cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Views

Author

Patrick De Geest, Jan 15 2000

Keywords

Comments

a(16) > 6.9*10^12. - Giovanni Resta, Jul 02 2018

Crossrefs

Cf. A052076, A052073, A052074.

Programs

  • Mathematica
    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 *)
  • Python
    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

Extensions

More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Sep 12 2006
Offset corrected by N. J. A. Sloane, Jul 13 2016
a(12)-a(15) from Giovanni Resta, Jul 02 2018
Definition clarified by Chai Wah Wu, Jan 20 2022

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

Views

Author

Patrick De Geest, Jan 15 2000

Keywords

Examples

			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, ...

Crossrefs

Cf. A052073, A052075, A052076, A274932.

Programs

  • Mathematica
    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 *)

Extensions

a(8) from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Sep 04 2006
Definition clarified and offset changed by N. J. A. Sloane, Jul 13 2016
a(9) from Giovanni Resta, May 24 2018

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

Views

Author

Patrick De Geest, Jan 15 2000

Keywords

Links

Crossrefs

Cf. A052075, A052073, A052074, A274932.

Programs

  • Mathematica
    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 *)

Extensions

More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Sep 12 2006
Definition clarified by Charles R Greathouse IV, Dec 24 2014
a(12) from Giovanni Resta, Jul 02 2018

A294087 Least prime p_k such that (p_k)^n has p_{k+1} as substring.

Original entry on oeis.org

23, 11, 37, 2, 7, 5, 3, 41, 3, 13, 3, 3, 2, 2, 2, 2, 5, 5, 5, 3, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 17, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2
Offset: 2

Views

Author

Paolo P. Lava, Feb 09 2018

Keywords

Comments

It appears that a(n) = 2 for n>153. In other words, for n>153, 3 is always a substring of 2^n. Is there any proof? See A035058.

Examples

			23^2 = 529 and 29 is the prime after 23.
11^3 = 1331 and 13 is the prime after 11.
37^4 = 1874161 and 41 is the prime after 37.

Crossrefs

Cf. A035058, A052073, A052075, A294088.

Programs

  • Maple
    P:=proc(q) local a,b,h,k,n,ok; for h from 2 to q do ok:=1; for n from 1 to q do
if ok=1 then a:=ithprime(n); b:=nextprime(a); for k from 1 to ilog10(a^h)-ilog10(b)+1 do
if b=trunc(a^h/10^(k-1)) mod 10^(ilog10(b)+1) then print(a); ok:=0; break;
fi; od; fi; od; od; end: P(10^6);

A294088 Least prime p_k such that (p_k)^n has p_{k-1} as substring.

Original entry on oeis.org

3701, 3, 43, 3, 3, 3, 5, 5, 7, 11, 11, 3, 3, 5, 3, 3, 3, 3, 5, 3, 5, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3
Offset: 2

Views

Author

Paolo P. Lava, Feb 09 2018

Keywords

Comments

It appears that a(n) = 3 for n>59. In other words, for n>59, 2 is always a substring of 3^n. Is there any proof? See A131625.

Examples

			3701^2 = 13697401 and 3697 is the prime before 3701.
3^3 = 27 and 2 is the prime before 3.
43^4 = 3418801 and 41 is the prime before 43.

Crossrefs

Cf. A052073, A052075, A131625, A294087.

Programs

  • Maple
    P:=proc(q) local a,b,h,k,n,ok; for h from 2 to q do ok:=1; for n from 1 to q do
if ok=1 then a:=ithprime(n); b:=prevprime(a); for k from 1 to ilog10(a^h)-ilog10(b)+1 do
if b=trunc(a^h/10^(k-1)) mod 10^(ilog10(b)+1) then print(a); ok:=0; break;
fi; od; fi; od; od; end: P(10^6);

A381969 Primes p with the property that PreviousPrime(p) is a substring of p^2.

Original entry on oeis.org

3701, 65442077, 8410957371097
Offset: 1

Views

Author

Giorgos Kalogeropoulos, Mar 11 2025

Keywords

Comments

a(4) <= 39835421121719177570419.
p = 1410901659681109388941762308365764228483 is also a member of this sequence and it is the only known term that is the greater of a twin prime pair.

Examples

			3701 is a term because PreviousPrime(3701) = 3697 is a substring of 3701^2 = 13697401.

Links

Crossrefs

Cf. A052073.

Programs

  • Mathematica
    Select[Prime@Range[10^4],StringContainsQ[ToString[#^2],ToString[NextPrime[#,-1]]]&]

A329270 List of primes p = prime(k) such that k-th composite is a substring of k-th prime.

Original entry on oeis.org

56591, 229547, 229583, 231131, 331999, 4333739, 7557833, 3160829137, 729005502091, 1627315917049
Offset: 1

Views

Author

Metin Sariyar, Nov 11 2019

Keywords

Comments

a(11) > 10^14, if it exists. - Giovanni Resta, Nov 12 2019

Examples

			56591 is a term because it is the 5738-th prime and 6591 is the 5738-th composite number.

Crossrefs

Cf. A000040, A002808, A052073.

Extensions

a(8)-a(9) from Giovanni Resta, Nov 11 2019
a(10) from Giovanni Resta, Nov 12 2019

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

Views

Author

Jean-Marc Rebert, May 01 2025

Keywords

Examples

			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, ...
  .., .., ..., ...., ...

Crossrefs

Cf. A052073, A052074, A052075, A052076, A274932.

Programs

  • PARI
    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
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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