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 1-4 of 4 results.

A156104 Primes p such that p+36 is also prime.

Original entry on oeis.org

5, 7, 11, 17, 23, 31, 37, 43, 47, 53, 61, 67, 71, 73, 101, 103, 113, 127, 131, 137, 157, 163, 191, 193, 197, 227, 233, 241, 257, 271, 277, 281, 311, 313, 317, 331, 337, 347, 353, 373, 383, 397, 421, 431, 443, 463, 467, 487, 521, 541, 557, 563, 571, 577, 607
Offset: 1

Views

Author

Vincenzo Librandi, Feb 08 2009

Keywords

Links

Crossrefs

Cf. A156112.
Cf. sequences of the type p+n are primes: A001359 (n=2), A023200 (n=4), A023201 (n=6), A023202 (n=8), A023203 (n=10), A046133 (n=12), A153417 (n=14), A049488 (n=16), A153418 (n=18), A153419 (n=20), A242476 (n=22), A033560 (n=24), A252089 (n=26), A252090 (n=28), A049481 (n=30), A049489 (n=32), A252091 (n=34), this sequence (n=36); A062284 (n=50), A049490 (n=64), A156105 (n=72), A156107 (n=144).

Programs

  • Magma
    [p: p in PrimesUpTo(1000) | IsPrime(p + 36)]; // Vincenzo Librandi, Oct 31 2012
  • Mathematica
    Select[Prime[Range[1000]], PrimeQ[(#+ 36)]&] (* Vincenzo Librandi, Oct 31 2012 *)

A257638 Numbers n such that n-25, n-1, n+1 and n+25 are consecutive primes.

Original entry on oeis.org

232962, 311712, 431832, 435948, 473352, 501342, 525492, 596118, 635388, 665922, 699792, 754182, 842448, 1013502, 1017648, 1036002, 1156848, 1255452, 1284738, 1306692, 1479912, 1516128, 1551732, 1560708, 1595928, 1659348, 1690572, 1745112
Offset: 1

Views

Author

Karl V. Keller, Jr., Nov 04 2015

Keywords

Comments

This is a subsequence of A014574 (average of twin prime pairs) and A256753.
The terms ending in 2 and 8 are congruent to 12 mod 30 and 18 mod 30 respectively.
The numbers n-25 and n+1 belong to A033560 (p and p+24 are primes) and A098974 (p where p+24 is the next prime).
The numbers n-25 and n-1 belong to A252089 (p and p+26 are primes).

Examples

			232962 is the average of the four consecutive primes 232937, 232961, 232963, 232987.
311712 is the average of the four consecutive primes 311687, 311711, 311713, 311737.

Links

Crossrefs

Cf. A014574, A077800 (twin primes), A249674, A256753.

Programs

  • Python
    from sympy import isprime,prevprime,nextprime
for i in range(0,1000001,6):
  if isprime(i-1) and isprime(i+1) and prevprime(i-1) == i-25 and nextprime(i+1) == i+25: print (i,end=', ')

A309392 Square array read by downward antidiagonals: A(n, k) is the k-th prime p such that p + 2*n is also prime, or 0 if that prime does not exist.

Original entry on oeis.org

3, 5, 3, 11, 7, 5, 17, 13, 7, 3, 29, 19, 11, 5, 3, 41, 37, 13, 11, 7, 5, 59, 43, 17, 23, 13, 7, 3, 71, 67, 23, 29, 19, 11, 5, 3, 101, 79, 31, 53, 31, 17, 17, 7, 5, 107, 97, 37, 59, 37, 19, 23, 13, 11, 3, 137, 103, 41, 71, 43, 29, 29, 31, 13, 11, 7, 149, 109
Offset: 1

Views

Author

Felix Fröhlich, Jul 28 2019

Keywords

Comments

The same as A231608 except that A231608 gives the upward antidiagonals of the array, while this sequence gives the downward antidiagonals.
Conjecture: All values are nonzero, i.e., for any even integer e there are infinitely many primes p such that p + e is also prime.
The conjecture is true if Polignac's conjecture is true.

Examples

			The array starts as follows:
3,  5, 11, 17, 29, 41, 59,  71, 101, 107, 137, 149, 179, 191
3,  7, 13, 19, 37, 43, 67,  79,  97, 103, 109, 127, 163, 193
5,  7, 11, 13, 17, 23, 31,  37,  41,  47,  53,  61,  67,  73
3,  5, 11, 23, 29, 53, 59,  71,  89, 101, 131, 149, 173, 191
3,  7, 13, 19, 31, 37, 43,  61,  73,  79,  97, 103, 127, 139
5,  7, 11, 17, 19, 29, 31,  41,  47,  59,  61,  67,  71,  89
3,  5, 17, 23, 29, 47, 53,  59,  83,  89, 113, 137, 149, 167
3,  7, 13, 31, 37, 43, 67,  73,  97, 151, 157, 163, 181, 211
5, 11, 13, 19, 23, 29, 41,  43,  53,  61,  71,  79,  83,  89
3, 11, 17, 23, 41, 47, 53,  59,  83,  89, 107, 131, 137, 173
7, 19, 31, 37, 61, 67, 79, 109, 127, 151, 157, 211, 229, 241
5,  7, 13, 17, 19, 23, 29,  37,  43,  47,  59,  73,  79,  83

Links

Crossrefs

Cf. A231608.
Cf. A001359 (row 1), A023200 (row 2), A023201 (row 3), A023202 (row 4), A023203 (row 5), A046133 (row 6), A153417 (row 7), A049488 (row 8), A153418 (row 9), A153419 (row 10), A242476 (row 11), A033560 (row 12), A252089 (row 13), A252090 (row 14), A049481 (row 15), A049489 (row 16), A252091 (row 17), A156104 (row 18), A271347 (row 19), A271981 (row 20), A271982 (row 21), A272176 (row 22), A062284 (row 25), A049490 (row 32), A020483 (column 1).

Programs

  • PARI
    row(n, terms) = my(i=0); forprime(p=1, , if(i>=terms, break); if(ispseudoprime(p+2*n), print1(p, ", "); i++))
array(rows, cols) = for(x=1, rows, row(x, cols); print(""))
array(12, 14) \\ Print initial 12 rows and 14 columns of the array

A365850 Numbers k for which k^2 + (k')^2 is a square, where k' is the arithmetic derivative of k (A003415).

Original entry on oeis.org

0, 1, 12, 15, 35, 81, 143, 323, 400, 441, 899, 1540, 1763, 2700, 3599, 4641, 5183, 5929, 9375, 10395, 10403, 11663, 13585, 18225, 19043, 21952, 22499, 30576, 32399, 35581, 36863, 39203, 48841, 51983, 57599, 72899, 79523, 97343, 121103, 148176, 166375, 175692, 176399
Offset: 1

Views

Author

Marius A. Burtea, Oct 09 2023

Keywords

Comments

If p and p + 2 are twin primes (A001359) then m = p*(p + 2) is a term. Indeed, m' = p + (p + 2) = 2*p + 2 and m^2 + (m')^2 = p^2*(p + 2)^2 + (2*p + 2)^2 = (p^2 + 2*p + 2)^2.
More generally, if p and p + 2*k, k >= 1, are prime numbers, then m = p^k*(p + 2*k)^k is a term. Indeed, m' = k*p^(k - 1)*(p + 2*k)^k+ k*p^k*(p + 2*k)^(k - 1) = k*p^(k - 1)*(p + 2*k)^(k-1)*(2*p + 2*k). Thus, m^2 + (m')^2 = p^(2*k)*(p + 2*k)^(2*k) + (k^2)*p^(2*k - 2)*(p + 2*k)^(2*k - 2)*(2*p + 2*k)^2 = p^(2*k - 2)*(p + 2*k)^(2*k - 2)*(p^2*(p + 2*k)^2 + k^2*(2*p + 2*k)^2) = p^(2*k - 2)*(p + 2*k)^(2*k - 2)*(2*k^2 + 2*k*p + p^2)^2.

Examples

			For k = 12, k' = 16 and 12^2 + 16^2 = 144 + 256 = 400 = 20^2, so 12 is a term.
For k = 15, k' = 8 and 15^2 + 8^2 = 225 + 64 = 289 = 17^2, so 15 is a term.
For k = 143, k' = 24 and 143^2 + 24^2 = 144 + 256 = 21025 = 145^2, so 143 is a term.

Crossrefs

Cf. A077800, A153417 - A153419, A156104, A242476, A252089 - A252091, A271347, A271981, A271982, A272176.

Programs

  • Magma
    f:=func; [n:n in [0..200000] |IsSquare( n^2+( Floor(f(n))^2))];
  • Maple
    ader:= proc(n) local f;   n*add(f[2]/f[1], f=ifactors(n)[2]) end proc:
select(t -> issqr(t^2 + ader(t)^2), [$0..10^6]; # Robert Israel, Oct 17 2023
  • Mathematica
    d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[0, 180000], IntegerQ[Sqrt[#^2 + d[#]^2]] &] (* Amiram Eldar, Oct 11 2023 *)
Showing 1-4 of 4 results.

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