A391957 Irregular triangle read by rows: exponents of the primes in the prime factorization of A391956(n) (primes in increasing order).

3, 1, 5, 13, 3, 7, 1, 9, 12, 10, 7, 4, 6, 5, 13, 41, 2, 3, 11, 27, 32, 1, 9, 32, 24, 18, 7, 16, 16, 14, 5, 21, 46, 31, 10, 3, 19, 26, 21, 6, 1, 17, 25, 33, 11, 2, 26, 15, 23, 11, 30, 27, 22, 13, 21, 113, 18, 21, 18, 11, 19, 83, 6, 15, 14, 9, 17, 33, 88, 64, 9, 10, 7, 15, 31

Offset

2,1

Comments

All prime factors p_i of A391956(n) satisfy p_i <= n, so the row length is A000720(n).

All columns show a similar regular pattern, being built up by the concatenation of subsequences of a decreasing arithmetic progression of length p_m for the m-th column. See also A391958 (p_1 = 2) and A391959 (p_2 = 3). The first p_m - 1 subsequences start with s*(2*p_m - 1) for s = 1..(p_m - 1); the corresponding arithmetic decrease of these subsequences is 2*s.

Links

A.H.M. Smeets, Table of n, a(n) for n = 2..10388, row indices 2..300

A.H.M. Smeets, First 200 rows, printed row by row

Formula

Row(n) = A067255(A391956(n)). - Michel Marcus, Feb 05 2026

Example

The irregular triangle a(n,m) begins:
   \   m   1   2   3   4   5   6   7   8   9  10  11
   n\p_m   2   3   5   7  11  13  17  19  23  29  31 = A000040(m)
   0
   1
   2       3
   3       1   5
   4      13   3
   5       7   1   9
   6      12  10   7
   7       4   6   5  13
   8      41   2   3  11
   9      27  32   1   9
  10      32  24  18   7
  11      16  16  14   5  21
  12      46  31  10   3  19
  13      26  21   6   1  17  25
  14      33  11   2  26  15  23
  15      11  30  27  22  13  21
  16     113  18  21  18  11  19
  17      83   6  15  14   9  17  33
  18      88  64   9  10   7  15  31
  19      56  48   3   6   5  13  29  37
  20     102  32  36   2   3  11  27  35
  21      66  57  28  39   1   9  25  33
  22      73  39  20  33  42   7  23  31
  23      35  21  12  27  38   5  21  29  45
  24     138  50   4  21  34   3  19  27  43
  25      94  30  94  15  30   1  17  25  41
  26     101  10  82   9  26  50  15  23  39
  27      55 149  70   3  22  46  13  21  37
  28     119 123  58  52  18  42  11  19  35
  29      69  97  46  44  14  38   9  17  33  57
  30      78 130  93  36  10  34   7  15  31  55
  31      26 102  79  28   6  30   5  13  29  53  61
  32     289  74  65  20   2  26   3  11  27  51  59
  33     227 111  51  12  63  22   1   9  25  49  57
  34     232  81  37   4  57  18  66   7  23  47  55
  35     168  51  92  65  51  14  62   5  21  45  53

Prog

(Python)
from sympy import prime
def Sum(n, p):
    s = 0
    while n > 0: s, n = s+n, n//p
    return s
def PrimePower(p, e):
    if e == 0: return 0
    else: return p*PrimePower(p, e-1) + 2*p**e - 1
def A391957(n, m):
    p, d = prime(m), []
    while n > 0: d, n = [n%p]+d, n//p
    L = len(d)-1
    a, k, ns = d[0]*PrimePower(p, L), 0, 0
    while k < L:
        ns, k = ns*p+d[k], k+1
        if d[k] > 0: a = a + d[k]*(PrimePower(p, L-k)-2*p**(L-k)*Sum(ns, p))
    return a

Crossrefs

Cf. A000040, A000720, A067255, A391956.

Sequence in context: A073496 A176122 A370380 * A091623 A215474 A348114

Adjacent sequences: A391954 A391955 A391956 * A391958 A391959 A391960

Keyword

nonn,tabf

Author

A.H.M. Smeets, Dec 23 2025

Status

approved