A391930 Irregular triangle T(n, k) = Sum_{i=1..n} (k mod prime(i))*A002110(i-1), with n >= 0, and 0 <= k < A002110(n), read by rows. Here A002110(n) is the n-th primorial number.

0, 0, 1, 0, 3, 4, 1, 2, 5, 0, 9, 16, 19, 26, 5, 6, 15, 22, 25, 2, 11, 12, 21, 28, 1, 8, 17, 18, 27, 4, 7, 14, 23, 24, 3, 10, 13, 20, 29, 0, 39, 76, 109, 146, 155, 186, 15, 52, 85, 92, 131, 162, 201, 28, 31, 68, 107, 138, 177, 184, 7, 44, 83, 114, 123, 160, 193, 20, 59, 60, 99, 136, 169, 206, 5, 36, 75, 112, 145, 152, 191

Offset

0,5

Comments

In his paper Matthias Schmitt defines PM(n), "the prime modular numbers of level n", that are tuples of length n, where the i-th term of the tuple is an integer in range 0 .. prime(i)-1. Then he defines a homomorphism f from a finite integral domain of size A002110(n) to PM(n) [which is of the same size] and proves that the homomorphism is injective and thus also an isomorphism. In this sequence we store the tuples of the prime modular numbers in the digits of the primorial base expansion (A049345) of a(n), thus each row n of length A002110(n) lists a permutation of integers in range 0 .. A002110(n)-1.

See also A391933 for another version of this sequence and A360407 for the inverse rows.

Links

Antti Karttunen, Table of n, a(n) for n = 0..32588; rows 0-6 of the triangle

Matthias Schmitt, A function to calculate all relative prime numbers up to the product of the first n primes, arXiv:1404.0706 [math.NT] (see pages 3-4).

Index entries for sequences related to primorial base.

Index entries for sequences related to primorial numbers.

Formula

a(n) = A276085(A391933(n)).

T(n, k) + T(n, (A002110(n)-1)-k) = A002110(n)-1.

Example

Irregular triangle begins as:
n\k| 0   1   2    3   4     5    6  7    8   9    ...
---+--------------------------------------------------
0  | 0
1  | 0,  1;
2  | 0,  3,  4,   1,  2,    5;
3  | 0,  9, 16,  19,  26,   5,   6, 15, 22, 25, 2, 11, 12, 21, 28, 1, 8, 17, 18, 27, 4, 7, 14, 23, 24, 3, 10, 13, 20, 29;
4  | 0, 39, 76, 109, 146, 155, 186, 15, 52, 85, 92, 131, 162, 201, 28, 31, 68, 107, 138, 177, 184, 7, 44, 83, 114, 123, ...

Prog

(PARI)
rows_up_to = 6;
A002110(n) = prod(i=1, n, prime(i));
A143293(n) = { if(n==0, return(1)); my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); s; };
A391930_tr(n, k) = sum(i=1, n, ((k % prime(i)) * A002110(i-1)));
A391930_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] = A391930_tr(row, k))); (v); };
v391930 = A391930_list(rows_up_to);
A391930(n) = if(!n, n, v391930[n]);

Crossrefs

Cf. A002110 (row lengths), A143293 (the second column), A276085, A391929 (number of fixed points on each row), A391931, A391933 [= A276086(a(n))].

Cf. also A049345, A240673, A360407 (triangle with inverse rows), A391934, A391935.

Sequence in context: A076942 A375567 A205127 * A360407 A343613 A333454

Adjacent sequences: A391927 A391928 A391929 * A391931 A391932 A391933

Keyword

nonn,base,tabf,look

Author

Antti Karttunen, Dec 29 2025

Status

approved