A388995 Square array A(n,k) = A017665(A389088(n,k)), read by falling antidiagonals; numerator of the abundancy index as applied onto special prime shift array A389088.

3, 7, 4, 2, 13, 6, 15, 8, 31, 8, 9, 40, 48, 57, 14, 7, 32, 156, 16, 183, 12, 12, 26, 84, 400, 168, 133, 18, 31, 56, 248, 96, 2380, 216, 307, 20, 13, 121, 72, 114, 252, 1464, 360, 381, 30, 21, 124, 781, 144, 2196, 240, 5220, 600, 871, 24, 18, 104, 342, 2801, 280, 2394, 540, 7240, 720, 553, 38, 5, 24, 434, 1464, 30941, 360, 6140, 480, 25260, 912, 1407, 32

Offset

1,1

Comments

Because A389088 is a permutation of natural numbers > 1, then, apart from 1/1, all possible abundancy ratios occur (possibly multiple times) in this array pair A388995/A388996, and no "abundancy outlaws" are present.

For any ratio A388995(n, k)/A388996(n, k), the ratio one step lower in the same column can be obtained as A388995(n+1, k)/A388996(n+1, k) = A388995(n, k)/A388996(n, k) * A389081(A389088(n, k))/A389082(A389088(n, k)).

Unlike in pair A341605 / A341606, here the ratio does not necessarily decrease monotonically when going down in each column.

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) = A389093(n, k) / A389094(n, k).

A(n, k) > A388996(n, k).

Example

The top left corner of the array:
k=  |  1    2    3      4     5      6     7       8      9     10    11      12
----+----------------------------------------------------------------------------
1   |  3,   7,   2,    15,    9,     7,   12,     31,    13,    21,   18,      5,
2   |  4,  13,   8,    40,   32,    26,   56,    121,   124,   104,   24,     16,
3   |  6,  31,  48,   156,   84,   248,   72,    781,   342,   434,   24,   1248,
4   |  8,  57,  16,   400,   96,   114,  144,   2801,  1464,   684,  240,    800,
5   | 14, 183, 168,  2380,  252,  2196,  280,  30941,  1862,  3294,  336,  28560,
6   | 12, 133, 216,  1464,  240,  2394,  360,  16105,  3684,   140,  456,  26352,
7   | 18, 307, 360,  5220,  540,  6140,  432,  88741,  6858,  9210,  576, 104400,
8   | 20, 381, 600,  7240,  480, 11430,   40, 137561, 17420,  9144,  840, 217200,
9   | 30, 871, 720, 25260, 1140, 20904,  960, 732541, 16590, 33098, 1320, 606240,
10  | 24, 553, 912, 12720,  768, 21014, 1008, 292561, 33768, 17696, 1296, 483360,

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); };
A017665(n) = { my(s=sigma(n)); (s/gcd(n, s)); };
A388995sq(row, col)  = A017665(A389088sq(row, col));
A388995list(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] = A388995sq(col, (a-(col-1))))); (v); };
v388995 = A388995list(up_to);
A388995(n) = v388995[n];

Crossrefs

Cf. A000203, A017665, A332818, A389081, A389082, A389088, A389093, A389094, A388996 (denominators).

Cf. also A341605.

Sequence in context: A367264 A266273 A341605 * A256676 A349010 A267412

Adjacent sequences: A388992 A388993 A388994 * A388996 A388997 A388998

Keyword

nonn,frac,tabl

Author

Antti Karttunen, Sep 24 2025

Status

approved