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erdős #1204

← #1203 · #1205 → (packet.json; erdosproblems.com)

We call a sequence of integers $0 \leq a_{1} < \dots < a_{k}$ admissible if it is missing at least one congruence class modulo every prime $p$ . Let $A (k) = min a_{k}$ . Estimate $A (k)$ - in particular, is it true that $A (k) \sim k lo g k ?$ Estimate $B (k) = min \frac{a _{1} + \dots + a _{k}}{k} .$

Worked, still open.

number theory · open · 0 attempts

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A008407 — Minimal difference s(n) between beginning and end of n consecutive large primes (n-tuplet) permitted by divisibility considerations.0,2,6,8,12,16,20,26,30,32,36,42,48,50,56,60,66,70,76,80,84,90,94,100,110,114,120,126,130,136,140,146,152,156,158,162,168 A023193 — a(n) gives the largest number k for which there is at least one admissible k-tuple taken from [0, 1, ..., n-1] if the tuple starts with 0. Admissibility is defined in a comment.1,1,2,2,2,2,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,12,13,13,14 A135311 — A greedy sequence of prime offsets.0,2,6,8,12,18,20,26,30,32,36,42,48,50,56,62,68,72,78,86,90,96,98,102,110,116,120,128,132,138,140,146,152,156,158,162,168

links

prime tuples conjecture · reference

Create a formalisation here · link

π (x)

counts the number of primes in

[1, x]

then is it true that (for large

x

y

π (x + y) \leq π (x) + π (y)?

status

open

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