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.

Previous Showing 11-17 of 17 results.

A291374 Primes p such that p^3*q^3 + p + q is prime, where q is next prime after p.

Original entry on oeis.org

11, 17, 41, 43, 47, 137, 313, 359, 389, 401, 491, 557, 577, 709, 757, 829, 863, 929, 937, 953, 1129, 1163, 1249, 1301, 1307, 1439, 1597, 1627, 1693, 1847, 2087, 2311, 2351, 2437, 2663, 2731, 2741, 3109, 3119, 3217, 3253, 4027, 4219, 4271, 4547, 4637, 5189, 5237
Offset: 1

Views

Author

K. D. Bajpai, Aug 23 2017

Keywords

Examples

			a(1) = 11 is prime; 13 is the next prime: 11^3*13^3 + 11 + 13 = 1331*2197 + 11 + 13 = 2924231 that is a prime.
a(2) = 17 is prime; 19 is the next prime: 17^3*19^3 + 17 + 19 = 4913*6859 + 17 + 19 = 33698303 that is a prime.

Links

Crossrefs

Cf. A000040, A001043, A006094, A030078, A096342, A120398, A126148, A152241, A291339.

Programs

  • Magma
    [p: p in PrimesUpTo(5000) | IsPrime(p^3*q^3 + p + q) where q is NextPrime(p)];
  • Maple
    select(p -> andmap(isprime, [p,(p^3*nextprime(p)^3+p+nextprime(p))]), [seq(p,p=1..10^4)]);
  • Mathematica
    Prime@Select[Range[1000], PrimeQ[Prime[#]^3* Prime[# + 1]^3 + Prime[#] + Prime[# + 1]] &]
  • PARI
    forprime(p=1,5000, q=nextprime(p+1); if(ispseudoprime(p^3*q^3 + p + q), print1(p, ", ")));
  • PARI
    list(lim)=my(v=List(),p=2,p3=8,q3); forprime(q=3,nextprime(lim\1+1), q3=q^3; if(isprime(p3*q3+p+q), listput(v,p)); p=q; p3=q3); Vec(v) \\ Charles R Greathouse IV, Aug 23 2017

A132261 Main diagonal of array in A132260.

Original entry on oeis.org

13, 107, 101, 491, 8039, 9349
Offset: 1

Views

Author

Jonathan Vos Post, Aug 15 2007

Keywords

Examples

			a(1) = 13 because 13 is the first prime p such that 2^2^3 + p^2^3 is prime.
a(2) = 107 because 107 is the 2nd prime p such that 2^2^4 + p^2^4 is prime.
a(3) = 101 because 101 is the 3rd prime p such that 2^2^5 + p^2^5 is prime.

Crossrefs

Cf. A000040, A001016, A050997, A120398, A122616, A130873, A130555, A132214, A132215, A132216, A132260.

Formula

a(n) = A[n,n+2] = n-th prime p such that 2^2^(n+2) + p^2^(n+2) is prime.

A133514 Biquadrateful (i.e., not biquadrate-free) palindromes.

Original entry on oeis.org

272, 464, 656, 848, 2112, 2992, 4224, 6336, 8448, 14641, 21312, 21712, 23232, 23632, 25152, 25552, 25952, 27072, 27472, 27872, 29392, 29792, 31213, 40304, 40704, 42224, 42624, 44144, 44544, 44944, 46064, 46464, 46864, 48384, 48784, 61216, 61616, 62426, 63136
Offset: 1

Views

Author

Jonathan Vos Post, Nov 30 2007

Keywords

Comments

This is to A035133 as 4th powers are to cubes. To make an analogy between analogies, the preceding sentence is to "A130873 is to 4th powers as A120398 is to cubes" as palindromes are to sums of two distinct prime powers.

Examples

			a(10) = 14641 = 11^4 (the smallest odd value in this sequence).
a(11) = 21312 = 2^6 * 3^2 * 37.

Links

Crossrefs

Cf. A002113, A046101.

Programs

  • Maple
    isA046101 := proc(n) local ifs,f ; ifs := ifactors(n)[2] ; for f in ifs do if op(2,f) >= 4 then RETURN(true) ; fi ; od: RETURN(false) ; end: isA002113 := proc(n) local digs,i ; digs := convert(n,base,10) ; for i from 1 to nops(digs) do if op(i,digs) <> op(-i,digs) then RETURN(false) ; fi ; od: RETURN(true) ; end: isA133514 := proc(n) isA046101(n) and isA002113(n) ; end: for n from 1 to 100000 do if isA133514(n) then printf("%d, ",n) ; fi ; od: # R. J. Mathar, Jan 12 2008
# second Maple program:
q:= n->StringTools[IsPalindrome](""||n) and max(map(i->i[2], ifactors(n)[2]))>3:
select(q, [$1..70000])[];  # Alois P. Heinz, Sep 27 2023
  • Mathematica
    a = {}; For[n = 2, n < 100000, n++, If[FromDigits[Reverse[IntegerDigits[n]]] == n, b = 0; For[l = 1, l < Length[FactorInteger[n]] + 1, l++, If[FactorInteger[n][[l,2]] > 3, b = 1]]; If[b == 1, AppendTo[a, n]]]]; a (* Stefan Steinerberger, Dec 26 2007 *)
Select[Range@100000,PalindromeQ@#&&3Hans Rudolf Widmer, Sep 27 2023 *)

Formula

A002113 INTERSECTION A046101.

Extensions

More terms from Stefan Steinerberger, Dec 26 2007
More terms from R. J. Mathar, Jan 12 2008

A137632 Sums of 2 cubes of distinct odd primes.

Original entry on oeis.org

152, 370, 468, 1358, 1456, 1674, 2224, 2322, 2540, 3528, 4940, 5038, 5256, 6244, 6886, 6984, 7110, 7202, 8190, 9056, 11772, 12194, 12292, 12510, 13498, 14364, 17080, 19026, 24416, 24514, 24732, 25720, 26586, 29302, 29818, 29916, 30134
Offset: 1

Views

Author

M. F. Hasler, Apr 13 2008

Keywords

Examples

			3^3 + 5^3 = 152 = a(1).
3^3 + 7^3 = 370 = a(2).
5^3 + 7^3 = 468 = a(3).

Crossrefs

A subset of A120398 and A086119. Cf. A138853, A138854.

Programs

  • Maple
    A137632 := proc(amax) local a,p,q; a := {} ; p := 3 ; while p^3 < amax do q := nextprime(p) ; while p^3+q^3 < amax do a := a union {p^3+q^3} ; q := nextprime(q) ; od: p := nextprime(p) ; od: sort(convert(a,list)) ; end: A137632(80000) ; # R. J. Mathar, May 04 2008
  • Mathematica
    f[upto_]:=Module[{max=Ceiling[Power[upto-27, (3)^-1]],prs}, prs=Prime[Range[2,max]]; Select[Union[Total/@(Subsets[prs,{2}]^3)], #<=upto&]]; f[31000] (* Harvey P. Dale, Apr 20 2011 *)

Extensions

More terms from R. J. Mathar, Apr 13 2008, May 04 2008

A138855 Half-sum (or average) of cubes of two distinct odd primes.

Original entry on oeis.org

76, 185, 234, 679, 728, 837, 1112, 1161, 1270, 1764, 2470, 2519, 2628, 3122, 3443, 3492, 3555, 3601, 4095, 4528, 5886, 6097, 6146, 6255, 6749, 7182, 8540, 9513, 12208, 12257, 12366, 12860, 13293, 14651, 14909, 14958, 15067, 15561, 15624, 15994
Offset: 1

Views

Author

M. F. Hasler, Apr 13 2008

Keywords

Comments

Even terms of A120398, divided by two. (Terms in A120398 are even iff they are the sum of two odd prime cubes.) Also, even terms of A138854 divided by two minus 4

Links

Crossrefs

Cf. A120398, A138854.

Programs

  • Mathematica
    With[{nn=40},Take[Union[Mean/@(Subsets[Prime[Range[2,nn/2]],{2}]^3)],nn]] (* Harvey P. Dale, Nov 16 2013 *)
  • PARI
    for(n=1,10^5, isA120398(2*n) & print1(n", "))

Formula

A138855 = { ( prime(i)^3+prime(j)^3 )/2 ; i>j>1 } = (1/2) { even terms in A120398 } = { even terms in A138854 } / 2 - 4.

A291464 Primes p such that p^3*q^3 + p^2 + q^2 is prime, where q is next prime after p.

Original entry on oeis.org

2, 11, 13, 41, 97, 277, 389, 1093, 1229, 1409, 1429, 1627, 1823, 1931, 1979, 2437, 2521, 2549, 2657, 2689, 2719, 2729, 2731, 2969, 3019, 3413, 3539, 3593, 3613, 3623, 3697, 4003, 4027, 4289, 4327, 4583, 4751, 5051, 5323, 5503, 5657, 5783, 6143, 6221, 6299, 6329
Offset: 1

Views

Author

K. D. Bajpai, Aug 24 2017

Keywords

Examples

			a(1) = 2 is prime; 3 is the next prime: 2^3*3^3 + 2^2 + 3^2 = 8*27 + 4 + 9 = 229 that is a prime.
a(2) = 11 is prime; 13 is the next prime: 11^3*13^3 + 11^2 + 13^2 = 1331*2197 + 121 + 169 = 2924497 that is a prime.

Links

Crossrefs

Cf. A000040, A001043, A006094, A030078, A096342, A120398, A126148, A152241, A291339, A291374.

Programs

  • Magma
    [p: p in PrimesUpTo(5000) | IsPrime(p^3*q^3 + p^2 + q^2) where q is NextPrime(p)];
  • Maple
    select(p -> andmap(isprime,[p,(p^3*nextprime(p)^3+p^2+nextprime(p)^2)]), [seq(p, p=1..10^4)]);
  • Mathematica
    Select[Prime[Range[5000]], PrimeQ[#^3*NextPrime[#]^3 + #^2 + NextPrime[#]^2] &]
Select[Partition[Prime[Range[1000]],2,1],PrimeQ[#[[1]]^3 #[[2]]^3+#[[1]]^2+#[[2]]^2]&][[;;,1]] (* Harvey P. Dale, Sep 11 2023 *)
  • PARI
    forprime(p=1, 5000, q=nextprime(p+1); p3=p^3; p2=p^2; q3=q^3; q2=q^2; if(ispseudoprime(p3*q3 + p2 + q2), print1(p, ", ")));

A286836 Even numbers that are the sum of two odd prime cubes.

Original entry on oeis.org

54, 152, 250, 370, 468, 686, 1358, 1456, 1674, 2224, 2322, 2540, 2662, 3528, 4394, 4940, 5038, 5256, 6244, 6886, 6984, 7110, 7202, 8190, 9056, 9826, 11772, 12194, 12292, 12510, 13498, 13718, 14364, 17080, 19026, 24334, 24416, 24514, 24732, 25720, 26586, 29302
Offset: 1

Views

Author

XU Pingya, Jul 31 2017

Keywords

Comments

Subsequence of A003325.

Crossrefs

Cf. A030078, A000578, A003325, A024670, A045636, A120398.

Programs

  • Mathematica
    Do[If[Prime[i]^3 + Prime[j]^3 == 2n, Print[2n]], {n, 15000}, {i, 2, n^(1/3)}, {j, i, (2n - i^3)^(1/3)}]
Previous Showing 11-17 of 17 results.

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