

« I’ve put up a graph of 10000 points (dark blue) and their connections (cyan) here. The image is roughly fractal: the plot for 100000 points would be identical except for multiplying the x- and y-axis identifiers by 10. »

10000

« Représentation graphique de a(n) en fonction de n pour n=1 jusque n=10000 »

1000

« Représentation plus précise de a(n) pour n=1 jusque n=1000 »

100

« Le début de la suite ; par exemple au tout début :

1, 2, 3, 4, puis on redescend avec 7, 6, 5, etc. »

Jean-Marc :

J’ai les mêmes résultats que Christopher Gribble (y compris les mêmes fréquences d’apparition de 2, 3, 5, 7, 11, 13, et 17)

__________

Thanks everyone, merci beaucoup!

Best,

É.