cp's OEIS Frontend

Index of first occurrence of n in A069288.

First differs from A038547 at n = 12.

All the terms are odd since all the terms of A326835 are odd (as phi(1) = phi(2) = 1).

a(n) exists for any n since 3^(n-1) is a term of A326835 which has n divisors.

a(n) is the least number k such that A348341(k) = n.

Since A348341(k) depends only on the prime signature of k, all the terms of this sequence are in A025487.

The width pattern (A341969) of the symmetric representation of sigma for a number with k >= 1 odd divisors has length 2*k - 1.

a(p) = p for any prime number p is realized by the m+1 numbers 3^(p-1), ..., 2^m * 3^(p-1) which contain m+1-p duplicates, where m = floor(log_2(3^(p-1))). Each width pattern first increases to a level 1 <= i <= p and then alternates between i and i-1 up to the diagonal of the symmetric representation of sigma resulting in p distinct patterns.

For some numbers n = 2^m * q, q odd and not prime, that are the least instantiations of a width pattern their odd parts q may not be the least instantiations of a width pattern, examples are 78, 1014, 12246 and 171366 with 4, 6, 8 and 10 odd divisors, respectively (see row 2 of the table in A367377).

Conjecture: a(9) = 28.

The least number instantiating the 28th width pattern, 12345654345654321, is n = 43356672, found in a search up to 5*10^9.

Table of width pattern counts of the symmetric representation of sigma and of all possible symmetric patterns:

# odd divisors 1 2 3 4 5 6 7 8 9 10 11 12

pattern count 1 2 3 6 5 16 7 40 28? >=47 11 >=223

A001405 1 2 3 6 10 20 35 70 126 252 462 924

The 4 symmetric patterns 10123232101, 10123432101, 12101010121 and 12123432121 cannot be instantiated as width patterns of numbers with 6 odd divisors.

30 of the 70 possible symmetric patterns of numbers n = 2^m * q, m>=0 and q odd, with 8 odd divisors cannot be instantiated as width patterns of the symmetric representation of sigma(n) since their sequence of widths contradicts the order of the odd divisors d_i of n and of the numbers 2^(m+1) * d_i and the positions of their corresponding 1's in the rows of the triangle of widths in A249223.

The length of row n in the triangle is A001227(n) and its first column T(n, 1) is ordered. Also, A001227(n) = number of 1s in row n of the triangle of A237048 = length of row n in the triangle of A280851. The records of row lengths in the triangle form sequence A038547.

T(i, j) = -1 for i >= 1 odd, nonprime, j even with 1 < j < i; also for i prime and all j with 1 < j < i.

The single value T(10, 4) = -1 has been verified; see the conjecture below.

T(i, i) <= 3^(i-1) for all i >=1 . Equality holds for all primes i. T(i, i) = A318843(i), for all i >= 1.

A038547(i) is the smallest number with exactly i odd divisors. Thus odd number A038547(i) occurs in row i of triangle T(i, j) so that A038547 is a subsequence of this sequence. For i prime, A038547(i) = T(i, i). For 4 <= i <= 10^9 nonprime, A038547(i) is in the third column, T(i, 3), except for i=8; furthermore, the first part of SRS(A038547(i)) has width 1 and size (A038547(i)+1)/2.

T(i, 1) <= 2 * 3^(i-1) and it is even for all i >1. Equality holds for all primes i.

T(i, 2) <= 2 * 3^(i/2-1) * p for all even i where p is the smallest prime greater than 4 * 3^(i/2-1). Equality holds when i = 2 * h where h is prime.

The positive numbers in columns 1..6 are subsequences of A174973, A239929, A279102, A280107, A320066, A320511, respectively.

Conjectures:

All entries T(i, j) in columns j >= 3 are odd.

T(i, 1)/2 is odd for all i > 1.

T(i, 1) = 2 * T(i, 3) for all nonprime i > 3, for i = 3, but not for i = 8.

T(i, 2)/2 is odd for all even i > 2.

T(i, 3) = A038547(i) for all nonprime i > 3, except i = 8.

T(2*i, 2*j) = -1 for j >= 2 and all prime i satisfying i >= prime(j+1).

From Omar E. Pol, Jun 08 2025: (Start)

T(i,j) is also the smallest number k whose symmetric representation of sigma(k) has i subparts and j parts, or -1 if no such k exists.

Observations:

At least for i < 12 if i is prime then T(i,1) = 2*T(i,i).

At least for i < 12 if i is prime then all terms in row i are -1's except the first and the last term. (End)

All terms are even.

a(n) is the smallest number such that A069283(n) == n*(n-1)/2 == A142150(n) (mod n). Or equivalently, a(n) is the smallest number of the form 2^t*s, where s is an odd number with exactly n + 1 divisors and divisible by A000265(n), t = 0 for odd n and A007814(n) - 1 for even n.

Let n = 2^e_0*Product_{i=1..m} p_i^e_i where p_i are odd primes; n + 1 = Product_{j=1..s} q_j where q_j are primes, then a(n) != 0 iff there is an injection f from {1,2,..,m} to {1,2,...,s} such that q_f(i) >= e_i + 1 for all 1 <= i <= m, implying s >= m. If such an injection does exist, then the number of k having exactly n ways to be represented as sum of at least two consecutive positive integers and expressible as sum of n consecutive positive integers is finite iff s = m, in which case the number of k is equal to the number of injections such that if i < j and e_i = e_j then q_f(i) <= q_f(j).

If A038547(n) is divisible by A000265(n), then a(n) = 2^t*A038547(n), t defined as above.

If n + 1 is a Fermat prime, then a(n) = (n/2)*3^n. If n = p - 1 = 2^e*q with p, q primes, then a(n) = 2^(e-1)*q^n, which is relatively large.