# possible seq: Partial sum of A061712

Jonathan Post jvospost3 at gmail.com
Tue Jul 1 19:58:15 CEST 2008

```Partial sum of A061712  Smallest prime with Hamming weight n (i.e.
with exactly n 1's when written in binary).

2, 5, 12, 35, 66, 377, 504, 887, 1878, 3917, 7500, 13643, 21834,
95561, 159048, 288071, 419142, 941381, 1465668, 3431747, 7560514,
23811393, 38491456, 72037695, 139146046, 340210493, 600257340,
1136997179, 2210476858, 7545631545, 9693115192, 18274661175,
35186094902, 67398349621, 118937957172, 325079610163, 591367582514,
1622158684977, 2446792405808, 4643668177711

I'm interested in which values of the partial sum (beyond the first
two, a(8), a(10), and a(14)) are themselves prime. The sequence,
obviously,  is strictly increasing, and alternates between even and
odd.

Example:

a(40) =
2 + 3 + 7 + 23 + 31 + 311 + 127 + 383 + 991 + 2039 + 3583 + 6143 +
8191 + 73727 + 63487+ 129023 + 131071+ 522239 + 524287+ 1966079 +
4128767 + 16250879 + 14680063 + 33546239 + 67108351 + 201064447 +
260046847 + 536739839 + 1073479679 + 5335154687 + 2147483647 +
8581545983 + 16911433727 + 32212254719 + 51539607551 + 206141652991 +
266287972351 + 1030791102463 + 824633720831 + 2196875771903 =
4643668177711 is prime.

```