A129293 -id:A129293 - cp's OEIS frontend

A070155 Numbers k such that k-1, k+1 and k^2+1 are prime numbers.

4, 6, 150, 180, 240, 270, 420, 570, 1290, 1320, 2310, 2550, 2730, 3360, 3390, 4260, 4650, 5850, 5880, 6360, 6780, 9000, 9240, 9630, 10530, 10890, 11970, 13680, 13830, 14010, 14550, 16230, 16650, 18060, 18120, 18540, 19140, 19380, 21600, 21840

Offset: 1

Labos Elemer, Apr 23 2002

Essentially the same as A129293. - R. J. Mathar, Jun 14 2008
Solutions to the equation: A000005(n^4-1) = 8. - Enrique Pérez Herrero, May 03 2012
Terms > 6 are multiples of 30. Subsequence of A070689. - Zak Seidov, Nov 12 2012
{a(n)-1} is a subsequence of A157468; for n>1, {a(n)^2+2} is a subsequence of A242720. - Vladimir Shevelev, Aug 31 2014

			150 is a term since 149, 151 and 22501 are all primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000005, A000720, A001359, A006512, A014574, A002496, A005574, A070156, A070689, A129293, A157468, A242720.

select(n -> isprime(n-1) and isprime(n+1) and isprime(n^2+1), [seq(2*i,i=1..10000)]); # Robert Israel, Sep 02 2014

Do[s=n; If[PrimeQ[s-1]&&PrimeQ[s+1]&&PrimeQ[1+s^2], Print[n]], {n, 1, 1000000}]
Select[Range[22000],AllTrue[{#+1,#-1,#^2+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 19 2014 *)

is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+1); \\ Amiram Eldar, Apr 15 2024

For n>1, a(n)^2 = A242720(pi(a(n)-2)) - 2, where pi(n) is the prime counting function (A000720). - Vladimir Shevelev, Sep 02 2014

A129297 Nonnegative integers m such that m^2-1 has no divisors d with 1

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, 312, 348, 420, 432, 462, 522, 570, 600, 618, 642, 660, 810, 822, 828, 858, 882, 1020, 1032, 1050, 1062, 1092, 1152, 1230, 1278, 1290, 1302, 1320, 1428, 1452, 1482, 1488

Offset: 1

Reinhard Zumkeller, Apr 09 2007

nonn

Since m^2-1 = (m+1)(m-1), this sequence is just 0,1,2,3, and the average of twin prime pairs A014574.

			{1,41,43,1763} is the set of divisors of 42^2-1, therefore 42 is a term, A129296(42) = #{1,41} = 2.

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

Cf. A014574, A129293, A129295, A129296.

nniQ[n_]:=Count[Rest[Divisors[n^2-1]],?(#<n-1&)]==0; Join[{0,1}, Select[ Range[2,1500],nniQ]] (* _Harvey P. Dale, Aug 09 2015 *)
Select[Range[0, 1500], #<4 || And @@ PrimeQ[# + {-1, 1}] &] (* Amiram Eldar, Jun 18 2022 *)

isA129297(n) = (n <= 3) || divisors(n^2-1)[2] >= n-1

A129296(a(n)) = #{1, a(n)-1} = 2;

a(n) = A014574(n-4) for n>4.

A129292 Number of divisors of n^4 - 1 that are not greater than n.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 4, 5, 3, 8, 3, 10, 4, 6, 4, 12, 4, 13, 4, 11, 6, 14, 3, 10, 6, 12, 6, 17, 3, 16, 7, 10, 9, 13, 4, 18, 7, 11, 4, 22, 3, 26, 8, 9, 7, 23, 5, 18, 7, 13, 6, 25, 4, 24, 8, 21, 6, 18, 3, 18, 10, 12, 14, 16, 4, 26, 8, 17, 7, 31, 5, 30, 6, 11, 13, 26, 7, 25, 6, 16, 10, 35, 4, 18, 11

Offset: 1

Reinhard Zumkeller, Apr 09 2007

nonn

a(n) = #{d: d<=n and A123865(n) mod d = 0};

a(n)>1 for n>2, see A129293 for m such that a(m)=2: a(A129293(n))=2.

			a(100) = #{1,3,9,11,33,73,99} = 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A129294, A129296.

f:= n -> nops(select(`<=`,numtheory:-divisors(n^4-1),n)):
1,seq(f(n), n=2..100); # Robert Israel, Sep 21 2014

Table[Count[Divisors[n^4-1],?(#<=n&)],{n,90}] (* _Harvey P. Dale, Aug 13 2014 *)

a(n) = if (n==1, 1, sumdiv(n^4-1, d, d <= n)); \\ Michel Marcus, Sep 21 2014

A129295 Numbers m such that m^3 - 1 has no divisors d with 1 < d < m - 1.

Original entry on oeis.org

3, 4, 6, 8, 12, 14, 20, 24, 38, 54, 62, 80, 90, 110, 138, 150, 164, 168, 192, 194, 272, 278, 314, 332, 348, 398, 402, 434, 500, 572, 642, 644, 720, 728, 762, 798, 812, 860, 864, 878, 920, 992, 1020, 1022, 1070, 1092, 1098, 1118, 1130, 1182, 1202, 1230, 1260, 1308

Offset: 1

Reinhard Zumkeller, Apr 09 2007

nonn

Numbers m such that A129294(m) = #{1,m-1} = 2.

Essentially the same as A096175. Note that m^3 - 1 = (m - 1)*(m^2 + m + 1), so m - 1 must be prime. For m > 4, the smallest divisor > 1 of m^2 + m + 1 is no larger than sqrt(m^2 + m + 1) < m + 1 unless m^2 + m + 1 is also prime. Also note that gcd(m, m^2 + m 1 ) = gcd(m - 1, m^2 + m + 1) = 1, so m^2 + m + 1 must also be prime, making m^3 - 1 a semiprime. - Jianing Song, Aug 01 2018

			{1,11,157,1727} is the set of divisors of 12^3 - 1, therefore 12 is a term, since A129294(12) = #{1,11} = 2.

Cf. A096175, A129293, A129294.

a(n) = A096175(n-2) for n > 2. - Jianing Song, Aug 01 2018

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A070155 Numbers k such that k-1, k+1 and k^2+1 are prime numbers.

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A129297 Nonnegative integers m such that m^2-1 has no divisors d with 1

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A129292 Number of divisors of n^4 - 1 that are not greater than n.

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A129295 Numbers m such that m^3 - 1 has no divisors d with 1 < d < m - 1.

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