A049532 -id:A049532 - cp's OEIS frontend

A218716 a(n) is smallest number such that a(n)^2 + 1 is divisible by 61^n.

0, 11, 682, 51412, 6304056, 144762466, 9435321777, 988322434636, 71294762793847, 3138611770750343, 283798117998769727, 15409745938584647495, 320007169218635518122, 45443939732277600209579, 207359227164430355867160, 59053635973003478214807486

Offset: 0

Michel Lagneau, Nov 04 2012

nonn

			a(3) = 51412 because 51412^2+1 = 5 * 17 * 61 ^ 3 * 137.

Cf. A002522, A049532, A034939, A218709, A218710, A218712, A218713, A218714, A218715.

b=11;n61=61;jo=Join[{0,b},Table[n61=61*n61;b=PowerMod[b,61,n61];b=Min[b,n61-b],{99}]]

A069187 Numbers k such that core(k) = ceiling(sqrt(k)) where core(k) is the squarefree part of k (the smallest integer such that k*core(k) is a square).

Original entry on oeis.org

1, 2, 20, 90, 272, 650, 1332, 4160, 6642, 10100, 14762, 20880, 28730, 38612, 50850, 65792, 83810, 130682, 160400, 194922, 234740, 280370, 332352, 391250, 457652, 532170, 615440, 708122, 810900, 924482, 1187010, 1337492, 1501850, 1680912, 1875530, 2314962, 2561600

Offset: 1

Benoit Cloitre, Apr 14 2002

nonn

Conjecture: sequence is A071253 minus those entries of A071253 that have their index in A049532, i.e., a(n) is of form n^2*(n^2+1) for all n not in A049532. - Ralf Stephan, Aug 18 2004

Amiram Eldar, Table of n, a(n) for n = 1..300

Cf. A007913, A049532, A071253.

core[n_] := Times @@ Apply[ Power, {#[[1]], Mod[#[[2]], 2]}& /@ FactorInteger[n], {1}]; Select[Range[500000], core[#] == Ceiling[Sqrt[#]]&] (* Jean-François Alcover, Jul 26 2011 *)

More terms from Amiram Eldar, Sep 10 2020

A124808 Number of numbers k <= n such that k^2 + 1 is squarefree.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 35, 36, 37, 37, 38, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 61, 62, 62, 63, 64

Offset: 0

Reinhard Zumkeller, Nov 08 2006

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A008683, A049532, A059592, A335963.

ListTools:-PartialSums([seq(numtheory:-mobius(k^2+1)^2, k=0..100)]); # Robert Israel, Jul 15 2015

Accumulate[Table[If[SquareFreeQ[k^2+1],1,0],{k,0,80}]] (* Harvey P. Dale, Mar 04 2014 *)
Table[Sum[MoebiusMu[k^2 + 1]^2, {k, 0, n}], {n, 0, 100}] (* Wesley Ivan Hurt, Jul 15 2015 *)

a(n)={my(k,r=0);for(k=0,n,if(issquarefree(k^2+1),r++));return(r);}
main(size)=my(n);vector(size,n,a(n-1)) /* Anders Hellström, Jul 15 2015 */

a(0) = 1, a(n) = a(n-1) + 0^(A059592(n) - 1).
a(n) = Sum_{k=0..n} mu(k^2+1)^2, where mu(n) is the Mobius function (A008683). - Wesley Ivan Hurt, Jul 15 2015
a(n) ~ c*n where c = Product_{p prime, p == 1 (mod 4)} (1 - 2/p^2) = 0.894841... (A335963). - Amiram Eldar, Feb 23 2021

A217798 Numbers n such that n^2 + 1 and (n+1)^2 + 1 are divisible by a square.

Original entry on oeis.org

117, 407, 606, 775, 943, 1193, 1252, 1482, 1743, 1957, 2267, 2563, 3217, 3281, 3309, 3457, 3506, 3618, 3718, 3817, 4007, 4632, 4831, 5168, 5742, 5743, 5845, 6031, 6182, 6492, 6768, 7506, 7843, 8042, 8118, 8331, 8368, 8418, 8707, 8782, 8857, 9056, 9292, 9393

Offset: 1

Michel Lagneau, Oct 12 2012

nonn

Also numbers n such that mu(n^2+1) = mu((n+1)^2+1)=0, where mu is the Moebius-function (A008683).

			117 is in the sequence because 117^2+1 = 2*5*37^2 and 118^2+1 = 5^2*557.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A002522, A068781.

Subsequence of A049532.

A002522:=func; [n: n in [1..10^4]| not IsSquarefree(A002522(n)) and not IsSquarefree(A002522(n+1))]; // Bruno Berselli, Oct 15 2012

with(numtheory):for n from 1 to 10000 do :x:=n^2+1:y:=(n+1)^2+1:if issqrfree(x)=false and issqrfree(y)=false then printf(`%d, `,n):else fi:od:

Select[ Range[2, 10000], Max[ Transpose[ FactorInteger[ #^2+1 ]] [[2]]] > 1 && Max[ Transpose[ FactorInteger[ (#+1)^2 + 1]] [[2]]] > 1 &]

A218564 Numbers n such that n^2 + 1 is divisible by a 5th power.

Original entry on oeis.org

1068, 2057, 4193, 5182, 7318, 8307, 10443, 11432, 13568, 14557, 16693, 17682, 19818, 20807, 22943, 23932, 26068, 27057, 29193, 30182, 32318, 33307, 35443, 36432, 38568, 39557, 41693, 42682, 44818, 45807, 47943, 48932, 51068, 52057, 54193, 55182, 57318, 58307

Offset: 1

Michel Lagneau, Nov 02 2012

nonn

For each prime p == 1 (mod 4), there are two values of x (mod p^5) that solve x^2 + 1 == 0 (mod p^5), and then x + k*p^5 is in the sequence for every k. Thus the asymptotic density of this sequence should be 1 - Product_p (1 - 2/p^5), where the product is over all primes p == 1 (mod 4). - Robert Israel, Sep 04 2018

			1068 is in the sequence because 1068^2+1 = 1140625 = 5^6*73;
143044 is in the sequence because 143044^2+1 = 20461585937 = 13^5*55109;
390112 is in the sequence because 390112^2+1 = 152187372545 = 5*13*17^6*97.

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

Cf. A002522, A049532, A034939, A218562, A218563.

N:= 10^5: # to get all terms <= N
P:= select(isprime,[seq(i,i=5..floor((N^2+1)^(1/5)),4)]):
g:= proc(x,r,N) local t; t:= rhs(op(x)); seq(t+r*k,k=0..(N-t)/r) end proc:
R:= `union`(seq(map(g, {msolve(n^2+1,p^5)},p^5,N),p=P)):
sort(convert(R,list)); # Robert Israel, Sep 04 2018

Select[Range[2,20000],Max[Transpose[FactorInteger[#^2+1]][[2]]]>4&]

isok(n) = vecmax(factor(n^2+1)[,2]) >= 5; \\ Michel Marcus, Sep 04 2018

A218717 a(n) is smallest number such that a(n)^2 + 1 is divisible by 73^n.

Original entry on oeis.org

0, 27, 776, 153765, 6459524, 404034898, 41865466758, 3219884218827, 239822883201307, 9110883894036198, 991706090146518323, 142813358470363920740, 8641533837443707913816, 586811715371303018585730, 2756887299416274753296336, 729513196939063257288876118

Offset: 0

Michel Lagneau, Nov 04 2012

nonn

			a(3) = 153765 because 153765^2+1 = 2 * 73 ^ 3 * 30389.

Cf. A002522, A049532, A034939, A218709, A218710, A218712, A218713, A218714, A218715, A218716.

b=27;n73=73;jo=Join[{0,b},Table[n73=73*n73;b=PowerMod[b,73,n73];b=Min[b,n73-b],{99}]]

A124895 Triangle read by rows, 1 <= k <= n: T(n,k) = mu(n^2 + k^2) with mu = A008683.

Original entry on oeis.org

-1, -1, 0, 1, -1, 0, -1, 0, 0, 0, 1, -1, 1, -1, 0, -1, 0, 0, 0, -1, 0, 0, -1, 1, 1, 1, 1, 0, 1, 0, -1, 0, -1, 0, -1, 0, 1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 1, 0, -1, -1, 1, -1, -1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, -1, -1, 1, 1, 1, 1, 1, -1, 0, -1, -1, -1, 0, -1, 0, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, 1, 0, 1, -1, 0, -1, 0, 0, 1, 0, 0, 0, 1, 0, 1, -1, 0

Offset: 1

Reinhard Zumkeller, Nov 12 2006

			Triangle begins:
 -1
 -1,  0
  1, -1,  0
 -1,  0,  0,  0
  1, -1,  1, -1,  0
 -1,  0,  0,  0, -1, 0
  0, -1,  1,  1,  1, 1,  0
  1,  0, -1,  0, -1, 0, -1, 0
  1,  1,  0, -1,  1, 0, -1, 1,  0
 -1,  0, -1,  0,  0, 0, -1, 0, -1, 0

Cf. A000007, A008683, A049532, A049533, A070216, A124897.

row[n_] := Table[MoebiusMu[n^2 + k^2], {k, 1, n}]; Array[row, 15] // Flatten (* Amiram Eldar, May 12 2025 *)

row(n) = vector(n, k, moebius(n^2 + k^2)); \\ Amiram Eldar, May 12 2025

T(n,k) = A008683(A070216(n,k)).
T(n,1) = A124897(1); T(A049533(n),1) <> 0; T(A049532(n),1) = 0.
T(n,n) = -A000007(n-1).
A124896(n) = Sum_{k=1..n} abs(T(n,k)), row sums of absolute values.

A218565 Numbers k such that k^2 + 1 is divisible by a 6th power.

Original entry on oeis.org

1068, 14557, 16693, 30182, 32318, 45807, 47943, 61432, 63568, 77057, 79193, 92682, 94818, 108307, 110443, 123932, 126068, 139557, 141693, 155182, 157318, 170807, 172943, 186432, 188568, 202057, 204193, 217682, 219818, 233307, 235443, 248932, 251068, 264557

Offset: 1

Michel Lagneau, Nov 02 2012

nonn

			1068 is in the sequence because 1068^2 + 1 = 5^6 * 73.
390112 is in the sequence because 390112^2 + 1 = 5 * 13 * 17 ^ 6 * 97.
1999509 is in the sequence because 1999509^2 + 1 = 2 * 13 ^ 6 * 29 * 14281.

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

Cf. A002522, A049532, A034939, A218562, A218563, A218564.

Cf. A001014 (6th powers).

Select[Range[2,27000],Max[Transpose[FactorInteger[#^2+1]][[2]]]>5&]

A218718 a(n) is smallest number such that a(n)^2 + 1 is divisible by 89^n.

Original entry on oeis.org

0, 34, 3861, 344464, 20099637, 2153335831, 102666405913, 4867146503697, 923990886302412, 50251663587824641, 5655954122907587985, 909925832091926912414, 85120439454684773642745, 2631773999763198769695986, 41332517834853462204330752

Offset: 0

Michel Lagneau, Nov 04 2012

nonn

			a(3) = 344464 because 344464^2+1 = 37 * 89 ^ 3 * 4549.

Cf. A002522, A049532, A034939, A218709, A218710, A218712, A218713, A218714, A218715, A218716, A218717.

b=34;n89=89;jo=Join[{0,b},Table[n89=89*n89;b=PowerMod[b, 89,n89];b=Min[b,n89-b],{99}]]

A260824 Least positive integer b such that b^(2^n)+1 is not squarefree.

Original entry on oeis.org

3, 7, 110, 40, 392, 894, 315, 48

Offset: 0

Jeppe Stig Nielsen, Aug 04 2015

For any n, a(n) <= A261117(n).
The smallest square in the factors of b^(2^n)+1 are 2^2, 5^2, 17^2, 17^2, 769^2. - Robert Price, Mar 07 2017; edited by Jeffrey Shallit, May 10 2017
a(8) <= 50104 (corresponding square 10753^2). - Jeffrey Shallit, May 10 2017
Some better bounds than A261117(n): a(9) <= 65863 (factor 13313^2), a(12) <= 265801 (factor 65537^2), a(16) <= 1493667 (factor 1179649^2), a(18) <= 15834352 (factor 7340033^2), a(19) <= 15786037 (factor 23068673^2), a(21) <= 78597313 (factor 230686721^2), a(22) <= 13753565041 (factor 469762049^2), a(23) <= 6276931961 (factor 469762049^2). - Max Alekseyev, Feb 20 2018

			a(1) = A049532(1) = 7.
For n=4, we consider b^16+1. The first time it is not squarefree is for b=392, where 392^16+1 is divisible by 769^2. So a(4)=392.

Subsequence of A248214.

Cf. A261117, A049532.

a(n) = for(b=1,10^42, !issquarefree(b^(2^n)+1) & return(b) );

from sympy.ntheory.factor_ import core
def a(n):
  b, pow2, t = 1, 2**n, 2
  while core(t, 2) == t:
    b += 1
    t = b**(pow2) + 1
  return b
print([a(n) for n in range(4)]) # Michael S. Branicky, Mar 07 2021

a(n) = A248214(2^n).

Edited and a(5)-a(7) added by Max Alekseyev, Feb 20 2018

cp's OEIS Frontend

A218716 a(n) is smallest number such that a(n)^2 + 1 is divisible by 61^n.

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A069187 Numbers k such that core(k) = ceiling(sqrt(k)) where core(k) is the squarefree part of k (the smallest integer such that k*core(k) is a square).

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A124808 Number of numbers k <= n such that k^2 + 1 is squarefree.

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A217798 Numbers n such that n^2 + 1 and (n+1)^2 + 1 are divisible by a square.

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A218564 Numbers n such that n^2 + 1 is divisible by a 5th power.

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A218717 a(n) is smallest number such that a(n)^2 + 1 is divisible by 73^n.

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A124895 Triangle read by rows, 1 <= k <= n: T(n,k) = mu(n^2 + k^2) with mu = A008683.

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A218565 Numbers k such that k^2 + 1 is divisible by a 6th power.

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