A049532 -id:A049532 - cp's OEIS frontend

A268641 Squarefree numbers k such that k^2 + 1 and k^2 - 1 are also squarefree.

2, 6, 14, 22, 30, 34, 42, 58, 66, 78, 86, 94, 102, 106, 110, 114, 130, 138, 142, 158, 166, 178, 186, 194, 202, 210, 214, 222, 230, 238, 254, 258, 266, 286, 302, 310, 322, 330, 346, 354, 358, 366, 390, 394, 398, 402, 410, 430, 434, 438, 446, 454, 462, 466, 470, 498

Offset: 1

K. D. Bajpai, Feb 09 2016

nonn

All the listed terms are even squarefree numbers.

Subsequence of A039956.

			a(2) = 6 = 2 * 3: 6^2 + 1 = 37 = 1 * 37; 6^2 - 1 = 35 = 5 * 7; 6, 37, 35 are all squarefree.

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

Cf. A005117, A007675, A039956, A049532, A049533, A069987, A173186.

[n : n in [1..1000]  |  IsSquarefree(n) and IsSquarefree(n^2+1) and IsSquarefree(n^2-1) ];

select(n -> andmap(issqrfree, [n, n^2+1, n^2-1]), [seq(n, n=2.. 10^3)]);

Select[Range[1000], SquareFreeQ[#] && SquareFreeQ[#^2 + 1] && SquareFreeQ[#^2 - 1] &]

for(n=2, 1000, issquarefree(n) & issquarefree(n^2 + 1) & issquarefree(n^2 - 1) & print1(n,", "))

A218574 Numbers k such that k^2 + 1 is divisible by a 7th power.

Original entry on oeis.org

32318, 45807, 110443, 123932, 188568, 202057, 266693, 280182, 344818, 358307, 422943, 436432, 501068, 514557, 579193, 592682, 657318, 670807, 735443, 748932, 813568, 827057, 891693, 905182, 969818, 983307

Offset: 1

Michel Lagneau, Nov 02 2012

nonn

			32318 is in the sequence because 32318^2 + 1 =  5 ^ 7 * 29 * 461.
6826318 is in the sequence because 6826318^2 + 1 = 5 ^ 3 * 13 ^ 8 * 457.

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

Cf. A002522, A049532, A034939, A218562, A218563, A218564, A218565.

Cf. A001015 (7th powers).

Select[Range[2,1500000],Max[Transpose[FactorInteger[#^2+1]][[2]]]>6&]

A218719 a(n) is smallest number such that a(n)^2 + 1 is divisible by 97^n.

Original entry on oeis.org

0, 22, 4052, 107551, 22709274, 331407850, 197177418061, 26457926739667, 2369608176604944, 76004727767164666, 25163629663367816827, 1965881512952938486496, 191165497320828772935835, 21700278688179406782082106, 560121950820639295011033922

Offset: 0

Michel Lagneau, Nov 04 2012

nonn

			a(3) = 107551 because 107551^2+1 = 2 * 97 ^ 3 * 6337.

Cf. A002522, A049532, A034939, A218709, A218710, A218712, A218713, A218714, A218715, A218716, A218717, A218718.

b=22;n97=97;jo=Join[{0,b},Table[n97=97*n97;b=PowerMod[b, 97,n97];b=Min[b,n97-b],{99}]]

A282092 Numbers m such that there exists at least one integer k < m such that m^2+1 and k^2+1 have the same prime factors.

Original entry on oeis.org

7, 18, 117, 239, 378, 843, 2207, 2943, 4443, 4662, 6072, 8307, 8708, 9872, 31561, 103682, 271443, 853932, 1021693, 3539232, 3699356, 6349657, 6907607, 7042807, 7249325, 9335094, 12623932, 12752043, 12813848, 22211431, 33385282, 42483057, 52374157, 105026693

Offset: 1

Michel Lagneau, Feb 06 2017

nonn

For the pairs (m, k), is k always unique?

The pairs (m, k) are (7, 3), (18, 8), (117, 43), (239, 5), (378, 132), (843, 377), (2207, 987), (2943, 73), (4443, 53), (4662, 1568), (6072, 5118), (8307, 743), (8708, 2112), (9872, 2738), ...

			7 is in the sequence because of the pair (m, k) = (7, 3), 7^2+1 = 2*5^2 and 3^2+1 = 2*5 with the same prime factors 2 and 5.

Subsequence of A049532 (numbers n such that n^2 + 1 is not squarefree).

Cf. A002522, A059591, A059592, A124809.

Select[Range@ 5000, Function[m, Total@ Boole@ Table[Function[w, And[SameQ[First@ w, #], SameQ[Last@ w, #]] &@ Union@ Flatten@ w]@ Map[FactorInteger[#][[All, 1]] &, {m^2 + 1, k^2 + 1}], {k, m - 1}] > 0]] (* Michael De Vlieger, Feb 07 2017 *)

isok(n)=ok = 0; vn = factor(n^2+1)[,1]; for (k=1, n-1, if (factor(k^2+1)[,1] == vn, ok = 1; break);); ok; \\ Michel Marcus, Feb 09 2017

squeeze(f)=factorback(f)\2
list(lim)=my(v=List(),m=Map(),t); for(n=1,lim, t=squeeze(factor(n^2+1)[,1]); if(mapisdefined(m,t), listput(v,n), mapput(m,t,0))); Vec(v) \\ Charles R Greathouse IV, Feb 12 2017

use ntheory qw(:all);
for (my ($m, %t) = 1 ; ; ++$m) {
my $k = vecprod(map{$_->[0]}factor_exp($m**2+1));
push @{$t{$k}}, $m;
if (@{$t{$k}} >= 2) {
print'('.join(', ',reverse(@{$t{$k}})).")\n";
}
} # Daniel Suteu, Feb 08 2017

a(15)-a(29) from Daniel Suteu, Feb 08 2017
a(30) from Daniel Suteu, Feb 10 2017
a(31)-a(34) from Joerg Arndt, Feb 11 2017

A218720 a(n) is smallest number such that a(n)^2 + 1 is divisible by 101^n.

Original entry on oeis.org

0, 10, 515, 296344, 35764191, 1108900220, 316411915250, 47023298541694, 3156215819652023, 310872228812491206, 28124944860980892220, 3783840171259076226254, 208193145695151069244665, 19364218657938636320485082, 663491749602035014400202724

Offset: 0

Michel Lagneau, Nov 04 2012

nonn

			a(3) = 296344 because 296344^2+1 = 101 ^ 3 * 85237.

Cf. A002522, A049532, A034939, A218709, A218710, A218712, A218713, A218714, A218715, A218716, A218717, A218718, A218719.

b=10;n101=101;jo=Join[{0,b},Table[n101=101*n101;b=PowerMod[b, 101,n101];b=Min[b,n101-b],{99}]]

A383672 Squarefree numbers k such that k^2+1 is not squarefree.

Original entry on oeis.org

7, 38, 41, 43, 57, 70, 82, 93, 107, 118, 143, 157, 182, 193, 218, 239, 251, 257, 282, 293, 307, 318, 327, 357, 382, 393, 407, 418, 437, 443, 457, 482, 493, 515, 518, 543, 557, 577, 582, 593, 606, 607, 618, 643, 682, 707, 718, 743, 746, 757, 782, 793, 807, 818, 829, 843, 857, 893

Offset: 1

Alexandre Herrera, May 04 2025

nonn

			38 = 2*19 is squarefree but 38*38 + 1 = 1445 = 5*17*17 is not squarefree.

Intersection of A005117 and A049532.
Includes A141932 and A141941.
Cf. A080666, A224718.

filter:= proc(n) numtheory:-issqrfree(n) and not numtheory:-issqrfree(n^2+1) end proc:
select(filter, [$1..1000]); # Robert Israel, May 04 2025

Select[Range[900],SquareFreeQ[#] && !SquareFreeQ[#^2+1] &] (* Stefano Spezia, May 04 2025 *)

isok(k) = issquarefree(k) && !issquarefree(k^2+1); \\ Michel Marcus, May 04 2025

from sympy import factorint
def is_squarefree(n):
    return all(exponent == 1 for exponent in factorint(n).values())
print([a for a in range(1,900) if is_squarefree(a) and not(is_squarefree(a*a + 1))])

cp's OEIS Frontend

A268641 Squarefree numbers k such that k^2 + 1 and k^2 - 1 are also squarefree.

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A218574 Numbers k such that k^2 + 1 is divisible by a 7th power.

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A218719 a(n) is smallest number such that a(n)^2 + 1 is divisible by 97^n.

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Mathematica

A282092 Numbers m such that there exists at least one integer k < m such that m^2+1 and k^2+1 have the same prime factors.

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A218720 a(n) is smallest number such that a(n)^2 + 1 is divisible by 101^n.

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Mathematica

A383672 Squarefree numbers k such that k^2+1 is not squarefree.

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Maple

Mathematica

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Python