A027862 -id:A027862 - cp's OEIS frontend

A230812 Smallest squarefree side lengths of primitive integer Soddyian triangles.

5, 13, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 1013, 1105, 1201, 1301, 1405, 1513, 1741, 1861, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3785, 3961, 4141, 4513, 4705, 5101, 5305, 5513, 5941

Offset: 1

Frank M Jackson, Oct 30 2013

nonn

A Soddyian triangle is a triangle whose outer Soddy circle has degenerated into a straight line. Its side lengths are related by the equation 1/sqrt(s-c) = 1/sqrt(s-b)+1/sqrt(s-a) where the sides a <= b <= c and s is the semiperimeter. It is Heronian. The smallest side length of a primitive Soddyian triangle is given as a = n^2((m+n)^2+m^2) for integers m >= n > 0 with GCD(m, n) = 1. If this side length is squarefree, then n = 1 and (m+1)^2+m^2 has to be squarefree. a(n) is the ordered sequence of squarefree integers t of the form t = (m+1)^2+m^2. Note that t uniquely determines the primitive Soddyian triple whenever the smallest side length is squarefree.

			a(3)=41 because the triangle with sides (41, 416, 425) is a primitive Soddyian triangles, 41 is squarefree and is the 3rd occurrence of such a squarefree integer.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
F. M. Jackson, Soddyian triangles, Forum Geom. 13 (2013), 1-6.

Supersequence of A027862.

lst = {}; Do[If[SquareFreeQ[(m+1)^2+m^2], AppendTo[lst, (m+1)^2+m^2]], {m, 1, 100}]; lst

select(issquarefree, vector(1000,m,(m+1)^2+m^2)) \\ Charles R Greathouse IV, Oct 31 2013

Squarefree integers of the form (m+1)^2+m^2 for any integer m > 0.

A385238 Numbers k such that A224787(k) - k is a square.

Original entry on oeis.org

8, 16, 20, 25, 95, 169, 221, 234, 295, 312, 323, 410, 543, 1027, 1681, 3071, 3419, 3721, 4183, 4352, 6649, 7448, 7979, 8188, 9047, 9200, 10108, 11203, 12769, 15732, 16240, 20303, 22819, 25351, 26291, 28769, 32761, 33728, 42880, 51198, 51338, 52206, 53613, 55303, 56800, 63731, 65567, 71531, 77550

Offset: 1

Will Gosnell and Robert Israel, Jul 28 2025

nonn

Numbers k such that the sum of the cubes of the prime factors of k, counted with multiplicity, is k plus a square.

Includes p^2 for p in A027862.

			a(3) = 20 = 2^2 * 5 is a term because 2*2^3 + 5^3 - 20 = 121 is a square.

Robert Israel, Table of n, a(n) for n = 1..2451

Cf. A027862, A224787, A386623, A386640.

filter:= proc(n) local t;
   issqr(add(t[1]^3*t[2], t=ifactors(n)[2]) - n)
end proc:
select(filter, [$1..10^5]);

A089619 a(n) = greatest prime factor of n^2 + (n+1)^2 for n >= 1.

Original entry on oeis.org

5, 13, 5, 41, 61, 17, 113, 29, 181, 17, 53, 313, 73, 421, 37, 109, 613, 137, 761, 29, 37, 1013, 17, 1201, 1301, 281, 89, 13, 1741, 1861, 397, 2113, 449, 2381, 2521, 41, 97, 593, 3121, 193, 53, 3613, 757, 233, 101, 173, 4513, 941, 29, 5101, 1061, 149, 229, 457, 101

Offset: 1

Cino Hilliard, Dec 31 2003

			2*7^2 - 2*7 + 1 = 85 = 5*17, so a(7) = 17.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001844, A006530, A027861, A027862.

a[n_] := FactorInteger[n^2 + (n+1)^2][[-1, 1]]; Array[a, 60] (* Amiram Eldar, Oct 29 2024 *)

xnpym1n(m) = { for(n=1,m, y = n^2+(n+1)^2; f = factor(y); l = length(component(f,1)); v = component(component(f,1),l); print1(v","); ) }

a(n) = A006530(A001844(n)).

Edited by Ray Chandler, Jan 03 2004

Offset corrected by Georg Fischer, May 27 2024

A093576 Smallest prime of the form n^j+(n+1)^k, with j,k integer, min(j,k)>1.

Original entry on oeis.org

5, 13, 43, 41, 61, 379, 113, 593, 181, 14741, 16369, 313, 2393, 421, 3631, 83777, 613, 105337, 761, 9661, 205129, 1013, 12743, 1201, 1301, 20359, 1146097, 615497, 1741, 1861, 1049537, 2113, 2522257, 2381, 2521, 51949, 3959297, 56393, 3121, 2561681

Offset: 1

Hugo Pfoertner, Apr 01 2004

nonn

			a(3)=43 because 3^3+4^2=43 is prime, whereas 3^2+4^2=25 is composite.

Cf. A027862 primes of the form n^2+(n+1)^2, A093574, A093575.

A158526 n and (1 + 2n + 2n^2) are primes.

Original entry on oeis.org

2, 5, 7, 17, 19, 29, 47, 79, 97, 109, 137, 139, 149, 157, 167, 199, 229, 347, 349, 389, 409, 467, 479, 547, 577, 599, 709, 719, 757, 857, 929, 937, 967, 1039, 1069, 1087, 1187, 1229, 1259, 1399, 1409, 1447, 1559, 1579, 1597, 1607, 1657, 1697, 1699, 1709

Offset: 1

Zak Seidov, Mar 20 2009

Numbers n such that A048395(n) is semiprime, or A048395(n)/n is prime.

Or, primes in A027861. Also, (1+2*n+2*n^2) are in A027862. - Zak Seidov, Sep 19 2015

			A048395(2)=26=2*13, A048395(5)=305=5*61, A048395(7)=791=7*113.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A048395 (sum of consecutive nonsquares), A001358 (semiprimes).

Cf. A027861, A027862.

[p: p in PrimesUpTo(2000) | IsPrime(2*p^2+2*p+1)]; // Vincenzo Librandi, Jun 22 2014

f[n_]:=PrimeQ[n^2+(n+1)^2];lst={};Do[p=Prime[n];If[f[p],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)

forprime(p=2,1e4,if(isprime(2*p*(p+1)+1),print1(p","))) \\ Charles R Greathouse IV, Sep 09 2009

A180263 Odd k such that (k^2 + 1)/2 is not prime.

Original entry on oeis.org

1, 7, 13, 17, 21, 23, 27, 31, 33, 37, 41, 43, 47, 53, 55, 57, 63, 67, 73, 75, 77, 81, 83, 87, 89, 91, 93, 97, 99, 103, 105, 107, 109, 111, 113, 115, 117, 119, 123, 125, 127, 129, 133, 135, 137, 143, 147, 149, 151, 153, 155, 157, 161, 163, 167, 173, 177, 179, 183, 185

Offset: 1

Alex Meiburg, Aug 21 2010

nonn

			a(2)=7: (7^2 + 1)/2 = 25, which is not prime, so 7 is in the sequence.
(9^2 + 1)/2 = 41, which is prime, so 9 is not in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A027862 (primes of the form (n^2+1)/2).

[ n: n in [1..200 by 2] | not IsPrime((n^2+1) div 2) ];

a={};For[i=1,i<100,i=i+2,If[PrimeQ[(i^2+1)/2],0,AppendTo[a,i]]]Print[a]
Select[Range[1,201,2],!PrimeQ[(#^2+1)/2]&] (* Harvey P. Dale, Jan 07 2016 *)

isok(n) = (n%2) && !isprime((n^2 + 1)/2); \\ Michel Marcus, Nov 23 2018

More terms from Vincenzo Librandi, Nov 18 2010

Example clarified by Harvey P. Dale, Jan 07 2016

A218213 Number of n-digit primes representable as sums of consecutive squares.

Original entry on oeis.org

1, 4, 13, 30, 69, 187, 519, 1401, 3889, 10861, 31640, 90735

Offset: 1

Martin Renner, Oct 23 2012

There are no common representations of two, three or six squares for n < 13, so

a(n) = A218207(n) + A218209(n) + A218211(n); n < 13.

Cf. A027861, A027862, A027863, A027864, A027866, A027867, A218207, A218209, A218211.

nn = 8; nMax = 10^nn; t = Table[0, {nn}]; Do[k = n; s = 0; While[s = s + k^2; s <= nMax, If[PrimeQ[s], t[[Ceiling[Log[10, s]]]]++];  k++], {n, Sqrt[nMax]}]; t (* T. D. Noe, Oct 23 2012 *)

a(n) = A218214(n) - A218213(n-1).

A227412 Primes of the form n^3 + (n+1)^3 + 2.

Original entry on oeis.org

11, 37, 191, 857, 2333, 3061, 4943, 6121, 9011, 22817, 33203, 89533, 105527, 114193, 341993, 421381, 536771, 931087, 1005041, 1294561, 1386443, 1583047, 1911493, 2416061, 4866481, 5086811, 5199427, 5429621, 7376141, 7814207, 8903071, 9399097, 9739811, 9913213

Offset: 1

K. D. Bajpai, Sep 22 2013

nonn

Primes which are sum of two consecutive cubes plus 2.

			a(2)=37: k^3+(k+1)^3+2= 2^3+3^3+2= 8+27+2= 37 which is prime.
a(3)=191: k^3+(k+1)^3+2= 4^3+5^3+2= 64+125+2= 191 which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..5000

Cf. A027862, A227340.

KD:= proc() local a; a:= (k)^3+(k+1)^3+2; if isprime(a) then RETURN(a): fi;end: seq(KD(),k=1..500);

Select[Table[(x^3+(x+1)^3+2), {x, 1000}], PrimeQ]

Primes of the form 2*n^3 + 3*n^2 + 3*n + 3.

A275530 Smallest positive integer m such that (m^(2^n) + 1)/2 is prime.

Original entry on oeis.org

3, 3, 3, 9, 3, 3, 3, 113, 331, 513, 827, 799, 3291, 5041, 71, 220221, 23891, 11559, 187503, 35963

Offset: 0

Walter Kehowski, Jul 31 2016

The terms of this sequence with n > 11 correspond to probable primes which are too large to be proven prime currently. - Serge Batalov, Apr 01 2018

a(15) is a statistically significant outlier; the sequence (m^(2^15)+1)/2 may require a double-check with software that is not GWNUM-based. - Serge Batalov, Apr 01 2018

			a(7) = 113 since 113 is the smallest positive integer m such that (m^(2^7)+1)/2 is prime.

Richard Fischer, Generalized Fermat numbers with odd base
Wikipedia, Fermat number

Cf. A056993, A027862.

a:= proc(n) option remember; local m; for m by 2
      while not isprime((m^(2^n)+1)/2) do od; m
    end:
seq(a(n), n=0..8);

Table[m = 1; While[! PrimeQ[(m^(2^n) + 1)/2], m++]; m, {n, 0, 9}] (* Michael De Vlieger, Sep 23 2016 *)

a(n) = {my(m = 1); while (! isprime((m^(2^n)+1)/2), m += 2); m;} \\ Michel Marcus, Aug 01 2016

from sympy import isprime
def a(n):
  m, pow2 = 1, 2**n
  while True:
    if isprime((m**pow2 + 1)//2): return m
    m += 2
print([a(n) for n in range(9)]) # Michael S. Branicky, Mar 03 2021

a(13)-a(14) from Robert Price, Sep 23 2016
a(15) from Serge Batalov, Mar 29 2018
a(16) from Serge Batalov, Mar 30 2018
a(17) from Serge Batalov, Apr 01 2018
a(18)-a(19) from Ryan Propper, Aug 16 2022. These correspond to 1382288- and 2388581-digit PRPs, respectively, found using an exhaustive search with Jean Penne's LLR2.

A293958 Smallest odd prime divisor of (2n+1)^2 + 1.

Original entry on oeis.org

5, 13, 5, 41, 61, 5, 113, 5, 181, 13, 5, 313, 5, 421, 13, 5, 613, 5, 761, 29, 5, 1013, 5, 1201, 1301, 5, 17, 5, 1741, 1861, 5, 2113, 5, 2381, 2521, 5, 29, 5, 3121, 17, 5, 3613, 5, 17, 41, 5, 4513, 5, 13, 5101, 5, 37, 5, 13, 61, 5, 17, 5, 73, 7321, 5, 13, 5, 53, 8581, 5, 13, 5, 9661, 9941, 5

Offset: 1

N. J. A. Sloane, Nov 04 2017, following a suggestion from Zoran Sunic

nonn

If the map "x -> smallest odd prime divisor of n^2+1" is iterated, does it always terminate in the 2-cycle (5 <-> 13)? - Zoran Sunic, Oct 25 2017

A027862 is a subsequence. - David A. Corneth, Nov 04 2017

David A. Corneth, Table of n, a(n) for n = 1..10000

A bisection of A125256. Cf. A027862, A069894, A078701, A256970.

sod[n_]:=With[{fi=FactorInteger[n]},If[fi[[1,1]]==2,fi[[2,1]],fi[1,1]]]; sod/@(Range[3,151,2]^2+1) (* Harvey P. Dale, Dec 23 2023 *)

a(n) = factor((2*n+1)^2 + 1)[2,1]; \\ Michel Marcus, Nov 04 2017

a(n) = A078701(A069894(n)). - Michel Marcus, Nov 04 2017

cp's OEIS Frontend

A230812 Smallest squarefree side lengths of primitive integer Soddyian triangles.

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A385238 Numbers k such that A224787(k) - k is a square.

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Maple

A089619 a(n) = greatest prime factor of n^2 + (n+1)^2 for n >= 1.

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A093576 Smallest prime of the form n^j+(n+1)^k, with j,k integer, min(j,k)>1.

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A158526 n and (1 + 2*n + 2*n^2) are primes.

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Magma

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PARI

A180263 Odd k such that (k^2 + 1)/2 is not prime.

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Programs

Magma

Mathematica

PARI

Extensions

A218213 Number of n-digit primes representable as sums of consecutive squares.

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Comments

Crossrefs

Programs

Mathematica

Formula

A227412 Primes of the form n^3 + (n+1)^3 + 2.

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A158526 n and (1 + 2n + 2n^2) are primes.