A316625 Terms in A259663, in ascending order.

1, 3, 5, 7, 11, 13, 15, 19, 21, 23, 31, 35, 47, 53, 55, 63, 79, 85, 87, 95, 99, 127, 143, 151, 191, 213, 223, 227, 255, 271, 319, 341, 351, 383, 407, 483, 511, 575, 663, 739, 767, 783, 853, 863, 895, 1023, 1175, 1251, 1279, 1365, 1407, 1535, 1599, 1807, 1887, 2047

Offset: 1

Author

Bob Selcoe, Jul 08 2018

Keywords

• nonn

Comments

See A259663 for discussion of these terms in relation to Collatz sequences.

There are k terms in the interval [2^k, 2^(k+1)], k >= 1; terms in each interval are of the form 2^k + a(n) for some n.

The sequence is a permutation (without repeating terms) of the following numbers:

2^i-1 and 7*2^i-1 when i is odd, i >= 1;

3^2^i-1 and 5^2^i-1 when i is even, i >= 2;

For fixed k >= 4: least residues of 3^j*(2^(2^(k-3) + i*2^(k-2) - j)) - 1 mod 2^(2^(k-3) + i*2^(k-2) + k-j), i >= 0, 0 <= j < 2^(k-3) + i*2^(k-2) . (See example).

Examples

			k=5, i=1 -- terms are least residues of 3^j*2^(12-j)-1 mod 2^(17-j), 0 <= j < 12:
j=0: 4096-1 mod 131072 = 4095;
j=1: 3*2048-1 mod 65536 = 6143;
j=2: 9*1024-1 mod 32768 = 9215;
j=3: 27*512-1 mod 16384 = 13823;
j=4: 81*256-1 mod 8192 = 20735 mod 8192 == 4351;
j=5: 243*128-1 mod 4096 = 31103 mod 4096 == 2431;
j=6: 729*64-1 mod 2048 = 46655 mod 2048 == 1599;
j=7: 2187*32-1 mod 1024 = 69983 mod 1024 == 351;
j=8: 6561*16-1 mod 512 = 104975 mod 512 == 15;
j=9: 19683*8-1 mod 256 = 157463 mod 256 == 23;
j=10: 59049*4-1 mod 128 = 236195 mod 128 == 35;
j=11: 177147*2-1 mod 64 = 354293 mod 64 == 53.
Note: k=5, i=0 is equivalent to starting with j=0: 15 mod 512.
		

Crossrefs

Cf. A259663.

Extensions

More terms from Michel Marcus, Jul 10 2018