A382747 -id:A382747 - cp's OEIS frontend

A382748 Primitive exponents for the greedy convolution of length 4.

1, 5, 6, 7, 8, 9, 11, 13, 17, 19, 23, 25, 27, 29, 30, 31, 32, 35, 36, 37, 40, 41, 42, 43, 45, 47, 48, 49, 53, 55, 56, 59, 61, 63, 65, 66, 67, 71, 73, 77, 78, 79, 83, 85, 88, 89, 91, 95, 97, 99, 101, 102, 103, 104, 107, 109, 113, 114, 115, 117, 119, 121, 125, 127, 131, 133, 135, 136, 137, 138, 139

Offset: 1

Jan Snellman, Apr 24 2025

Integers n such that p^n is primitive for the "greedy convolution of length 4" when p is a prime number.
Smallest elements in the blocks of the greedy partition of the positive integers into parts of length at most 4.
First column of A382747.
Strict definition:
Form an infinite rooted tree T on the nonnegative integers in the following way.
1. 0 is the root
2. Form a branch 0 - 1 - 2 -3 - 4
3. Proceed inductively. Add n to end of an existing branch as either
  0 - k=n
  0 - k - 2k=n
  0 - k - 2k - 3k=n
  0 - k - 2k - 3k - 4k=n
with a preference for smaller k.
The primitive elements are the integers at distance one from the root.

			Up to n=15 the branches of the aforementioned tree looks like
  0 - 1 - 2 - 3 - 4
  0 - 5 - 10 - 15
  0 - 6 - 12
  0 - 7 - 14
  0 - 8
  0 - 9
  0 - 11
  0 - 13
so the primitive elements <= 15 are 1, 5, 6, 7, 8, 9, 11, 13.

Jan Snellman, Table of n, a(n) for n = 1..2598
W. Narkiewicz, On a class of arithmetical convolutions, Colloq. Math. 10, 1963, pp 81--94.
Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.

First column of A382747.

Cf. A121537.

SageMath
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# See A382747.
```

A382749 Position in block for greedy partition of length 4.

Original entry on oeis.org

1, 2, 3, 4, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 4, 3, 2, 1, 4, 1, 2, 1, 4, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 2, 3, 4, 1, 2, 1, 1, 3, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 3, 2, 1, 2, 1, 2, 3, 4, 1, 1, 1, 2, 3, 2, 1, 2, 1, 2, 3, 1, 1, 3, 1, 4, 3, 2, 1, 3, 1, 2, 1, 4

Offset: 1

Jan Snellman, Apr 24 2025

nonn

a(n) = height of n with respect to the greedy convolution of length 4.
Position of n i block containing it for the greedy partition of length 4 of the positive integers.
a(n) = the column (one-indexed) in which n occurs in the infinite matrix A382747.

Jan Snellman, Table of n, a(n) for n = 1..5000
W. Narkiewicz, On a class of arithmetical convolutions, Colloq. Math. 10, 1963, pp 81--94.
Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.

a(n) = column index of the unique position in the infinite matrix A382747 in which n occurs.

a(n) = 1 iff n is "primitive" with respect to the greedy convolution of length 4 i.e. if n occurs in A382748.

cp's OEIS Frontend

A382748 Primitive exponents for the greedy convolution of length 4.

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