A388996 Square array A(n,k) = A017666(A389088(n,k)), read by falling antidiagonals; denominator of the abundancy index as applied onto special prime shift array A389088.

2, 4, 3, 1, 9, 5, 8, 5, 25, 7, 5, 27, 35, 49, 13, 3, 21, 125, 13, 169, 11, 7, 15, 65, 343, 143, 121, 17, 16, 39, 175, 77, 2197, 187, 289, 19, 6, 81, 55, 91, 221, 1331, 323, 361, 29, 10, 75, 625, 119, 1859, 209, 4913, 551, 841, 23, 11, 63, 245, 2401, 247, 2057, 493, 6859, 667, 529, 37, 2, 17, 325, 1183, 28561, 319, 5491, 437, 24389, 851, 1369, 31

Offset

1,1

Links

Antti Karttunen, Table of n, a(n) for n = 1..22155; the first 210 antidiagonals of the array

Index entries for sequences computed from indices in prime factorization.

Index entries for sequences related to sigma(n).

Formula

A(n, k) = A389088(n, k) / A389094(n, k).

Example

The top left corner of the array:
k=  |  1    2    3      4     5      6     7       8      9     10    11      12
----+----------------------------------------------------------------------------
1   |  2,   4,   1,     8,    5,     3,    7,     16,     6,    10,   11,      2,
2   |  3,   9,   5,    27,   21,    15,   39,     81,    75,    63,   17,      9,
3   |  5,  25,  35,   125,   65,   175,   55,    625,   245,   325,   19,    875,
4   |  7,  49,  13,   343,   77,    91,  119,   2401,  1183,   539,  203,    637,
5   | 13, 169, 143,  2197,  221,  1859,  247,  28561,  1573,  2873,  299,  24167,
6   | 11, 121, 187,  1331,  209,  2057,  319,  14641,  3179,   121,  407,  22627,
7   | 17, 289, 323,  4913,  493,  5491,  391,  83521,  6137,  8381,  527,  93347,
8   | 19, 361, 551,  6859,  437, 10469,   37, 130321, 15979,  8303,  779, 198911,
9   | 29, 841, 667, 24389, 1073, 19343,  899, 707281, 15341, 31117, 1247, 560947,
10  | 23, 529, 851, 12167,  713, 19573,  943, 279841, 31487, 16399, 1219, 450179,

Prog

(PARI)
up_to = 22155; \\ = binomial(210+1, 2)
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f);
A108546list(up_to) = { my(v=vector(up_to), p, q); v[1] = 2; v[2] = 3; v[3] = 5; for(n=4, up_to, p = v[n-2]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[n] = q); (v); };
v108546 = A108546list(up_to);
A108546(n) = v108546[n];
A108548(n) = { my(f=factor(n)); f[, 1] = apply(A108546, apply(primepi, f[, 1])); factorback(f); };
A332806list(up_to) = { my(v=vector(2), xs=Map(), lista=List([]), p, q, u); v[2] = 3; v[1] = 5; mapput(xs, 1, 1); mapput(xs, 2, 2); mapput(xs, 3, 3);  for(n=4, up_to, p = v[2-(n%2)]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[2-(n%2)] = q; mapput(xs, primepi(q), n)); for(i=1, oo, if(!mapisdefined(xs, i, &u), return(Vec(lista)), listput(lista, prime(u)))); };
v332806 = A332806list(up_to);
A332806(n) = v332806[n];
A332808(n) = { my(f=factor(n)); f[, 1] = apply(A332806, apply(primepi, f[, 1])); factorback(f); };
A332818(n) = A108548(A003961(A332808(n)));
A389088sq(row, col) = { my(x=2*col); for(i=2, row, x = A332818(x)); (x); };
A017666(n) = (n/gcd(n, sigma(n)));
A388996sq(row, col)  = A017666(A389088sq(row, col));
A388996list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, for(col=1, a, i++; if(i > up_to, return(v)); v[i] = A388996sq(col, (a-(col-1))))); (v); };
v388996 = A388996list(up_to);
A388996(n) = v388996[n];

Crossrefs

Cf. A000203, A017666, A332818, A389088, A389093, A389094, A388995 (numerators).

Cf. also A341606.

Sequence in context: A322398 A395599 A341606 * A109158 A307500 A049245

Adjacent sequences: A388993 A388994 A388995 * A388997 A388998 A388999

Keyword

nonn,frac,tabl

Author

Antti Karttunen, Sep 24 2025

Status

approved