A389088 Prime shift array for A332818: Square array read by antidiagonals: A(1,col) = 2*col, and for row > 1, A(row,col) = A332818(A(row-1,col)).

2, 4, 3, 6, 9, 5, 8, 15, 25, 7, 10, 27, 35, 49, 13, 12, 21, 125, 91, 169, 11, 14, 45, 65, 343, 143, 121, 17, 16, 39, 175, 77, 2197, 187, 289, 19, 18, 81, 55, 637, 221, 1331, 323, 361, 29, 20, 75, 625, 119, 1859, 209, 4913, 551, 841, 23, 22, 63, 245, 2401, 247, 2057, 493, 6859, 667, 529, 37, 24, 51, 325, 1183, 28561, 319, 5491, 437, 24389, 851, 1369, 31

Offset

1,1

Comments

Permutation of natural numbers > 1.

In any column containing any of the terms of A228058 (like column 6), they alternate with some terms of A388983 \ A228058. Compare also to array A389095.

Links

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

Index entries for sequences related to prime indices in the factorization of n.

Index entries for sequences that are permutations of the natural numbers.

Example

The top left corner of the array:
k= |  1     2     3      4     5      6     7        8      9     10    11
---+-------------------------------------------------------------------------
1  |  2,    4,    6,     8,   10,    12,   14,      16,    18,    20,   22,
2  |  3,    9,   15,    27,   21,    45,   39,      81,    75,    63,   51,
3  |  5,   25,   35,   125,   65,   175,   55,     625,   245,   325,   95,
4  |  7,   49,   91,   343,   77,   637,  119,    2401,  1183,   539,  203,
5  | 13,  169,  143,  2197,  221,  1859,  247,   28561,  1573,  2873,  299,
6  | 11,  121,  187,  1331,  209,  2057,  319,   14641,  3179,  2299,  407,
7  | 17,  289,  323,  4913,  493,  5491,  391,   83521,  6137,  8381,  527,
8  | 19,  361,  551,  6859,  437, 10469,  703,  130321, 15979,  8303,  779,
9  | 29,  841,  667, 24389, 1073, 19343,  899,  707281, 15341, 31117, 1247,
10 | 23,  529,  851, 12167,  713, 19573,  943,  279841, 31487, 16399, 1219,
11 | 37, 1369, 1147, 50653, 1517, 42439, 1591, 1874161, 35557, 56129, 1739,

Prog

(PARI)
up_to = 26921;
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
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); };
A389088list(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] = A389088sq(col, (a-(col-1))))); (v); };
v389088 = A389088list(up_to);
A389088(n) = v389088[n];

Crossrefs

Cf. A228058, A332818, A388983, A389081, A389082.

Cf. A108546 (leftmost column), A005843 (topmost row, after the initial 0).

Derived arrays (in parentheses the applied function): A389093 (sigma), A389094 (A009194), A389095 (A336937, 2-adic valuation of sigma), A388995 (A017665, numerator of abundancy ratio), A388995 (A017666, denominator of abundancy ratio).

Cf. also A246278.

Sequence in context: A255127 A083221 A246278 * A359299 A363473 A386625

Adjacent sequences: A389085 A389086 A389087 * A389089 A389090 A389091

Keyword

nonn,tabl

Author

Antti Karttunen, Sep 23 2025

Status

approved