A118235 -id:A118235 - cp's OEIS frontend

A379632 Irregular triangle read by rows in which row n lists the smallest parts of the partitions of n into consecutive parts (with the partitions ordered by increasing number of parts).

1, 2, 3, 1, 4, 5, 2, 6, 1, 7, 3, 8, 9, 4, 2, 10, 1, 11, 5, 12, 3, 13, 6, 14, 2, 15, 7, 4, 1, 16, 17, 8, 18, 5, 3, 19, 9, 20, 2, 21, 10, 6, 1, 22, 4, 23, 11, 24, 7, 25, 12, 3, 26, 5, 27, 13, 8, 2, 28, 1, 29, 14, 30, 9, 6, 4, 31, 15, 32, 33, 16, 10, 3, 34, 7, 35, 17, 5, 2, 36, 11, 1, 37, 18, 38, 8, 39, 19, 12, 4, 40, 6, 41, 20

Offset: 1

Omar E. Pol, Dec 31 2024

Row n gives the first A001227(n) terms of the n-th row of A379630.

			Triangle begins:
   1;
   2;
   3,  1;
   4;
   5,  2;
   6,  1;
   7,  3;
   8;
   9,  4,  2;
  10,  1;
  11,  5;
  12,  3;
  13,  6;
  14,  2;
  15,  7,  4,  1;
  16;
  17,  8;
  18,  5,  3;
  19,  9;
  20,  2;
  21, 10,  6,  1;
  22,  4;
  23, 11;
  24,  7;
  25, 12,  3;
  26,  5;
  27, 13,  8,  2;
  28,  1;
  ...
Illustration of initial terms:
                                                         _
                                                       _|1|
                                                     _|2 _|
                                                   _|3  |1|
                                                 _|4   _| |
                                               _|5    |2 _|
                                             _|6     _| |1|
                                           _|7      |3  | |
                                         _|8       _|  _| |
                                       _|9        |4  |2 _|
                                     _|10        _|   | |1|
                                   _|11         |5   _| | |
                                 _|12          _|   |3  | |
                               _|13           |6    |  _| |
                             _|14            _|    _| |2 _|
                           _|15             |7    |4  | |1|
                         _|16              _|     |   | | |
                       _|17               |8     _|  _| | |
                     _|18                _|     |5  |3  | |
                   _|19                 |9      |   |  _| |
                 _|20                  _|      _|   | |2 _|
               _|21                   |10     |6   _| | |1|
             _|22                    _|       |   |4  | | |
           _|23                     |11      _|   |   | | |
         _|24                      _|       |7    |  _| | |
       _|25                       |12       |    _| |3  | |
     _|26                        _|        _|   |5  |  _| |
   _|27                         |13       |8    |   | |2 _|
  |28                           |         |     |   | | |1|
  ...
The diagram is also the left part of the diagram of A379630.
The geometrical structure is the same as the diagram of A237591.

Positive terms of A211343.
Absolute values of A341971.
Column 1 gives A000027.
Right border gives A118235.
Row lengths give A001227.
Row sums give A286014.
Subsequence of A286001 and of A299765 and of A379630.
For the largest parts see A379633.
Cf. A196020, A235791, A236104, A237048, A237591, A237593, A245092, A262626, A379631, A379634.

A218621 a(n) = unique divisor d of n such that d + (n/d - 1)/2 is minimal and integral.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 1, 8, 3, 2, 1, 4, 1, 2, 3, 16, 1, 2, 1, 4, 3, 2, 1, 8, 5, 2, 3, 4, 1, 6, 1, 32, 3, 2, 5, 4, 1, 2, 3, 8, 1, 6, 1, 4, 5, 2, 1, 16, 7, 10, 3, 4, 1, 6, 5, 8, 3, 2, 1, 4, 1, 2, 7, 64, 5, 6, 1, 4, 3, 10, 1, 8, 1, 2, 5, 4, 7, 6, 1, 16, 9, 2, 1, 4, 5

Offset: 1

L. Edson Jeffery, Feb 18 2013

nonn

Differs from A079891 starting at a(18).
For integers M, k, with 0<=k<=M, consider a representation of n as n = T(M) - T(M-k) = M + (M-1) + ... + (M-k+1), in which k is maximal, where T(r) = r*(r+1)/2 is the r-th triangular number. Then k = A109814(n), and M = A212652(n) = a(n) + (n/a(n) - 1)/2 is minimal.
Conjecture. For n, p, v, j natural numbers, the conditions on a(n) seem to be the following:
1. If n is an odd prime, then a(n) = 1.
2. If n is odd and composite, then
     a(n) = max(p : p | n, p <= sqrt(n), p is a prime).
3. If n is equal to a power of 2, then a(n) = n.
4. If n = 2^j*v, with v odd, v>1 and j>1, then a(n) = 2^j.
5. If n = 2*v, with v odd and composite, then
     a(n) = 2*p, where p is the least prime such that p | v.
6. If n = 2*p, for p an odd prime, then a(n) = 2.

T. D. Noe, Table of n, a(n) for n = 1..2048

Cf. A000217, A109814, A118235, A141419, A209260, A212652.

Table[d = Divisors[n]; mn = Infinity; best = 0; Do[q = i + (n/i - 1)/2; If[IntegerQ[q] && q < mn, mn = q; best = i], {i, d}]; best, {n, 100}] (* T. D. Noe, Feb 21 2013 *)

A345708 a(n) is the least positive number starting an interval of consecutive integers whose product of elements is n.

Original entry on oeis.org

1, 1, 3, 4, 5, 1, 7, 8, 9, 10, 11, 3, 13, 14, 15, 16, 17, 18, 19, 4, 21, 22, 23, 1, 25, 26, 27, 28, 29, 5, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 6, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 7, 57, 58, 59, 3, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 1

Rémy Sigrist, Jun 24 2021

nonn

This sequence is similar to A118235; here we multiply, there we add.
a(n) is the index of the first row of A068424 (interpreted as a square array) containing n.
If n is the product of k consecutive integers, then k! divides n.

			The square array A068424(n, k) begins:
  n\k|   1    2     3      4       5        6
  ---+---------------------------------------
    1|   1    2     6     24     120      720
    2|   2    6    24    120     720     5040
    3|   3   12    60    360    2520    20160
    4|   4   20   120    840    6720    60480
- so a(1) = a(2) = a(6) = a(24) = a(120) = a(720) = 1,
     a(3) = a(12) = a(60) = a(360) = 3,
     a(4) = a(20) = 4.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A000142, A001710, A045619, A068424, A118235, A246655.

a(n) = { fordiv (n, d, my (r=n); for (k=d, oo, if (r==1, return (d), r%k, break, r/=k))) }

a(n) = { for (i=2, oo, if (n%i!, forstep (j=i-1, 2, -1, my (d=sqrtnint(n,j)); while (d-1 && n%(d-1)==0, d--); while (n%d==0, my (r=n); for
(k=d, oo, if (r==1, return (if (d==2, 1, d)), r%k, break, r/=k)); d++)); break)); return (n) }

from sympy import divisors
def a(n):
    if n%2 == 0: return n
    divs = divisors(n)
    for i, d in enumerate(divs[:len(divs)//2]):
        p = lastj = d
        for j in divs[i+1:]:
            if p >= n or j - lastj > 1: break
            p, lastj = p*j, j
        if p == n: return d
    return n
print([a(n) for n in range(1, 70)]) # Michael S. Branicky, Jun 29 2021

a(n) = 1 iff n is a factorial number (A000142).
a(n) <> 2.
a(n) = 3 iff n >= 3 and n belongs to A001710.
a(n) <= n.
a(p! / (n-1)!) = n for any n >= 3 and any prime number p >= n.
a(q) = q for any prime power q > 2.
a(n) = n for any odd number n.
a(n) < n iff n belongs to A045619.

cp's OEIS Frontend

A379632 Irregular triangle read by rows in which row n lists the smallest parts of the partitions of n into consecutive parts (with the partitions ordered by increasing number of parts).

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A218621 a(n) = unique divisor d of n such that d + (n/d - 1)/2 is minimal and integral.

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A345708 a(n) is the least positive number starting an interval of consecutive integers whose product of elements is n.

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