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.

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A168540 Natural numbers n for which 100n^3 + 27 is prime.

Original entry on oeis.org

1, 2, 4, 5, 7, 13, 17, 25, 29, 32, 44, 55, 61, 76, 77, 80, 92, 106, 109, 112, 116, 121, 124, 136, 137, 142, 143, 149, 152, 154, 158, 161, 169, 170, 178, 190, 191, 196, 200, 208, 221, 223, 224, 227, 230, 245, 254, 259, 260, 262
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 29 2009

Keywords

Comments

It is conjectured that sequence is infinite.

Examples

			(1) 3^3+10^2*1^3=127=prime(31) gives a(1)=1
(2) 3^3+10^2*2^3=827=prime(144) gives a(2)=2
(3) 3^3+10^2*13^3=219727=prime(19588) gives a(6)=13

References

  • Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
  • Friedhelm Padberg, Elementare Zahlentheorie, Spektrum Akademischer Verlag, 2. Auflage 1991

Crossrefs

Cf. A168147 Primes of the form p = 1 + 10*n^3 for a natural number n
Cf. A168327 Primes of concatenated form p= "1 n^3"
Cf. A168375 Natural numbers n for which the concatenation p= "1 n^3"is prime

Programs

  • Mathematica
    Select[Range[300],PrimeQ[100#^3+27]&] (* Harvey P. Dale, May 10 2013 *)

Extensions

Edited by Charles R Greathouse IV, Apr 28 2010

A173874 Primes in A173836.

Original entry on oeis.org

29, 41, 101, 173, 191, 197, 383, 1019, 1049, 1091, 1163, 1409, 1481, 1613, 1637, 1721, 1823, 1913, 1973, 2027, 2099, 2243, 2339, 2351, 2447, 2729, 2837, 2897, 2999, 3023, 3089, 3137, 3167, 3203, 3251, 3407, 3881, 4019, 4349, 4397, 4451, 4457
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Mar 01 2010

Keywords

Comments

For a prime p and its k-digit cube p^3 we need to check if q = 11^3 * 10^k + p^3 is a prime.
11^3*10^k is congruent to 2 (mod 3), so p^3 must be congruent to 2 (mod 3) because otherwise the sum q cannot become a prime.
In turn, all p in the sequence are also congruent to 2 (mod 3) (see A003627).

Examples

			The prime 29 is in the sequence because 29^3=24389, and the concatenation 133124389=prime(7545294) is a prime number.

References

  • K. Haase and P. Mauksch: Spass mit Mathe, Urania-Verlag Leipzig, Verlag Dausien Hanau, 2. Auflage 1985

Links

Crossrefs

Cf. A000578, A168147, A168219, A173579, A173733, A173836

Programs

  • Maple
    cat2 := proc(a,b) ndgs := max(1, ilog10(b)+1) ; a*10^ndgs+b ; end proc:
for i from 1 to 800 do p := ithprime(i) ; if isprime(cat2(1331,p^3)) then printf("%d,",p) ; end if; end do: # R. J. Mathar, Mar 26 2010
  • Mathematica
    Select[Prime[Range[2000]],PrimeQ[FromDigits[Join[{1,3,3,1}, IntegerDigits[ #^3]]]]&] (* Harvey P. Dale, Oct 14 2011 *)

Extensions

Definition simplified, missing numbers 2243, 2339 etc. inserted, numbers like 2621, 2693 removed - R. J. Mathar, Mar 26 2010

A265181 Prime numbers resulting from the concatenation of at least two copies of a cubic number followed by a trailing "1.".

Original entry on oeis.org

881, 27271, 7297291, 133113311, 337533751, 19683196831, 42875428751, 68921689211, 1038231038231, 1574641574641, 2053792053791, 2744274427441, 4218754218751, 6859685968591, 7290007290001, 7297297297291, 106120810612081, 224809122480911, 274400027440001, 280322128032211, 317652331765231, 500021150002111, 812060181206011, 1251251251251251, 1757617576175761, 1968319683196831, 5931959319593191
Offset: 1

Views

Author

Thomas S. Pedigo, Dec 03 2015

Keywords

Comments

Subsequence of A030430 (primes of the form 10n+1). - Michel Marcus, Dec 04 2015
If m is a term then (m-1)/10 is divisible by a cube (A000578) and the resulting quotient, different from 1, is in A076289. - Michel Marcus, Dec 05 2015
Without the "repeated at least twice" constraint, A168147 would be a subsequence. - Michel Marcus, Dec 05 2015

Examples

			8 = 2^3; 881 is prime.
27 = 3^3; 27271 is prime.

Links

Crossrefs

Cf. A000578, A030430, A066592, A076289, A168147 , A232066.

Programs

  • Maple
    N:= 20: # to get all terms with at most N digits
M:= floor((N-1)/2):
res:= {}:
for s from 1 to floor(10^(M/3)) do
   x:= s^3;
   m:= 1+ilog10(x);
   for k from 2 to floor((N-1)/m) do
     p:= x*add(10^(1+m*i),i=0..k-1)+1;
     if isprime(p) then res:= res union {p} fi;
   od
od:
sort(convert(res,list)); # Robert Israel, Jan 13 2016
  • Mathematica
    Take[Sort@ Flatten[Select[#, PrimeQ] & /@ Table[FromDigits@ Append[Flatten@ IntegerDigits@ Table[n^3, {#}], 1] & /@ Range[2, 20], {n, 1, 300}] /. {} -> Nothing], 27] (* Michael De Vlieger, Jan 05 2016 *)
  • Python
    from itertools import count, islice
from sympy import isprime
def A265181_gen(): # generator of terms
    return filter(isprime,(int(str(k**3)*2)*10+1 for k in count(1)))
A265181_list = list(islice(A265181_gen(),20)) # Chai Wah Wu, Feb 20 2023

A169586 Primes p in A168540 for which q = 3^3 + 10^2*p^3 (A168487) is prime.

Original entry on oeis.org

2, 5, 7, 13, 17, 29, 61, 109, 137, 149, 191, 223, 227, 269, 311, 331, 337, 359, 389, 397, 409, 433, 457, 467, 491, 587, 619, 653, 661, 709, 727
Offset: 1

Views

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Dec 02 2009

Keywords

Comments

It is conjectured that sequence is infinite

Examples

			(1) 3^3+10^2*2^3=827=prime(144) gives a(1)=2=prime(1)
(2) 3^3+10^2*5^3=12527=prime(1496) gives a(2)=5=prime(3)
(3) 3^3+10^2*13^3=219727=prime(19588) gives a(4)=13=prime(6)

References

  • Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
  • Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005
  • Arnold Scholz, Bruno Schoeneberg: Einführung in die Zahlentheorie, Walter de Gruyter, 5. Auflage 1973

Crossrefs

A000040 The prime numbers
A167535 Concatenation of two square numbers which give a prime
A168147 Primes of the form p = 1 + 10*n^3 for a natural number n
A168327 Primes of concatenated form p= "1 n^3"
A168375 Naturals n for which the concatenation p= "1 n^3"is prime
A168487 Primes of form p = 3^3 + 10^2*n^3 with a natural number n
A168540 Naturals n for which the concatenation p = 3^3 + 10^2*n^3 is prime
Previous Showing 11-14 of 14 results.

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