A391934 - OEIS

%I #13 Jan 05 2026 16:55:22

%S 0,0,0,0,0,9,9,26,9,0,0,39,39,146,39,155,39,162,186,15,39,68,39,85,52,

%T 136,39,7,39,114,92,201,39,88,92,31,193,136,39,0,0,249,249,986,249,

%U 1205,249,372,1446,1485,249,1118,249,1975,1732,2236,249,2107,249,534,2192,621,249,88,2192,871,1243,2236,249,1989

%N Irregular triangle T(n, k) = Sum_{i=1..1+n} (A003415(k) mod prime(i))*A002110(i-1), with n >= 0, and 0 <= k < A002110(n), read by rows.

%H Antti Karttunen, <a href="/A391934/b391934.txt">Table of n, a(n) for n = 0..32588; rows 0-6 of the triangle</a>

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>.

%e Irregular triangle begins as:

%e n\k| 0  1    2    3    4    5     6    7    8     9    ...

%e ---+--------------------------------------------------------

%e 0  | 0;

%e 1  | 0, 0;

%e 2  | 0, 0,   9,   9,  26,   9;

%e 3  | 0, 0,  39,  39, 146,  39,  155,  39, 162,  186, 15, 39, 68, 39, 85, 52, 136, 39, 7, 39, 114, 92, 201, 39, 88, 92, 31, 193, 136, 39;

%e 4  | 0, 0, 249, 249, 986, 249, 1205, 249, 372, 1446, 1485, 249, 1118, 249, 1975, 1732, 2236, 249, 2107, 249, 534, 2192, 621, 249, 88, 2192, ...

%o (PARI)

%o rows_up_to = 6;

%o A002110(n) = prod(i=1,n,prime(i));

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A143293(n) = { if(n==0, return(1)); my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); s; };

%o A391934_tr(n, k) = { my(x=A003415(k)); sum(i=1,1+n,((x % prime(i)) * A002110(i-1))); };

%o A391934_list(rows_up_to) = { my(v = vector(A143293(rows_up_to)-1), j=0); for(row=1,rows_up_to, for(k=0, A002110(row)-1, j++; v[j] = A391934_tr(row, k))); (v); };

%o v391934 = A391934_list(rows_up_to);

%o A391934(n) = if(!n, n, v391934[n]);

%Y Cf. A002110 (row lengths), A003415, A143293.

%Y Cf. also A391930, A391935.

%K nonn,tabf

%O 0,6

%A _Antti Karttunen_, Jan 05 2026