A391929 Number of fixed points on row n of triangle A391930.

1, 2, 2, 10, 10, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22

Offset

0,2

Comments

The first n with a(n) > 22 is at n = A341805(4)+1 = 9823713, where a(9823713) = 2*prime(9823713) = 352156586, because the row 9823713 of A391928 is an ascending run of terms 0, 1, 2, 3, ..., prime(9823713)-1. See Jeppe Stig Nielsen's Feb 20 2021 comment in A079276.

Links

Table of n, a(n) for n=0..79.

Rémy Sigrist, Illustration of the terms of A343405 (term a(n)-1 essentially gives the width of that tree at depth n)

Index entries for sequences related to primorial base.

Formula

a(0) = 1, a(1) = 2, and for n > 1, if A002110(n-1)-1 is a multiple of prime(n), then a(n) = 2*prime(n), otherwise a(n) = a(n-1).

a(n) = Sum_{k=0 .. A002110(n)-1} [k == A360407(n, k)], where [ ] is the Iverson bracket.

a(n) = Sum_{k=0 .. A002110(n)-1} [k == (Sum_{i=1..n} ((k mod prime(i)) * A002110(i-1)))].

Example

On row 3 of A391930 (and equally on row 3 of A360407) the fixed points are 0, 5, 6, 11, 12, 17, 18, 23, 24, 29, therefore a(3) = 10.
On row 4 of A391930 the fixed points are: 0, 36, 65, 72, 101, 108, 137, 144, 173, 209, therefore a(4) = 10.
On row 5 the fixed points are: 0, 209, 210, 419, 420, 629, 630, 839, 840, 1049, 1050, 1259, 1260, 1469, 1470, 1679, 1680, 1889, 1890, 2099, 2100, 2309, therefore a(5) = 22.
On row 6 the fixed points are: 0, 1469, 2939, 4200, 5670, 7140, 8609, 10079, 11340, 12810, 14280, 15749, 17219, 18689, 19950, 21420, 22889, 24359, 25829, 27090, 28560, 30029, therefore a(6) = 22.
On row 10 the fixed points are: 0, 75741330, 92099070, 869259720, 1306416510, 1400639309, 1568479710, 2194157699, 2631314489, 2782797150, 2799154890, 3670538339, 3686896079, 3838378740, 4275535530, 4901213519, 5069053920, 5163276719, 5600433509, 6377594159, 6393951899, 6469693229, therefore a(10) = 22.

Prog

(PARI)
\\ Slow program:
A002110(n) = prod(i=1, n, prime(i));
is_fixed_point_of_A391930_row(n, k) = { my(orgk=k); for(i=1, n, if(((orgk-k)%prime(i))!=0, return(0)); k \= prime(i)); (1); };
A391929(n) = sum(k=0, A002110(n)-1, is_fixed_point_of_A391930_row(n, k));
(PARI)
\\ Use this for obtaining the actual fixed points:
memo_fp_lists = Map();
A002110(n) = prod(i=1, n, prime(i));
A391930_fp_lists(n) = { my(v); if(mapisdefined(memo_fp_lists, n, &v), v, v = if(!n, [0], my(prev_fps = A391930_fp_lists(n-1), new_fps = List([]), w = A002110(n-1)); for(i=1, #prev_fps, if((w%prime(n))>1, listput(new_fps, prev_fps[i] + w*lift(Mod(prev_fps[i]/(1-w), prime(n)))), for(d=0, prime(n)-1, if(d==((prev_fps[i]+(d*w))%prime(n)), listput(new_fps, (prev_fps[i]+(d*w))))))); Vec(new_fps)); mapput(memo_fp_lists, n, v); (v)); };
A391929(n) = #A391930_fp_lists(n);
(PARI)
\\ And this for computing just their count:
A002110(n) = prod(i=1, n, prime(i));
A391929(n) = if(n<2, 1+n, if(!((A002110(n-1)-1)%prime(n)), 2*prime(n), A391929(n-1)));

Crossrefs

Cf. A002110, A079276, A341805, A343405, A360407, A391928, A391930.

Sequence in context: A249152 A216708 A032005 * A147801 A367545 A263053

Adjacent sequences: A391926 A391927 A391928 * A391930 A391931 A391932

Keyword

nonn

Author

Antti Karttunen, Jan 03 2026

Status

approved