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
A052076
List of cubes p^3 of primes with property that next prime after p is a substring of p^3.
Original entry on oeis.org
1331, 1030301, 11224377919, 40343019516073, 83964379378069, 153551511228149, 153548930496746303, 214889691497505989703913, 79943078473759892945966959, 16573430415736921632549592733, 22837138677705447754568672309, 940309072235302647342697969346063
Offset: 1
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f[0]=8;f[n_]:=Module[{i=PrimePi[f[n-1]^(1/3)]+1},
While[StringPosition[ToString[Prime[i]^3],ToString[NextPrime[Prime[i]]]]=={}, i++];Prime[i]^3];f/@Range[7] (* Ivan N. Ianakiev, Nov 16 2016 *)
#^3&/@Select[Prime[Range[10^6]],SequenceCount[IntegerDigits[#^3],IntegerDigits[ NextPrime[ #]]]> 0&] (* Harvey P. Dale, Jul 29 2023 *)
More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Sep 12 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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