A052073
Primes p with the property that nextprime(p) is a substring of p^2.
Original entry on oeis.org
23, 83, 113, 1123, 200003, 328127, 381289, 714597769, 4916552822383
Offset: 1
381289 is a term because nextprime(381289) = 381301 is a substring of 381289^2 = 145381301521.
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Select[Prime@Range[1000000],
StringContainsQ[ToString[#^2], ToString[NextPrime[#]]] &] (* Robert Price, Oct 12 2019 *)
a(8) from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Sep 04 2006
A052075
Primes p such that nextprime(p) is substring of p^3.
Original entry on oeis.org
11, 101, 2239, 34297, 43789, 53549, 535487, 59897017, 430784719, 2549592677, 2837138669, 97969345967, 100000000019, 328096840219, 4110739763869
Offset: 1
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Select[Prime[Range[1,10000]],StringContainsQ[ToString[#^3],ToString[NextPrime[#]]]&] (* Julien Kluge, Sep 19 2016 *)
Select[Prime[Range[45000]],SequenceCount[IntegerDigits[#^3],IntegerDigits[ NextPrime[ #]]]>0&] (* Requires Mathematica version 10 or later *) (* The program generates the first 7 terms of the sequence: to generate more, increase Range constant. *) (* Harvey P. Dale, Jan 25 2021 *)
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from itertools import count, islice
from sympy import prime, nextprime
def A052075_gen(): return filter(lambda p: str(nextprime(p)) in str(p**3), (prime(n) for n in count(1)))
A052075_list = list(islice(A052075_gen(),3)) # Chai Wah Wu, Jan 20 2022
More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Sep 12 2006
A052074
Squares of primes p^2 with the property that nextprime(p) is a substring of p^2.
Original entry on oeis.org
529, 6889, 12769, 1261129, 40001200009, 107667328129, 145381301521, 510649971459777361, 24172491655282243145798689
Offset: 1
The corresponding primes p are 23, 83, 113, 1123, 200003, 328127, ..., the "next primes" are 29, 89, 127, 1129, 200009, 328129, ..., and these numbers are indeed substrings of 529, 6889, 12769, 1261129, 40001200009, 107667328129, ...
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Select[Prime[Range[37*10^6]]^2,SequenceCount[IntegerDigits[#], IntegerDigits[ NextPrime[ Sqrt[#]]]]>0&] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, Jul 13 2016 *)
a(8) from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Sep 04 2006
A383607
Square array read by antidiagonals upwards: T(n,k) is the smallest k-digit prime p such that nextprime(p) is a substring of p^n; or -1 if no such prime exists, n>1, k>0.
Original entry on oeis.org
-1, -1, 23, -1, 11, 113, 2, 37, 101, 1123, 7, 17, 487, 2239, -1, 5, 47, -1, 5659, 34297, 200003, 3, 13, -1, 2399, 91801, 535487, -1, -1, 31, 607, 1279, 31627, 842483, -1, -1, 3, 41, 431, 3163, 12281, 825059, 6315629, 59897017, 714597769, -1, 37, 233, 1931, 15791, 179947, 5623421, -1, 430784719, -1
Offset: 2
T(2,3) = 113, because nextprime(113) = 127 is a substring in 113^2 = 12769, starting at position 1, and no smaller 3-digit prime satisfies this condition.
Top left corner begins at T(2,1):
-1, 23, 113, 1123, ...
-1, 11, 101, 2239, ...
-1, 37, 487, 5659, ...
2, 17, -1, 2399, ...
.., .., ..., ...., ...
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T(n,k) = forprime(p=10^(k-1), 10^k-1, if (#strsplit(Str(p^n), Str(nextprime(p+1))) >= 2, return(p));); return(-1); \\ Michel Marcus, May 02 2025
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