Markus Sigg - cp's OEIS frontend

A383026 Triangle T(n,k) read by rows whose n-th row is the lexicographically first n-tuple of ordered distinct positive integers with sum A382547(n) and product A382547(n) * 100^(n-1), or an n-tuple of zeros when A382547(n) = 0.

1, 180, 225, 150, 175, 200, 125, 160, 175, 184, 125, 127, 150, 160, 200, 100, 125, 140, 150, 175, 192, 80, 100, 125, 150, 160, 173, 250, 80, 100, 110, 125, 140, 150, 200, 250, 50, 100, 112, 125, 150, 155, 160, 200, 250, 50, 80, 100, 125, 128, 150, 170, 175, 200, 250

Offset: 1

Markus Sigg, Apr 13 2025

Because A382547(n) > 0 for only finitely many n, the triangle has only finitely many nonzero rows.

			Triangle begins:
    1,
  180, 225,
  150, 175, 200,
  125, 160, 175, 184,
  125, 127, 150, 160, 200,
  100, 125, 140, 150, 175, 192,
   80, 100, 125, 150, 160, 173, 250,
   80, 100, 110, 125, 140, 150, 200, 250,
   50, 100, 112, 125, 150, 155, 160, 200, 250,
   50,  80, 100, 125, 128, 150, 170, 175, 200, 250,
   50,  65,  75, 100, 125, 128, 150, 175, 200, 250, 320,
   25,  50,  80, 100, 125, 128, 150, 200, 225, 230, 250, 300,
  ...
For n = 6 there are three 6-tuples with sum A382547(6) = 882 and product 100^5 * 882, namely (100, 125, 140, 150, 175, 192), (100, 125, 147, 150, 160, 200), (112, 120, 125, 150, 175, 200). The first of these is the lexicographically smallest and thus is row 6 of the triangle.

Markus Sigg, Table of n, a(n) for n = 1..231, rows 1..21, flattened.

Cf. A382547, A380887, A381187.

A382547 a(n) is the smallest positive integer s that can be partitioned into n distinct positive integers whose product is s * 100^(n-1), or 0 if no such s exists.

Original entry on oeis.org

1, 405, 525, 644, 762, 882, 1038, 1155, 1302, 1428, 1638, 1863, 2079, 2187, 2457, 2673, 3078, 3213, 3402, 3861, 4374, 5103, 5103, 6174

Offset: 1

Markus Sigg, Mar 31 2025

a(n) >= A380887(n) in case of a(n) > 0.
There are only finitely many positive a(n): If x_1 < ... < x_n are positive integers with the required properties, then x_k >= k, and (n-1)! * x_n <= x_1 * ... * x_n = 100^(n-1) * (x_1 + ... + x_n) <= 100^(n-1) * n * x_n gives (n-1)! <= 100^(n-1) * n, hence n <= 274. In fact, n <= 273 must hold, see Mathematics StackExchange link. A more elaborate argumentation in the same discussions shows n <= 269.
By restricting the search space, solution tuples have been found for 25 <= n <= 42. These tuples are not guaranteed to have the smallest possible sum and thus only give upper bounds for a(n). For example, the tuple (1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 20, 25, 30, 32, 40, 50, 60, 64, 75, 80, 100, 120, 125, 128, 150, 160, 200, 225, 250, 400, 625, 800, 1000, 1250, 2500, 3125, 5000, 6250, 12500, 78125, 1953125) shows a(42) <= 2066715.
Programs used for A380887 can be adapted for this sequence.

			a(2) = 405 because 180 + 225 = 405 and 180 * 225 = 405 * 100^1 and no positive integer smaller than 405 exists with the requested properties.

Mathematics StackExchange user Servaes, What is the maximum number of distinct prices whose product equals their sum?, answer dated May 29 2025.

Cf. A380887, A381187, A383026.

dfs(rs, rp, i, r, tp) = if(r==1, return(rs==rp&&setsearch(d, rs)>i)); if((rs/r)^r<=rp, return(0)); for(j=i+1, oo, if(tp>rp, return(0)); if(rp%d[j]==0, if(dfs(rs-d[j], rp/d[j], j, r-1, tp/d[j]), return(1))); tp*=d[j+r]/d[j]);
a(n) = if(n>1, my(p); for(s=100*n, oo, d=divisors(p=s*100^(n-1)); if(dfs(s, p, 0, n, prod(i=1, n, d[i])), return(s))), 1); \\ Jinyuan Wang, May 14 2025

a(17)-a(21) from Markus Sigg, Apr 21 2025

a(22)-a(24) from Jinyuan Wang, May 14 2025

A381619 Sorted list of sums of 3 prices in minor currency units for a currency that has a 2-decimal minor unit, such that the riddle "sum of prices equals product of prices" has a solution, with prices expressed as floating point numbers with 2 decimals.

Original entry on oeis.org

525, 540, 546, 549, 555, 561, 567, 570, 585, 588, 600, 612, 630, 642, 660, 660, 663, 675, 726, 735, 744, 750, 759, 765, 783, 792, 798, 810, 819, 825, 840, 840, 891, 897, 900, 930, 945, 957, 966, 966, 975, 981, 996, 1050, 1050, 1071, 1080, 1092, 1125, 1134, 1155, 1155, 1170

Offset: 1

Hugo Pfoertner and Markus Sigg, Mar 02 2025

The sequence has 622 terms. See linked files for all solutions.

A natural number s occurs k times in the list if there exist k multisets {x,y,z} of natural numbers with s = x + y + z and 10000*s = x*y*z.

			a(1) = 525 because 1.50 + 1.75 + 2.00 = 1.50*1.75*2.00 = 5.25 is the solution with minimum sum;
a(15) = a(16) = 660 because there are 2 solutions:
  0.80 + 2.50 + 3.30 = 0.80*2.50*3.30 = 6.60 and
  1.10 + 1.50 + 4.00 = 1.10*1.50*4.00 = 6.60;
a(31) = a(32) = 840:
  0.60 + 2.80 + 5.00 = 0.60*2.80*5.00 = 8.40 and
  1.00 + 1.40 + 6.00 = 1.00*1.40*6.00 = 8.40;
a(622) = 100030002 is the largest term:
  0.01 + 100.01 + 1000200.00 = 0.01*100.01*1000200.00 = 1000300.02.

Hugo Pfoertner, Table of n, a(n) for n = 1..622
Hugo Pfoertner, Solution triples sorted by sum. (2025)
Hugo Pfoertner, Solution triples sorted by smallest price. (2025)
Eric Snyder and others, Finding solutions of sum a_i = product a_i = n, where the a_i are "price rationals", question in Mathematics StackExchange, Jun 29, 2022.

Cf. A380887, A381187, A381620, A381621, A382510.

A381187 Triangle T(n,k) read by rows whose n-th row is the lexicographically first n-tuple of ordered positive integers with sum A380887(n) and product A380887(n) * 100^(n-1).

Original entry on oeis.org

1, 200, 200, 150, 175, 200, 125, 160, 175, 184, 125, 125, 160, 165, 184, 125, 125, 144, 150, 160, 160, 125, 125, 128, 144, 150, 150, 150, 110, 125, 125, 125, 128, 150, 150, 176, 125, 125, 125, 125, 128, 128, 132, 150, 150, 120, 120, 125, 125, 125, 125, 128, 128, 150, 150

Offset: 1

Markus Sigg, Feb 16 2025

			Triangle begins:
    1,
  200, 200,
  150, 175, 200,
  125, 160, 175, 184,
  125, 125, 160, 165, 184,
  125, 125, 144, 150, 160, 160,
  125, 125, 128, 144, 150, 150, 150,
  110, 125, 125, 125, 128, 150, 150, 176,
  125, 125, 125, 125, 128, 128, 132, 150, 150,
  120, 120, 125, 125, 125, 125, 128, 128, 150, 150,
  115, 122, 125, 125, 125, 125, 125, 125, 128, 128, 160,
  104, 125, 125, 125, 125, 125, 125, 125, 128, 128, 128, 145,
  ...
For n = 8 there are three 8-tuples with sum A380887(8) = 1089 and product 100^7 * 1089, namely (110, 125, 125, 125, 128, 150, 150, 176), (120, 125, 125, 125, 125, 128, 165, 176), (121, 125, 125, 125, 125, 128, 160, 180). The first of these is the lexicographically smallest and thus is row 8 of the triangle.

David A. Corneth, Table of n, a(n) for n = 1..45150 (first 528 terms from Hugo Pfoertner and Markus Sigg)

Cf. A380887, A382547, A383026.

A380887 a(n) is the smallest positive integer s that can be partitioned into n positive integers whose product is s * 100^(n-1).

Original entry on oeis.org

1, 400, 525, 644, 759, 864, 972, 1089, 1188, 1296, 1403, 1508, 1612, 1722, 1827, 1932, 2040, 2145, 2250, 2354, 2457, 2565, 2668, 2772, 2880, 2988, 3087, 3192, 3294, 3399, 3498, 3604, 3705, 3810, 3915, 4018, 4116, 4221, 4323, 4425, 4536, 4635, 4732, 4836, 4940

Offset: 1

Markus Sigg, Feb 07 2025

nonn

The AM-GM inequality shows a(n) >= 100 * n^(n/(n-1)). This bound gives a(2) >= 400, a(3) >= 520, a(4) >= 635, a(5) >= 748, a(6) >= 859.
a(n) is 100 times the smallest price r such that n prices exist whose sum and product both are equal to r. For example (see Guardian article link) 7.11 = 1.20 + 1.25 + 1.50 + 3.16 = 1.20 * 1.25 * 1.50 * 3.16.
Upper bounds for the next terms a(31)-a(40) are 3498, 3604, 3705, 3810, 3915, 4018, 4116, 4221, 4323, 4425. - Karl-Heinz Hofmann, Mar 26 2025
a(n) is well-defined: For n > 1, the sum of the numbers 1, ..., 1, k+1, k*(k+n-1), where the first n-2 numbers are 1 and k = 100^(n-1), is an example (possibly the largest one) of a positive integer s with the required properties. - Markus Sigg, Mar 30 2025
Better upper bounds can be given for specific values of n: Let s > 1 be an integer number and n = 2^s-s. Then the n-tuple of n-s times the number 100 and s times the number 200 has the required properties, hence a(2^s-s) <= 100*(n-s) + 200*s = 100 * 2^s. For s = 6, together with the lower bound from above, this gives 6229 <= a(58) <= 6400. - Markus Sigg, Apr 22 2025
For every n, the n-tuple (100, ..., 100, 10100, n + 99) has the required properties, hence a(n) <= 101*n + 9999. - Markus Sigg, May 30 2025

			a(2) = 400 because 200 + 200 = 400 and 200 * 200 = 400 * 100^1 and no positive integer smaller than 400 exists with the requested properties.
For a(3) the sum is 525 = 150 + 175 + 200.
For a(4) it is 644 = 125 + 160 + 175 + 184.
For a(5) it is 759 = 125 + 125 + 160 + 165 + 184.

David A. Corneth, Table of n, a(n) for n = 1..300
Alex Bellos, Can you solve it? Sexy maths, 2. The 7-Eleven, The Guardian, 3 Feb 2025.
David A. Corneth, PARI program
Markus Sigg, PARI program suitable for calculating a(n) for up to about n = 30.

Cf. A381187, A381619, A381621, A382547.

PARI
```
\\ See Sigg link
```
PARI
```
\\ See Corneth link
```

a(8)-a(9) from Hugo Pfoertner, Feb 13 2025
a(10)-a(12) from Hugo Pfoertner, Feb 16 2025
a(13) from Karl-Heinz Hofmann, Mar 02 2025
a(14)-a(30) from Markus Sigg, Mar 27 2025
a(31)-a(32) from Markus Sigg, Apr 23 2025
a(33)-a(45) from Jinyuan Wang, May 01 2025

A375206 T(n,k) for n >= 2, k < n is the distance of n and k in the Collatz graph, where T(n,k) is a triangle read by rows.

Original entry on oeis.org

1, 7, 6, 2, 1, 5, 5, 4, 2, 3, 8, 7, 1, 6, 3, 16, 15, 11, 14, 11, 12, 3, 2, 4, 1, 2, 5, 13, 19, 18, 14, 17, 14, 15, 3, 16, 6, 5, 1, 4, 1, 2, 10, 3, 13, 14, 13, 9, 12, 9, 10, 2, 11, 5, 8, 9, 8, 2, 7, 4, 1, 13, 6, 16, 3, 11, 9, 8, 4, 7, 4, 5, 7, 6, 10, 3, 5, 6, 17, 16, 12, 15, 12, 13, 1, 14, 2, 11, 3, 14, 8, 17, 16, 12, 15, 12, 13, 17, 14, 20, 11, 15, 14, 10, 18

Offset: 2

Markus Sigg, Oct 16 2024

The restriction k < n is there to avoid the trivial values T(n,n) = 0. Consequently, the first term is T(2,1).

			The triangle begins
  1,
  7, 6,
  2, 1, 5,
  5, 4, 2, 3,
  8, 7, 1, 6, 3,
  ...

Markus Sigg, Table of n, a(n) for n = 2..10012 (rows 2..142, flattened)
Hugo Pfoertner, Animated 3-d view of rows 2..200, view angle tilted by 15 degrees against the n-k plane, 2024.

Cf. A006370, A008908, A070165.

C(n) = {
    my(L = List([n]));
    while(n > 1, n = if(n % 2 == 0, n/2, 3*n+1); listput(L, n));
    L
};
a375206(n,k) = {
    my(Cn = C(n), Ck = C(k));
    for(i = 1, #Cn,
        for(j = 1, #Ck,
            if(Cn[i] == Ck[j], return(i+j-2))
        )
    )
};

A376079 a(n) is the largest difference of adjacent elements in the sorted list of the Collatz trajectory elements of n.

Original entry on oeis.org

1, 6, 2, 8, 6, 12, 4, 12, 6, 12, 4, 20, 12, 54, 8, 14, 12, 30, 6, 32, 12, 54, 8, 18, 14, 1944, 12, 36, 54, 1944, 16, 18, 12, 54, 12, 56, 30, 66, 20, 1944, 22, 48, 8, 68, 54, 1944, 24, 38, 18, 78, 14, 80, 1944, 1944, 12, 24, 30, 66, 54, 54, 1944, 1944, 32, 48, 12

Offset: 2

Markus Sigg, Sep 09 2024

Trajectories end when they reach 1.

			The trajectory of 3 is (3,10,5,16,8,4,2,1), the sorted list of the trajectory elements is (1,2,3,4,5,8,10,16), the list of differences is (1,1,1,1,3,2,6) with maximum 6, so a(3) = 6.

Markus Sigg, Table of n, a(n) for n = 2..10001

Cf. A220237 (sorted trajectory), A008908, A025586.

Table[Max[Differences[Sort[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]]]],{n,2,70}] (* Harvey P. Dale, Aug 24 2025 *)

a(n) = my(L = List([n])); while(n > 1, n = if(n % 2 == 0, n/2, 3*n + 1); listput(L, n)); listsort(L); vecmax(vector(#L - 1, i, L[i+1] - L[i]));

A375955 T(n,k) for n >= 1, k <= n is the maximum value in the intersection of the Collatz trajectories of n and k, where a trajectory ends when it reaches 1. T(n,k) is a triangle read by rows.

Original entry on oeis.org

1, 1, 2, 1, 2, 16, 1, 2, 4, 4, 1, 2, 16, 4, 16, 1, 2, 16, 4, 16, 16, 1, 2, 16, 4, 16, 16, 52, 1, 2, 8, 4, 8, 8, 8, 8, 1, 2, 16, 4, 16, 16, 52, 8, 52, 1, 2, 16, 4, 16, 16, 16, 8, 16, 16, 1, 2, 16, 4, 16, 16, 52, 8, 52, 16, 52, 1, 2, 16, 4, 16, 16, 16, 8, 16, 16, 16, 16

Offset: 1

Markus Sigg, Sep 03 2024

			The triangle begins:
       k=1  2   3  4   5   6    7  8
  n=1:   1;
  n=2:   1, 2;
  n=3:   1, 2, 16;
  n=4:   1, 2,  4, 4;
  n=5:   1, 2, 16, 4, 16;
  n=6:   1, 2, 16, 4, 16, 16;
  n=7:   1, 2, 16, 4, 16, 16, 52;
  n=8:   1, 2,  8, 4,  8,  8,  8, 8;
  ...
T(20,3) = 16 since the trajectory of 20 is (20,10,5,16,8,4,2,1), the trajectory of 3 is (3,10,5,16,8,4,2,1), and their intersection has the maximum 16. This example shows that T(n,k) does not necessarily denote the start of the common trajectory of n and k.

Markus Sigg, Table of n, a(n) for n = 1..11325 (row 1..150).

Cf. A025586 (main diagonal)

C(n) = my(L = List([n])); while(n > 1, n = if(n % 2 == 0, n/2, 3*n + 1); listput(L, n)); Set(L);
a375955_row(n) = my(Cn = C(n)); vector(n, k, vecmax(setintersect(Cn, C(k))));

T(n,n) = A025586(n).

A375937 Odd numbers which are the largest odd number in their Collatz trajectory.

Original entry on oeis.org

1, 5, 13, 17, 21, 29, 33, 37, 45, 49, 53, 61, 65, 69, 77, 81, 85, 93, 101, 113, 117, 133, 141, 149, 157, 173, 177, 181, 197, 205, 209, 213, 229, 237, 241, 245, 261, 269, 273, 277, 289, 301, 305, 309, 317, 321, 325, 341, 349, 357, 369, 373, 385, 397, 401, 405

Offset: 1

Markus Sigg, Sep 03 2024

nonn

a(n) == 1 (mod 4) because the trajectory of 4x+3 is (4x+3, 12x+10, 6x+5, ...) and 6x+5 > 4x+3.

			The odd elements of the Collatz trajectory (3,10,5,16,8,4,2,1) are {3,5,1} with maximum 5 > 3, so 3 is not a term. The odd elements of the Collatz trajectory (13,40,20,10,5,16,8,4,2,1) are {13,5,1} with maximum 13, so 13 is a term.

Markus Sigg, Table of n, a(n) for n = 1..10000

Cf. A033496, A176869.

makeEntries(count) = {
    my(L = List(), k = 1);
    while(#L < count,
        my(m = k);
        while(m > 1 && m <= k,
            m = 3*m + 1;
            while(m % 2 == 0, m = m / 2);
        );
        if(m == 1, listput(L, k));
        k += 2
    );
    L
};
print(Vec(makeEntries(56)));

a(n) = (A176869(n) - 1) / 3 for n > 1.

A375650 a(n) is the cardinality of the sumset of the Collatz trajectory of n.

Original entry on oeis.org

1, 3, 23, 6, 18, 24, 69, 10, 71, 22, 68, 25, 41, 69, 125, 15, 61, 73, 104, 28, 36, 68, 110, 33, 115, 48, 3060, 69, 95, 131, 2951, 21, 133, 67, 92, 76, 108, 108, 297, 37, 3007, 45, 203, 76, 105, 117, 2914, 45, 147, 119, 183, 57, 70, 3081, 3060, 82, 228, 102, 284

Offset: 1

Markus Sigg, Aug 24 2024

nonn

"Sumset" of a set S = {s_i} means the set of sums of pairs, s_i + s_j with i <= j.

			The Collatz trajectory of 3 is {3,10,5,16,8,4,2,1}, which has the sumset {2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,24,26,32} of size 23, so a(3) = 23.

Markus Sigg, Table of n, a(n) for n = 1..10000

A375006 is the list of those n for which a(n) < A008908(n) * (A008908(n) + 1) / 2.

a(n) = {
  my(T = List([n]), S = Set());
  while(n > 1, n = if(n % 2 == 0, n/2, 3*n+1); listput(T, n));
  for(i = 1, #T,
    for(j = i, #T,
      S = setunion(S, Set([T[i] + T[j]]));
    )
  );
  #S
};
print(vector(59, n, a(n)));

def a(n):
    T, S = [n], set()
    while n > 1:
        if n & 1 == 0: n >>= 1
        else: n = 3 * n + 1
        T.append(n)
    for i in range(len(T)):
        for j in range(i, len(T)):
            S.add(T[i] + T[j])
    return len(S)
print([a(n) for n in range(1, 60)]) # Darío Clavijo, Aug 24 2024

cp's OEIS Frontend

User: Markus Sigg

A383026 Triangle T(n,k) read by rows whose n-th row is the lexicographically first n-tuple of ordered distinct positive integers with sum A382547(n) and product A382547(n) * 100^(n-1), or an n-tuple of zeros when A382547(n) = 0.

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A382547 a(n) is the smallest positive integer s that can be partitioned into n distinct positive integers whose product is s * 100^(n-1), or 0 if no such s exists.

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A381619 Sorted list of sums of 3 prices in minor currency units for a currency that has a 2-decimal minor unit, such that the riddle "sum of prices equals product of prices" has a solution, with prices expressed as floating point numbers with 2 decimals.

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A381187 Triangle T(n,k) read by rows whose n-th row is the lexicographically first n-tuple of ordered positive integers with sum A380887(n) and product A380887(n) * 100^(n-1).

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A380887 a(n) is the smallest positive integer s that can be partitioned into n positive integers whose product is s * 100^(n-1).

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PARI

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A375206 T(n,k) for n >= 2, k < n is the distance of n and k in the Collatz graph, where T(n,k) is a triangle read by rows.

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A376079 a(n) is the largest difference of adjacent elements in the sorted list of the Collatz trajectory elements of n.

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Mathematica

PARI

A375955 T(n,k) for n >= 1, k <= n is the maximum value in the intersection of the Collatz trajectories of n and k, where a trajectory ends when it reaches 1. T(n,k) is a triangle read by rows.

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A375937 Odd numbers which are the largest odd number in their Collatz trajectory.