A372285 Array read by upward antidiagonals: A(n,k) is the number of terms of A086893 in the interval [b(n, k), b(n+1, k)], n,k >= 1, where b = A372282.

5, 9, 4, 17, 9, 7, 33, 17, 13, 2, 65, 33, 25, 5, 4, 129, 65, 49, 10, 6, 3, 257, 129, 97, 22, 13, 6, 6, 513, 257, 193, 45, 26, 14, 13, 3, 1025, 513, 385, 89, 54, 29, 25, 4, 4, 2049, 1025, 769, 177, 109, 57, 49, 9, 10, 3, 4097, 2049, 1537, 353, 217, 113, 97, 22, 21, 6, 9, 8193, 4097, 3073, 705, 433, 225, 193, 45, 41, 13, 17, 2

Offset: 1

Author

Antti Karttunen, Apr 27 2024

Keywords

• nonn
• tabl

Examples

			Array begins:
n\k|    1     2      3     4     5     6      7     8      9    10     11
---+----------------------------------------------------------------------
1  |    5,    4,     7,    2,    4,    3,     6,    3,     4,    3,     9,
2  |    9,    9,    13,    5,    6,    6,    13,    4,    10,    6,    17,
3  |   17,   17,    25,   10,   13,   14,    25,    9,    21,   13,    33,
4  |   33,   33,    49,   22,   26,   29,    49,   22,    41,   26,    65,
5  |   65,   65,    97,   45,   54,   57,    97,   45,    81,   54,   129,
6  |  129,  129,   193,   89,  109,  113,   193,   89,   161,  109,   257,
7  |  257,  257,   385,  177,  217,  225,   385,  177,   321,  217,   513,
8  |  513,  513,   769,  353,  433,  449,   769,  353,   641,  433,  1025,
9  | 1025, 1025,  1537,  705,  865,  897,  1537,  705,  1281,  865,  2049,
10 | 2049, 2049,  3073, 1409, 1729, 1793,  3073, 1409,  2561, 1729,  4097,
11 | 4097, 4097,  6145, 2817, 3457, 3585,  6145, 2817,  5121, 3457,  8193,
12 | 8193, 8193, 12289, 5633, 6913, 7169, 12289, 5633, 10241, 6913, 16385,
etc.
The count includes also the starting and/or ending point, if either of them is a term of A086893. For example, when going from A372282(2,1) = 21 to A372282(3,1) = 5461, we count terms A086893(5..13) = [21, 53, 85, 213, 341, 853, 1365, 3413, 5461], nine in total, therefore A(2,1) = 9.
When going from A371102(1,8) = 15 to A371102(2,8) = 93, we count terms 21, 53, 85 of A086893 in the interval [15, 93], therefore A(1,8) = 3.
		

Crossrefs

Cf. A086893, A372282, A372286.

Programs

PARI

up_to = 78;
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3); \\ From A086893
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372282sq(n,k) = if(1==n,2*k-1,A371094(A372282sq(n-1,k)));
A372286(n) = { my(u=A371094(n), k1); for(i=1,oo,if(A086893(i)>=n,k1=i-1; break)); for(i=k1,oo,if(A086893(i)>u,return(i-k1-1))); };
A372285sq(n,k) = A372286(A372282sq(n,k));
A372285list(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] = A372285sq((a-(col-1)),col))); (v); };
v372285 = A372285list(up_to);
A372285(n) = v372285[n];

Formula

A(n, k) = A372286(A372282(n, k)).