A391957 - OEIS

%I #24 Feb 05 2026 09:25:56

%S 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,

%T 16,14,5,21,46,31,10,3,19,26,21,6,1,17,25,33,11,2,26,15,23,11,30,27,

%U 22,13,21,113,18,21,18,11,19,83,6,15,14,9,17,33,88,64,9,10,7,15,31

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

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

%C 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.

%H A.H.M. Smeets, <a href="/A391957/b391957.txt">Table of n, a(n) for n = 2..10388, row indices 2..300</a>

%H A.H.M. Smeets, <a href="/A391957/a391957.txt">First 200 rows, printed row by row</a>

%F Row(n) = A067255(A391956(n)). - _Michel Marcus_, Feb 05 2026

%e The irregular triangle a(n,m) begins:

%e    \   m   1   2   3   4   5   6   7   8   9  10  11

%e    n\p_m   2   3   5   7  11  13  17  19  23  29  31 = A000040(m)

%e    0

%e    1

%e    2       3

%e    3       1   5

%e    4      13   3

%e    5       7   1   9

%e    6      12  10   7

%e    7       4   6   5  13

%e    8      41   2   3  11

%e    9      27  32   1   9

%e   10      32  24  18   7

%e   11      16  16  14   5  21

%e   12      46  31  10   3  19

%e   13      26  21   6   1  17  25

%e   14      33  11   2  26  15  23

%e   15      11  30  27  22  13  21

%e   16     113  18  21  18  11  19

%e   17      83   6  15  14   9  17  33

%e   18      88  64   9  10   7  15  31

%e   19      56  48   3   6   5  13  29  37

%e   20     102  32  36   2   3  11  27  35

%e   21      66  57  28  39   1   9  25  33

%e   22      73  39  20  33  42   7  23  31

%e   23      35  21  12  27  38   5  21  29  45

%e   24     138  50   4  21  34   3  19  27  43

%e   25      94  30  94  15  30   1  17  25  41

%e   26     101  10  82   9  26  50  15  23  39

%e   27      55 149  70   3  22  46  13  21  37

%e   28     119 123  58  52  18  42  11  19  35

%e   29      69  97  46  44  14  38   9  17  33  57

%e   30      78 130  93  36  10  34   7  15  31  55

%e   31      26 102  79  28   6  30   5  13  29  53  61

%e   32     289  74  65  20   2  26   3  11  27  51  59

%e   33     227 111  51  12  63  22   1   9  25  49  57

%e   34     232  81  37   4  57  18  66   7  23  47  55

%e   35     168  51  92  65  51  14  62   5  21  45  53

%o (Python)

%o from sympy import prime

%o def Sum(n,p):

%o     s = 0

%o     while n > 0: s, n = s+n, n//p

%o     return s

%o def PrimePower(p,e):

%o     if e == 0: return 0

%o     else: return p*PrimePower(p,e-1) + 2*p**e - 1

%o def A391957(n,m):

%o     p, d = prime(m), []

%o     while n > 0: d, n = [n%p]+d, n//p

%o     L = len(d)-1

%o     a, k, ns = d[0]*PrimePower(p,L), 0, 0

%o     while k < L:

%o         ns, k = ns*p+d[k], k+1

%o         if d[k] > 0: a = a + d[k]*(PrimePower(p,L-k)-2*p**(L-k)*Sum(ns,p))

%o     return a

%Y Cf. A000040, A000720, A067255, A391956.

%K nonn,tabf

%O 2,1

%A _A.H.M. Smeets_, Dec 23 2025