Remoteness for put-or-take-a-prime positions through 150
(according to the not-necessarily-optimal strategy of
if there is a choice of winning moves taking the subtraction).
For P-positions both losing moves are listed,
as are intermediate positions arising from this strategy.

**P**

0 1

**N**

2 3 4 5 6 7 8 11 12 13 14 17 18 19 20 23 24 29 30 31 32 37 38 41 42 43
44 47 48 53 54 59 60 61 62 67 68 71 72 73 74 79 80 83 84 89 90 97 98 107
108 109 110 113 114 127 128 131 132 137 138 139 140 149 150 151 152
(more generally these consist exactly of prime and prime+1)

**P**

10:(3,17) 16:(3,29) 22:(3,41) 25:(2,48) 36:(5,67) 46:(3,89) 50:(3,97)
51:(4,98) 55:(2,108) 56:(3,109) 57:(4,110) 66:(5,127) 70:(3,137)
76:(3,149) 77:(4,150) 78:(5,151) 85:(2,168) 93:(4,182) 100:(3,197)
115:(2,228) 116:(3,229) 117:(4,230) 120:(7,233) 121:(8,234) 126:(13,239)
130:(3,257) 133:(2,264) 142:(3,281) 143:(4,282) 144:(5,283) 145:(6,284)
160:(3,317) 161:(4,318) 170:(3,337) 196:(3,389) 232:(3,461) 235:(2,468)
280:(3,557) 286:(3,569) 287:(4,570) 470:(3,937) 471:(4,938) 507:(4,1010)
528:(5,1051) 529:(6,1052) 667:(6,1328) 1190:(3,2377) 1191:(4,2378)
11509:(6,23012) 92046:(5,184087) 576590:(9,1153171) 736320:(41,1472599)

**N**

9:16 27:50 28:51 35:66 39:76 40:77 65:126 81:160 82:161 87:170 88:171
99:196 119:232 122:235 123:10 141:280 147:286 148:287 209:10 221:10
237:470 238:471 256:507 265:528 266:529 336:667 597:1190 598:1191
46025:92046 5760:11509 288297:576590 368167:736320

**P**

15:(2,28) 21:(2,40) 34:(3,65) 45:(2,88) 52:(5,99) 58:(5,111) 75:(2,148)
106:(3,209) 112:(3,221) 124:(11,237) 125:(12,238) 129:(2,256) 134:(3,265)
135:(4,266) 169:(2,336) 220:(9,431) 304:(11,597) 305:(12,598) 2881:(2,5760)
23014:(3,46025) 144158:(19,288297) 184086:(5,368167)

**N**

64:125 111:220 63:124 86:169 153:304 1442:2881 8404:15 11511:23014
72081:144158 92045:184086

**P**

33:(2,64) 154:305 723:(4,1442) 4203:(2,8404) 5762:(13,11511)
36044:(7,72081) 46024:(3,92045)

**N**

364:723 2104:4203 2883:5762 18031:36044 23013:46024

**P**

183:(2,364) 1053:(2,2104) 1444:(5,2883) 9018:(5,18031) 11510:(7,23013)

**N**

94:183 530:1053 725:1444 4511:9018 5761:11510

**P**

267:(4,530) 366:(7,725) 2260:(9,4511) 2882:(3,5761)

**N**

136:267 185:366 1131:2260 1443:2882

**P**

69:(2,136) 96:(7,185) 568:(5,1131) 724:(5,1443)

**N**

49:96 285:568 365:724

**P**

26:(3,49) 146:(7,285) 184:(3,365)

**N**

95:184




****N****
105

****P****
118



There are 31 P-positions less than 100.

Is this like tribulations?
I.e. all positions either P or N,
with ratio P/N tending towards golden ratio?
Is the ratio affected by even/odd considerations?

Should this sort of behavior happen with any "put or take" game
that has "dense enough" values to be added/subtracted
and that doesn't get "unlucky" and have an infinite loop early on?

Can we say anything about relative numbers of positions at different levels?