A049532 -id:A049532 - cp's OEIS frontend

A218709 a(n) is smallest number such that a(n)^2 + 1 is divisible by 13^n.

0, 5, 70, 239, 239, 143044, 1999509, 6826318, 6826318, 822557039, 52199939826, 603633907222, 11356596271444, 11356596271444, 1828607235824962, 13920898306972194, 13920898306972194, 2675587335039691558, 49226908181248336040, 513050126578538629605

Offset: 0

Michel Lagneau, Nov 04 2012

nonn

			a(4) = 239 because 239^2+1 = 2*13^4.

Cf. A002522, A049532, A034939.

b=5;n13=13;jo=Join[{0,b},Table[n13=13*n13;b=PowerMod[b,13,n13];b=Min[b,n13-b],{99}]]

A049533 Numbers k such that k^2+1 is squarefree.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 39, 40, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Labos Elemer

nonn

Estermann proved that a(n) ~ kn with k = 1.117...; more precisely, there are cx + O(x^(2/3) log x) terms up to x, where c = 1/k = Product (1 - 2/p^2) where the product is over primes p which are 1 mod 4. Heath-Brown improves the error term to O(x^(7/12) log x). - Charles R Greathouse IV, Oct 16 2017, corrected by Amiram Eldar, Jul 08 2020

There are 89489 terms up to 10^5, 894856 terms up to 10^6, 8948417 up to 10^7, 89484102 up to 10^8, and 894841314 up to 10^9. - Charles R Greathouse IV, Nov 26 2017, corrected and extended by Amiram Eldar, Jul 08 2020

			10 is a member because 10^2 + 1 = 100 + 1 = 101 is squarefree.
Reasons why certain numbers are excluded: 7^2+1 = 2*5^2, 18^2+1 = 13*5^2, 32^2+1 = 41*5^2, 38^2+1 = 5*17^2, 41^2+1 = 2*29^2, 43^2+1 = 74*5^2, 57^2+1 = 130*5^2, 82^2+1 = 269*5^2. - Neven Juric, Oct 06 2008

Michael De Vlieger, Table of n, a(n) for n = 1..10000
T. Estermann, Einige Sätze über quadratfreie Zahlen, Math. Ann. 105 (1931), pp. 653-662.
D. R. Heath-Brown, Square-free values of n^2 + 1, Acta Arithmetica 155:1 (2012), pp. 1-13; arXiv:1010.6217 [math.NT], 2010-2012.

Complement of A049532.

Cf. A059592, A069987, A335963.

[ n: n in [1..100] | IsSquarefree(n^2+1) ]; // Vincenzo Librandi, Dec 25 2010

Select[Range@ 80, SquareFreeQ[#^2 + 1] &] (* Michael De Vlieger, Aug 09 2017 *)

isok(n) = issquarefree(n^2+1); \\ Michel Marcus, Feb 09 2016

Numbers k such that A059592(k) = 1. - Reinhard Zumkeller, Nov 08 2006

A218710 a(n) is smallest number such that a(n)^2 + 1 is divisible by 17^n.

Original entry on oeis.org

0, 4, 38, 1985, 27493, 390112, 390112, 96940388, 3379649772, 24306922095, 450044583893, 5597937117454, 28673959190179, 3524407382568745, 13428985415474682, 13428985415474682, 5711417117604156904, 91610966633729580058, 6709533061724423693474

Offset: 0

Michel Lagneau, Nov 04 2012

nonn

			a(4) = 27493 because 27493^2+1 =  2 * 5 ^ 2 * 17 ^ 4 * 181.

Cf. A002522, A049532, A034939, A218709.

b=4;n17=17;jo=Join[{0,b},Table[n17=17*n17;b=PowerMod[b,17,n17];b=Min[b,n17-b],{99}]]

A218712 a(n) is the smallest number such that a(n)^2 + 1 is divisible by 29^n.

Original entry on oeis.org

0, 12, 41, 10133, 34522, 7745569, 253879357, 7986582530, 61012922706, 4563230639355, 67972499239990, 1330094199140593, 47471944863682723, 5000878909740249297, 5000878909740249297, 590115586441858677665, 77072583141941801290876, 423420364218752896284166

Offset: 0

Michel Lagneau, Nov 04 2012

nonn

			a(4) = 34522 because 34522^2+1 =  5 * 29 ^ 4 * 337.

Robert Israel, Table of n, a(n) for n = 0..682

Cf. A002522, A049532, A034939, A218709, A218710.

R:= 0,12: U:= [12,17]:
for n from 2 to 30 do
  qs:= map(u -> (u^2+1)/29^(n-1), U);
  ys:= [seq(-qs[i]/(2*U[i]) mod 29,i=1..2)];
  U:= U + ys*29^(n-1) mod 29^n;
  R:= R,min(U);
od:
R; # Robert Israel, Jan 13 2025

b=12;n29=29;jo=Join[{0,b},Table[n29=29*n29;b=PowerMod[b,29,n29];b=Min[b,n29-b],{99}]]

A124809 Numbers of the form (square + 1) that are not squarefree.

Original entry on oeis.org

50, 325, 1025, 1445, 1682, 1850, 3250, 4625, 4901, 6725, 8650, 9802, 11450, 13690, 13925, 17425, 20450, 24650, 28225, 33125, 37250, 42850, 47525, 53825, 57122, 59050, 63002, 66050, 71825, 79525, 85850, 94250, 101125, 106930, 110225, 117650

Offset: 1

Reinhard Zumkeller, Nov 08 2006

nonn

The sequence is infinite (see comment in A049532). - Emmanuel Vantieghem, Oct 25 2016

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002522, A013929, A049532, A069987.

[n^2+1: n in [1..500]| not IsSquarefree(n^2+1)]; // Vincenzo Librandi, Oct 26 2016

Select[Range[400]^2+1,!SquareFreeQ[#]&] (* Harvey P. Dale, Sep 15 2016 *)

is(n)=issquare(n-1) && !issquarefree(n) \\ Charles R Greathouse IV, Nov 05 2017

a(n) = A049532(n)^2 + 1.

A218713 a(n) is smallest number such that a(n)^2 + 1 is divisible by 37^n.

Original entry on oeis.org

0, 6, 117, 9466, 800982, 6423465, 756360062, 24900904028, 1019349744435, 15069267560119, 794839706330581, 71333925879937154, 2419512779032508628, 116073623326088126430, 359642847542169431827, 144552623583462302226851, 3523356323886506075746572

Offset: 0

Michel Lagneau, Nov 04 2012

nonn

			a(3) = 9466 because 9466^2+1 =  29 * 37 ^ 3 * 61.

Cf. A002522, A049532, A034939, A218709, A218710, A218712.

b=6;n37=37;jo=Join[{0,b},Table[n37=37*n37;b=PowerMod[b,37,n37];b=Min[b,n37-b],{99}]]

A218714 a(n) is smallest number such that a(n)^2 + 1 is divisible by 41^n.

Original entry on oeis.org

0, 9, 378, 11389, 1251967, 46464143, 2363588163, 92615568742, 287369842623, 112076323050317, 2403749863808044, 56094387104417648, 1156752450536914530, 43970228150195457632, 10132163897314954464899, 503212117431217218892992, 19164391897329672149556204

Offset: 0

Michel Lagneau, Nov 04 2012

nonn

			a(3) = 11389 because 11389^2+1 =  2 * 41 ^ 3 * 941.

Cf. A002522, A049532, A034939, A218709, A218710, A218712, A218713.

b=9;n41=41;jo=Join[{0,b},Table[n41=41*n41;b=PowerMod[b,41,n41];b=Min[b,n41-b],{99}]]

A218562 Numbers k such that k^2 + 1 is divisible by a cube.

Original entry on oeis.org

57, 68, 182, 193, 239, 307, 318, 432, 443, 557, 568, 682, 693, 807, 818, 932, 943, 1057, 1068, 1182, 1193, 1307, 1318, 1432, 1443, 1557, 1568, 1682, 1693, 1807, 1818, 1932, 1943, 1958, 1985, 2057, 2068, 2182, 2193, 2307, 2318, 2432, 2436, 2443, 2557, 2568

Offset: 1

Michel Lagneau, Nov 02 2012

nonn

			239 is in the sequence because 239^2 + 1 =  2 * 13 ^ 4.
1985 is in the sequence because 1985^2 + 1 = 2 * 17 ^ 3 * 401.

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

Cf. A002522, A049532, A034939.

Cf. A000578 (cubes).

Select[Range[2,2600],Max[Transpose[FactorInteger[#^2+1]][[2]]]>2&]

A218715 a(n) is smallest number such that a(n)^2 + 1 is divisible by 53^n.

Original entry on oeis.org

0, 23, 500, 27590, 623098, 23048345, 5041394261, 416081467190, 11331029931180, 50928660480181, 6548598523124085, 2441875986594058601, 76594163421571591377, 7783548304686046882879, 252583670951378815076851, 4392422457122810120236558, 1165802007767335105471573954

Offset: 0

Michel Lagneau, Nov 04 2012

nonn

			a(3) = 27590 because 27590^2+1 =  53 ^ 3 * 5113.

Cf. A002522, A049532, A034939, A218709, A218710, A218712, A218713, A218714.

b=23;n53=53;jo=Join[{0,b},Table[n53=53*n53;b=PowerMod[b,53,n53];b=Min[b,n53-b],{99}]]

A218563 Numbers n such that n^2 + 1 is divisible by a 4th power.

Original entry on oeis.org

182, 239, 443, 807, 1068, 1432, 1693, 2057, 2318, 2682, 2943, 3307, 3568, 3932, 4193, 4557, 4818, 5182, 5443, 5807, 6068, 6432, 6693, 7057, 7318, 7682, 7943, 8307, 8568, 8932, 9193, 9557, 9818, 10182, 10443, 10807, 11068, 11432, 11693, 12057, 12318, 12682

Offset: 1

Michel Lagneau, Nov 02 2012

nonn

Includes all n == 182 or 443 (mod 625). In particular, the sequence has positive asymptotic density. # Robert Israel, Oct 06 2016

			239 is in the sequence because 239^2+1 = 57122 = 2*13^4;
27493 is in the sequence because 27493^2+1 = 755865050 = 2*5^2*17^4*181.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002522, A049532, A034939, A218562.

N:= 100000: # to get all terms <= N
res:= {}:
p:= 2;
while p^4 <= N^2+1 do
  for v in map(t -> subs(t,n), [msolve(n^2+1, p^4)]) do
    res:= res union {seq(k*p^4+v, k = 0 .. (N-v)/p^4)}
  od;
  p:= nextprime(p);
od:
sort(convert(res,list)); # Robert Israel, Oct 06 2016

Select[Range[2,13000],Max[Transpose[FactorInteger[#^2+1]][[2]]]>3&]

cp's OEIS Frontend

A218709 a(n) is smallest number such that a(n)^2 + 1 is divisible by 13^n.

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A049533 Numbers k such that k^2+1 is squarefree.

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A218710 a(n) is smallest number such that a(n)^2 + 1 is divisible by 17^n.

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A218712 a(n) is the smallest number such that a(n)^2 + 1 is divisible by 29^n.

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Maple

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A124809 Numbers of the form (square + 1) that are not squarefree.

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A218713 a(n) is smallest number such that a(n)^2 + 1 is divisible by 37^n.

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A218714 a(n) is smallest number such that a(n)^2 + 1 is divisible by 41^n.

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A218562 Numbers k such that k^2 + 1 is divisible by a cube.

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A218715 a(n) is smallest number such that a(n)^2 + 1 is divisible by 53^n.

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