Jan Snellman - cp's OEIS frontend

A382751 Numbers k for which the 3-adic valuation A007949(k) == 0 (mod 3).

1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 54, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95

Offset: 1

Jan Snellman, May 12 2025

Positive integers k for which the number of trailing 0 digits, which written in ternary, is a multiple of 3.
"Selective sifting" of the positive integers w.r.t. S={3,9}, where s(S) = {positive integers n: n cannot be written n = a*b with a in S, b in s(S), b < n}.
In other words, s(S) is determined by the fact that {s(S), S*s(S)} is a partition of the positive integers.
The asymptotic density of this sequence is 9/13. - Amiram Eldar, Jul 23 2025

			7 is a term since its 3-adic valuation is A007949(7) = 0 which is == 0 (mod 3).

Jan Snellman, Table of n, a(n) for n = 1..6923
Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.

Cf. A007949, A007417.

Select[Range[100], Divisible[IntegerExponent[#, 3], 3] &] (* Amiram Eldar, May 13 2025 *)

isok(k) = (valuation(k, 3) % 3) == 0; \\ Michel Marcus, Jun 03 2025

A382750 If k appears, 9*k does not.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Jan Snellman, May 09 2025

Integers k with val(k, 9) even, where val(k, 9) is the 9-adic valuation of k.
Natural density 9/10.
Differs from A168183: 81 for example is not in A168183 but in this sequence. - R. J. Mathar, May 26 2025

			18 = 9*2 is not a term because 2 is a term.
162 = 9*18 is a term since 18 is not a term.

Jan Snellman, Table of n, a(n) for n = 1..9000
Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.

Cf. A003159, A007417, A382744, A382745, A382746.

Select[Range[100], Mod[IntegerExponent[#, 3], 4] < 2 &] (* Amiram Eldar, May 12 2025 *)

from sympy import integer_log
def A382750(n):
    def f(x): return n+x-sum((k:=x//9**m)-k//9 for m in range(0,integer_log(x,9)[0]+1,2))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, May 24 2025

[ for  in range(1,100+1) if (valuation(_,3) % 4) < 2 ]

A382747 Greedy partition of the positive integers into arithmetic progressions of length at most 4.

Original entry on oeis.org

1, 2, 3, 4, 5, 10, 15, 20, 6, 12, 18, 24, 7, 14, 21, 28, 8, 16, 0, 0, 9, 0, 0, 0, 11, 22, 33, 44, 13, 26, 39, 52, 17, 34, 51, 68, 19, 38, 57, 76, 23, 46, 69, 92, 25, 50, 75, 100, 27, 54, 81, 108, 29, 58, 87, 116, 30, 60, 90, 120, 31, 62, 93, 124, 32, 64, 96, 128, 35, 70, 105, 140, 36, 72, 0, 0, 37, 74, 111, 148, 40, 80, 0, 0

Offset: 1

Jan Snellman, Apr 23 2025

Table by rows, rows have length 4.
Elements are a permutation of positive integers, intermixed with zeros.
Expanded description: Partition the positive integers into arithmetic progressions of the form k, 2k, 3k, 4k, putting every positive integer into the first progressions where it fits, allowing shortened progressions (which are padded with zeros):
  k, 0, 0, 0;
  k, 2k, 0, 0;
  k, 2k, 3k, 0.
Construction:
  1. Start by matrix M with rows indexed by positive integers, columns by 1,2,3,4.
  2. M_ij = i*j.
  3. Proceeding row by row, then column by column, if M_ij = M_rk with r < i, set M_is = 0 for i <= s <= 4; if j=0, remove entire row.
  4. Call the resulting matrix A.
So starting with
M = [ 1  2  3  4]
    [ 2  4  6  8]
    [ 3  6  9 12]
    [ 4  8 12 16]
    [ 5 10 15 20]
    [ 6 12 18 24]
    [ 7 14 21 28]
    [ 8 16 24 32]
    [ 9 18 27 36]
    [10 20 30 40]
    [11 22 33 44]
    [12 24 36 48]
    [13 26 39 52]
    [14 28 42 56]
    [15 30 45 60]
    [16 32 48 64]
    [17 34 51 68]
    [18 36 54 72]
    [19 38 57 76]
    [20 40 60 80]
we arrive at
A = [  1   2   3   4]
    [  5  10  15  20]
    [  6  12  18  24]
    [  7  14  21  28]
    [  8  16   0   0]
    [  9   0   0   0]
    [ 11  22  33  44]
    [ 13  26  39  52]
    [ 17  34  51  68]
    [ 19  38  57  76]
Every row in A is an arithmetic progression (even when 0 is adjoined), and every positive integer occurs at precisely one position.
Thus we get a partition of the positive integers into parts which are arithmetic progressions; using this partition for every prime number p yields a regular (in the sense of Narkiewicz) convolution product on the vector space of arithmetic functions.
The first column of A contains those b such that p^b is primitive.
Alternative construction:
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.
  4. After infinitely many steps you have constructed T.
  5. Read the positive integers branch by branch.

			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;
but since 20=5*4 the second branch may not be complete, so at this stage we only know the first row of the matrix A. Adding 16, 17, 18, 19, 20 we get
  0 -  1 -  2 -  3 -  4,
  0 -  5 - 10 - 15 - 20,
  0 -  6 - 12 - 18,
  0 -  7 - 14,
  0 -  8 - 16,
  0 -  9,
  0 - 11,
  0 - 13,
  0 - 17,
  0 - 19;
and now we know the first two rows of A.

Jan Snellman, Table of n, a(n) for n = 1..20788
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 yields A382748.
A382749(n) = column where n occurs in this matrix.
The case length=2 is A036552, when the latter is interpreted as a matrix with two columns.

def A382747_generator(blocklength=4):
    a_set = set()
    a0 = 1
    while 1:
        while a0 in a_set:
            a_set.remove(a0)
            a0 += 1
        for i in range(1,blocklength+1):
            a = i*a0
            if i != 1 and a in a_set:
                for j in range(blocklength-i+1): yield 0
                break
            yield a
            a_set.add(a) # Pontus von Brömssen, Apr 30 2025

def greedy_matrix(blocklength=2,initial_cols=20):
    m, n = blocklength, initial_cols
    A = matrix(ZZ, m,n, lambda i,j: (i+1)*(j+1))
    for c in range(2, n+1):
        for r in range(1, m+1):
            prev = set(flatten([list() for  in A.columns()[:(c-1)]]))
            v = A[r-1, c-1]
            if v in prev:
                for j in range(r, m+1):
                    A[j-1,c-1] = 0
                break
    return A
def pruned_greedy_matrix(blocklength=2, initial_cols=20):
    A = greedy_matrix(blocklength=blocklength, initial_cols=initial_cols)
    return matrix([ for  in A.columns() if add(_) > 0]).transpose()
pruned_greedy_matrix(blocklength=4, initial_cols=20)

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.

A382748 Primitive exponents for the greedy convolution of length 4.

Original entry on oeis.org

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
```
# See A382747.
```

A382746 If k appears, 8*k does not.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Jan Snellman, Apr 04 2025

Also: integers of the form 2^m*r, r odd, m congruent to 0, 1, 2 mod 6.

The asymptotic density of this sequence is 8/9. - Amiram Eldar, May 31 2025

			8, 16, ... , 56 are removed, but 8*8 = 64 remains.

Jan Snellman, Table of n, a(n) for n = 1..8889
Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.

Cf. A003159, A007417, A382744, A382745.

Select[Range[100], Mod[IntegerExponent[#, 2], 6] < 3 &] (* Amiram Eldar, Apr 04 2025 *)

isok(k) = valuation(k, 2) % 6 < 3; \\ Amiram Eldar, May 31 2025

from functools import lru_cache
@lru_cache(maxsize=None, typed=True)
def in_sieve(n, S):
    if n == 1:
        return True
    elif n in S:
        return False
    else:
        L = [s for s in S if (n % s) == 0]
        return all(not in_sieve(n/ell, S) for ell in L )
def nth_in_sieve(n, S):
    if n == 1:
        return 1
    else:
        i, m = 1, 1
        while i < n:
            m = m+1
            if in_sieve(m, S):
                i = i+1
    return m
def a(n):
    return nth_in_sieve(n, tuple([8]))

def A382746(n):
    def f(x): return n+x-sum((x>>m)+1>>1 for m in range(x.bit_length()+1) if m%6<3)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Apr 10 2025

[ for  in range(1,100) if (valuation(_,2) % 6) < 3]

A382744 If k appears, 5*k does not.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84

Offset: 1

Jan Snellman, Apr 04 2025

nonn

Also: numbers with an even number of 5's in their prime factorization.

Natural density 5/6.

			5 is removed since 5 = 5*1, 10 is removed, 15 is removed, 20 is removed, but 25 remains.

Jan Snellman, Table of n, a(n) for n = 1..8333
Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.

Cf. A003159, A007417, A382745, A382746.

select(t -> padic:-ordp(t,5)::even, [$1..100]); # Robert Israel, Apr 04 2025

Select[Range[100], EvenQ[IntegerExponent[#, 5]] &] (* Amiram Eldar, Apr 04 2025 *)

def ok(n):
    c = 0
    while n and n%5 == 0: n //= 5; c += 1
    return c&1 == 0
print([k for k in range(1, 82) if ok(k)]) # Michael S. Branicky, Apr 04 2025

from sympy import integer_log
def A382744(n):
    def f(x): return n+x-sum((k:=x//5**m)-k//5 for m in range(0,integer_log(x,5)[0]+1,2))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Apr 10 2025

[ for  in range(1,100) if (valuation(_,5) % 2) == 0]

a(n) ~ (6/5)*n.

A382745 If k appears, 7*k does not.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 85

Offset: 1

Jan Snellman, Apr 04 2025

Also numbers with an even number of 7's in their prime factorization.

Natural density 7/8.

			7 is removed since 7 = 7*1, 14, 21, 28, 35, 42 are removed, but 49 remains.

Jan Snellman, Table of n, a(n) for n = 1..8751
Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.

Cf. A003159, A007417, A382744, A382746.

q:= n-> is(irem(padic[ordp](n,7), 2)=0):
select(q, [$1..85])[];  # Alois P. Heinz, Apr 04 2025

Select[Range[100], EvenQ[IntegerExponent[#, 7]] &] (* Amiram Eldar, Apr 04 2025 *)

def ok(n):
    c = 0
    while n and n%7 == 0: n //= 7; c += 1
    return c&1 == 0
print([k for k in range(1, 86) if ok(k)]) # Michael S. Branicky, Apr 04 2025

from sympy import integer_log
def A382745(n):
    def f(x): return n+x-sum((k:=x//7**m)-k//7 for m in range(0,integer_log(x,7)[0]+1,2))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Apr 10 2025

a(n) ~ (8/7)*n.

A086159 Number of partitions of n into the first three triangular numbers, 1, 3 and 6.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 4, 4, 4, 6, 6, 6, 9, 9, 9, 12, 12, 12, 16, 16, 16, 20, 20, 20, 25, 25, 25, 30, 30, 30, 36, 36, 36, 42, 42, 42, 49, 49, 49, 56, 56, 56, 64, 64, 64, 72, 72, 72, 81, 81, 81, 90, 90, 90, 100, 100, 100, 110, 110, 110, 121, 121, 121, 132, 132, 132

Offset: 0

Jan Snellman, Aug 25 2003

nonn

Jan Snellman and Michael Paulsen, Enumeration of Concave Integer Partitions, J. Integer Seq., Vol. 7 (2004), Article 04.1.3.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,0,1,-1,0,-1,1).

Cf. A000217, A008620.

LinearRecurrence[{1, 0, 1, -1, 0, 1, -1, 0, -1, 1}, {1, 1, 1, 2, 2, 2, 4, 4, 4, 6}, 100] (* Amiram Eldar, Feb 14 2023 *)

G.f.: 1/((1-x)*(1-x^3)*(1-x^6)).

Sum_{n>=0} 1/a(n) = Pi^2/2 + 3. - Amiram Eldar, Feb 14 2023

A084913 Number of monomial ideals in two variables that are Artinian, integrally closed and of colength n.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 11, 17, 23, 28, 39, 48, 59, 79, 100, 121, 152, 185, 225, 280, 338, 404, 492, 584, 696, 835, 983, 1162, 1385, 1612

Offset: 0

Jan Snellman and Michael Paulsen, Jul 03 2003

Alternatively, "concave partitions" of n, where a concave partition is defined by demanding that the monomial ideal, generated by the monomials whose exponents do not lie in the Ferrers diagram of the partition, is integrally closed.

			a(4) = 4 because the Artinian monomial ideals in two variables that have colength 4 are (x^4,y), (x^3,y^2), (x^2, y^2), (x^2,xy,y^3), (x,y^4), corresponding to the partitions (1,1,1,1), (3,1), (2,2), (2,1,1), (4); the ideal (x^2,y^2) is not integrally closed, hence the partition (2,2) is not concave.

G. E. Andrews, The Theory of Partitions, Addison-Wesley Publishing Company, 1976.
M. Paulsen & J. Snellman, Enumerativa egenskaper hos konkava partitioner (in Swedish), Department of Mathematics, Stockholm University.

V. Crispin Quinonez, Integrally closed monomial ideals and powers of ideals, Research Reports in Mathematics Number 7 2002, Department of Mathematics, Stockholm University.

Cf. A086161, A086162, A086163.

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User: Jan Snellman

A382751 Numbers k for which the 3-adic valuation A007949(k) == 0 (mod 3).

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A382750 If k appears, 9*k does not.

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A382747 Greedy partition of the positive integers into arithmetic progressions of length at most 4.

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A382749 Position in block for greedy partition of length 4.

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A382748 Primitive exponents for the greedy convolution of length 4.

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A382746 If k appears, 8*k does not.

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A382744 If k appears, 5*k does not.

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