[seqfan] Sum of the a(n) first digits of S is a prime -- an illusion?

[Eric Angelini]

Hello Seqfans -- here is another nightmare:

« Smallest integer not yet present in S such that the sum of the a(n) first digits of S is a prime »:

I get by hand:

S = 2,1,4,6,3,7,8,60,9,11,14,17,61,16,22,25,30,26,28,34,49,37,200,36,38,39,42,59,51,54,56,61,69,201,...

Sum of the first  2 digits is a prime : 2+1 = 3 
Sum of the first  1 digit  is a prime : 2 = 2
Sum of the first  4 digits is a prime : 2+1+4+6 = 13
Sum of the first  6 digits is a prime : 2+1+4+6+3+7= 23
Sum of the first  3 digits is a prime : 2+1+4 = 7
Sum of the first  7 digits is a prime : 2+1+4+6+3+7+8 = 31
Sum of the first  8 digits is a prime : 2+1+4+6+3+7+8+6 = 37
Sum of the first 60 digits is a prime :  
2+1+4+6+3+7+8+6+0+9+1+1+1+4+1+7+6+1+1+6+2+2+2+5+3+0+2+6+2+8+3+4+4+9+3+7+2+0+0+3+6+3+8+3+9+4+2+5+9+5+1+5+4+5+6+6+1+6+9+2 = 241
Sum of the first  9 digits is a prime : 2+1+4+6+3+7+8+6+0 = 37
Sum of the first 11 digits is a prime : 2+1+4+6+3+7+8+6+0+9+1 = 47
Sum of the first 14 digits is a prime : 2+1+4+6+3+7+8+6+0+9+1+1+1+4 = 53
Sum of the first 17 digits is a prime : 2+1+4+6+3+7+8+6+0+9+1+1+1+4+1+7+6 = 67
Sum of the first 61 digits is a prime : 
2+1+4+6+3+7+8+6+0+9+1+1+1+4+1+7+6+1+1+6+2+2+2+5+3+0+2+6+2+8+3+4+4+9+3+7+2+0+0+3+6+3+8+3+9+4+2+5+9+5+1+5+4+5+6+6+1+6+9+2+0 = 241
Sum of the first 16 digits is a prime : 2+1+4+6+3+7+8+6+0+9+1+1+1+4+1+7 = 61
...              ||
                 ||
--Above column is S

In computing this, one is forced to put constraints on the future of S.
See here, for instance:

S = 2,1,4,...

As we cannot write "3" after "1" (because the sum of the first "3" digits
would be 6 -- a composite), we write "4" -- leaving the future "open".
But this "4" forces the next term, 6:

S = 2,1,4,6,...

And again, this "6" puts a constraint on the future of S. "3" is forced,
as "3" is the « smallest integer not yet present in S such that the sum
of the a(n) first digits of S is a prime »; indeed, it works, "3" describes
the "past" of S and 2+1+4=7, a prime.

S = 2,1,4,6,3,7,...

"7" resolves the constraint put by "6"; indeed the sum of the first 6
digits of S is a prime -- "7" being the smallest available integer:
2+1+4+6+3+7= 23

This "7" puts another constraint on the future of S -- resolved by "8":

S = 2,1,4,6,3,7,8,... Check:  2+1+4+6+3+7+8 = 31, ok.

This "8", again, puts a constraint on the next term -- and we notice
quickly that we must take "60" as the next term! If we could have taken
"6" instead, we would have had a prime sum (31+6=37) -- but 6 was already
in S! We must try something else -- and after having checked that no
smallest integer fits, we write "60" after "8":

S = 2,1,4,6,3,7,8,60,...

This puts a huge constraint on S! We have to remember that the sum of the
first 60 digits of S is a prime!

Nevertheless, we proceed... (fast forward until):

S = 2,1,4,6,3,7,8,60,9,11,14,17,61,16,22,25,30,26,28,...

The above last digit ("8") is the 30st of S; the digit sum so far is 107;

What would be the next term "xy", after 28? All "past" prime sums have
been described so far -- thus we write "for the future" and try "31":

S = 2,1,4,6,3,7,8,60,9,11,14,17,61,16,22,25,30,26,28,31,...

No -- the "3" of "31" is the 31st digit of S; as 107+"3" is 110 (composite)
we cannot keep "31". We try "32":

S = 2,1,4,6,3,7,8,60,9,11,14,17,61,16,22,25,30,26,28,32,...

No again: the sum of the 32 first digits would now be 107+"3"+"2" = 112,
a composite.

"33" seems ok -- as the sum of the 33 first digits is prime (read the
"nth digit" vertically and the cumulative sums "CS" vertically too):

nth=1 2 3 4 5 6 7 89 1 11 11 11 11 12 22 22 22 22 23 33 33
nth                  0 12 34 56 78 90 12 34 56 78 90 12 34
S = 2,1,4,6,3,7,8,60,9,11,14,17,61,16,22,25,30,26,28,33,xy 
CS= 2 3 7 1 1 2 3 33 4 44 45 56 66 67 77 88 88 99 91 11 .
CS        3 6 3 1 77 6 78 93 41 78 95 79 16 99 17 90 11 .
                                                   7 03 .

Ouch... We have a problem with the next digit, "x"! No digit will
fit, as the closest prime to 113 is 127 -- 14 units apart! We see
that 33 puts too strong a constraint on the future of S -- a halting
one! We thus erase "33" and try "34" (shouting 'halleluya' here to
the inventor of the copy/paste technique):

nth=1 2 3 4 5 6 7 89 1 11 11 11 11 12 22 22 22 22 23 33 33
nth                  0 12 34 56 78 90 12 34 56 78 90 12 34
S = 2,1,4,6,3,7,8,60,9,11,14,17,61,16,22,25,30,26,28,34,xy  ...
CS= 2 3 7 1 1 2 3 33 4 44 45 56 66 67 77 88 88 99 91 11 ..
CS        3 6 3 1 77 6 78 93 41 78 95 79 16 99 17 90 11 ..
                                                   7 04 ..
 
... Now, with "34", we have 2 digits available to bridge the gap; the
smallest integer fitting will be 49:

nth=1 2 3 4 5 6 7 89 1 11 11 11 11 12 22 22 22 22 23 33 33
nth                  0 12 34 56 78 90 12 34 56 78 90 12 34
S = 2,1,4,6,3,7,8,60,9,11,14,17,61,16,22,25,30,26,28,34,49,  ...
CS= 2 3 7 1 1 2 3 33 4 44 45 56 66 67 77 88 88 99 91 11 11
CS        3 6 3 1 77 6 78 93 41 78 95 79 16 99 17 90 11 12
                                                   7 04 87

I don't now if such a seq can exist; there might be for ever the doubt
that we have to erase a lot of terms in order to proceed -- and "a lot
of terms" might force us, at some (far away) point, NOT to start with

S = 2,1,4,6,3,7,8,60,...

Best,
É.