A335910 - OEIS

%I #13 Jul 13 2020 21:31:52

%S 1,2,3,4,5,9,8,6,11,27,16,7,13,33,81,32,10,15,37,99,243,64,12,18,39,

%T 107,297,729,128,14,19,43,109,321,891,2187,256,17,21,45,111,327,963,

%U 2673,6561,512,20,22,53,117,333,981,2889,8019,19683,1024,24,23,54,121,351,999,2943,8667,24057,59049,2048,28,25,55,129,363,1053,2997,8829,26001,72171,177147

%N Square array where row n lists all numbers k for which A335885(k) = n, read by falling antidiagonals.

%C Array is read by descending antidiagonals with (n,k) = (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ... where A(n,k) is the (k+1)-th solution x to A335885(x) = n. The row indexing (n) starts from 0, and column indexing (k) also from 0.

%C For any odd prime p that appears on row n, either p-1 or p+1 appears on row n-1.

%C The e-th powers of the terms on row n form a subset of terms on row (e*n). More generally, a product of terms that occur on rows i_1, i_2, ..., i_k can be found at row (i_1 + i_2 + ... + i_k), because A335885 is completely additive.

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%e The top left corner of the array:

%e n\k |     0      1      2      3      4      5      6      7      8      9

%e ----+--------------------------------------------------------------------------

%e   0 |     1,     2,     4,     8,    16,    32,    64,   128,   256,   512, ...

%e   1 |     3,     5,     6,     7,    10,    12,    14,    17,    20,    24, ...

%e   2 |     9,    11,    13,    15,    18,    19,    21,    22,    23,    25, ...

%e   3 |    27,    33,    37,    39,    43,    45,    53,    54,    55,    57, ...

%e   4 |    81,    99,   107,   109,   111,   117,   121,   129,   131,   135, ...

%e   5 |   243,   297,   321,   327,   333,   351,   363,   387,   393,   405, ...

%e   6 |   729,   891,   963,   981,   999,  1053,  1089,  1161,  1177,  1179, ...

%e   7 |  2187,  2673,  2889,  2943,  2997,  3159,  3267,  3483,  3531,  3537, ...

%e   8 |  6561,  8019,  8667,  8829,  8991,  9477,  9801, 10449, 10593, 10611, ...

%e   9 | 19683, 24057, 26001, 26487, 26973, 28431, 29403, 31347, 31779, 31833, ...

%o (PARI)

%o up_to = 78-1; \\ = binomial(12+1,2)-1.

%o memoA335885 = Map();

%o A335885(n) = if(1==n,0,my(v=0); if(mapisdefined(memoA335885,n,&v), v, my(f=factor(n)); v = sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+min(A335885(f[k,1]-1),A335885(f[k,1]+1))))); mapput(memoA335885,n,v); (v)));

%o memoA335910sq = Map();

%o A335910sq(n, k) = { my(v=0); if((0==k), v = -1, if(!mapisdefined(memoA335910sq,[n,k-1],&v), v = A335910sq(n, k-1))); for(i=1+v,oo,if(A335885(1+i)==n,mapput(memoA335910sq,[n,k],i); return(1+i))); };

%o A335910list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0,oo, for(col=0,a, i++; if(i > #v, return(v)); v[i] = A335910sq(col,(a-(col))))); (v); };

%o v335910 = A335910list(up_to);

%o A335910(n) = v335910[1+n];

%o for(n=0,up_to,print1(A335910(n),", "));

%Y Cf. A335885.

%Y Cf. A000079, A335911, A335912 (rows 0-2), A000244 (is very like the leftmost column).

%Y Cf. also arrays A334100, A335430.

%K nonn,tabl

%O 0,2

%A _Antti Karttunen_, Jul 01 2020