A391933 Irregular triangle T(n, k) = Product_{i=1..n} prime(i)^(k mod prime(i)), with n >= 0, and 0 <= k < A002110(n), read by rows.

1, 1, 2, 1, 6, 9, 2, 3, 18, 1, 30, 225, 250, 1875, 18, 5, 150, 1125, 1250, 3, 90, 25, 750, 5625, 2, 15, 450, 125, 3750, 9, 10, 75, 2250, 625, 6, 45, 50, 375, 11250, 1, 210, 11025, 85750, 4501875, 302526, 588245, 150, 7875, 61250, 1029, 216090, 420175, 88236750, 5625, 14, 735, 154350, 300125, 63026250, 1058841, 10, 525

Offset

0,3

Comments

In contrast to A391930, in this sequence we store the tuples of the prime modular numbers in the prime exponents of the first n primes, whose product a(n) is. Therefore each row n is a permutation of the initial A002110(n) terms of A276086.

See further comments and references in A391930.

Links

Antti Karttunen, Table of n, a(n) for n = 0..2558; rows 0-5 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 numbers.

Formula

a(n) = A276086(A391930(n)).

Example

Irregular triangle begins as:
n\k|  0      1      2       3        4       5
---+--------------------------------------------
0  |  1
1  |  1,     2;
2  |  1,     6,     9,      2,       3,     18;
3  |  1,    30,   225,    250,    1875,     18,      5, 150, 1125, 1250, 3, 90, 25, 750, 5625, 2, 15, ...
4  |  1,   210, 11025,  85750, 4501875, 302526, 588245, 150, 7875, ...

Prog

(PARI)
up_to_row = 5;
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; };
A391933_list(up_to_row) = { my(v = vector(A143293(up_to_row)-1), j=0); for(row=1, up_to_row, for(k=0, A002110(row)-1, j++; v[j] = prod(i=1, row, prime(i)^(k % prime(i))))); (v); };
v391933 = A391933_list(up_to_row);
A391933(n) = if(!n, 1, v391933[n]);

Crossrefs

Cf. A000012 (the leftmost column), A002110 (the second column of the triangle, also the row lengths), A276086, A391930.

Sequence in context: A139526 A248927 A176013 * A387632 A263255 A145663

Adjacent sequences: A391930 A391931 A391932 * A391934 A391935 A391936

Keyword

nonn,tabf

Author

Antti Karttunen, Dec 29 2025

Status

approved