cp's OEIS Frontend

A353248 Irregular table, read by rows, where row n is the concatenation of all prime signatures leading to n divisors, in reverse lexicographic order.

%I A353248 #18 Apr 27 2022 11:54:57
%S A353248 1,2,3,1,1,4,5,2,1,6,7,3,1,1,1,1,8,2,2,9,4,1,10,11,5,1,3,2,2,1,1,12,
%T A353248 13,6,1,14,4,2,15,7,1,3,3,3,1,1,1,1,1,1,16,17,8,1,5,2,2,2,1,18,19,9,1,
%U A353248 4,3,4,1,1,20,6,2,21,10,1,22,23,11,1,7,2,5,3,5,1,1,3,2,1
%N A353248 Irregular table, read by rows, where row n is the concatenation of all prime signatures leading to n divisors, in reverse lexicographic order.
%C A353248 The number-of-divisors function d = A000005 is multiplicative with d(p^e) = e+1. Therefore, a prime signature (e1, e2, ..., eN) which yields a given number of divisors corresponds to a factorization (e1+1)*...*(eN+1) of that number. Following the usual convention, we only consider prime signatures with decreasing exponents (e1 >= e2 >= ... >= eN). Furthermore, we list them in reverse lexicographic order, i.e., largest e1 first, etc.
%C A353248 The sequence starts with row 1 which has length 0 (the only number having only 1 divisor is 1 which has the empty product as prime factorization), so the first term a(1) = 1 is the first element of row 2. Here and thereafter, the first element of row n is easily recognized as the first occurrence of n-1, which is the only element of the row (and therefore followed by n) iff n is prime.
%e A353248 Table begins:
%e A353248 row n | prime signatures
%e A353248    1  | ()
%e A353248    2  | (1)
%e A353248    3  | (2)
%e A353248    4  | (3), (1,1)
%e A353248    5  | (4)
%e A353248    6  | (5), (2,1)
%e A353248    7  | (6)
%e A353248    8  | (7), (3,1), (1,1,1)
%e A353248    9  | (8), (2,2)
%e A353248   10  | (9), (4,1)
%e A353248   11  | (10)
%e A353248   12  | (11), (5,1), (3,2), (2,1,1)
%o A353248 (PARI) A353248_row(n, M=n)={if(n>1, my(f=factor(n)~, m=f[1,#f], L=List()); fordiv(n, d, n < m*d && break; n > M*d || foreach(self()(d, n/d), S, listput(L,concat(n/d-1,S)))); Vec(L), [[]])}
%Y A353248 Cf. A000005 (d = tau = number-of-divisors function).
%Y A353248 Cf. A025487 (products of primorial numbers, representatives of prime signatures), A046523 (representative of prime signature of n), A118914, A212171 and A124010 (prime signature of n), A124832 (prime signatures listed in order of representatives A025487), A080577 (partitions in grad.rev.lex order), A036036 (partitions in rev.lex order).
%K A353248 nonn,tabf
%O A353248 1,2
%A A353248 _M. F. Hasler_, Apr 08 2022