Problems & Puzzles: Puzzles

Puzzle 302. Primes inside the sequence of primes.

Time ago I wonder if a prime p may appear K times in the string of the concatenated primes (SCP) from 2 to p, for any K value.

Below I show the earliest prime p with this property for the first three non trivial values of K.

The prime p appears K times in SCP(2->p):

p, K
23, 2
71, 3
971,4

2357111317192329313741434753596167717379838997101103107109113127131137139149151157163167173179181191
193197199211223227229233239241251257263269271277281283293307311313317331337347349353359367373379383389 397401409419421431433439443449457461463467479487491499503509521523541547557563569571577587593599601607
613617619631641643647653659661673677683691701709719727733739743751757761769773787797809811821823827829
839853857859863877881883887907911919929937941947953967971...

Find the earliest prime that appears K times in SCP, for K = 5, 6, & 7

Contributions came from: J. K Andersen, Mike Oakes, Giovanni Resta, Sambit Nayak, Edwin Clark & Faride Firoozbakht who got solutions uo to K equal 11, 10, 10, 7, 7 & 6 respectively.

Here are the Andresen's results (as you can see no more than two consecutive primes need to be considered in the search):

(K, p): (5, 9719), (6, 93719), (7, 911969), (8, 9379199), (9, 71497337), (10, 331119133), (11, 3331139333).

Primes between commas below are consecutive.

(5, 9719):
197 199, 709 719, 971 977, 1997 1999, 9719.

(6, 93719):
3709 3719, 9371 9377, 19937 19949, 37189 37199, 71993 71999, 93719.

(7, 911969):
11959 11969, 69119 69127, 96911 96931, 119689 119699, 196991 196993, 699119 699133, 911969.

(8, 9379199):
199379 199399, 379189 379199, 919937 919939, 937919 937927, 1999379 1999423, 7919893 7919921, 9199937 9199987, 9379199.

(9, 71497337):
1497317 1497337, 3371497 3371509, 3714973 3714979, 7149733 7149761, 14973337 14973373, 33771497 33771527, 37714973 37714979, 49733771 49733791, 71497337.

(10, 331119133):
1331119 1331123, 11191333 11191339, 13331119 13331167, 19133311 19133327, 33111913 33111917, 33311191 33311209, 91333111 91333157, 311191313 311191333 311191337, 331119133.
Note 311191333 is part of p in two ways (no digit is part in two ways).

(11, 3331139333):
1393331 1393333, 9333113 9333131, 33331139 33331153, 39333311 39333317, 93333113 93333131, 139333331 139333349, 333113933 333113951, 333331139 333331171, 1139333333 1139333399, 3311393303 3311393339, 3331139333.

The search ran to 10^10. There was no K=12 but 6 more (unchecked) p with K=11:
3348773333, 3371913133, 7774017347, 9133411319, 9937397399, 9977331899.

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