Dmitry Ezhov - cp's OEIS frontend

A306666 Positive integers x such that x(7-5x+x^2)(6-4x+x^2) is a square.

1, 2, 3, 7, 21

Offset: 1

Dmitry Ezhov, Mar 04 2019

No other terms below 10^6.

No other terms below 10^10. - Chai Wah Wu, Nov 24 2019

dxdy forum, x(7-5x+x^2)(6-4x+x^2)=y^2 in natural numbers (in Russian).

Select[Range[1000], IntegerQ[Sqrt[#*(7 - 5*# + #^2)*(6 - 4*# + #^2)]] &] (* Vaclav Kotesovec, Mar 10 2019 *)

isok(x)=issquare(x*(7-5*x+x^2)*(6-4*x+x^2));

from sympy.ntheory.primetest import is_square
A306666_list = [n for n in range(1,10**3) if is_square(n*(n*(n*(n*(n - 9) + 33) - 58) + 42))] # Chai Wah Wu, Nov 24 2019

A320076 a(n) is smallest positive integer i such that difference of numerator and denominator of sum of j^(-i), when j=1..n and n > 2, is prime.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 32, 1

Offset: 3

Dmitry Ezhov, Oct 05 2018

a(11) > 6360.
a(11) > 12000. - Chai Wah Wu, Nov 15 2018
a(19) = a(20) = a(26) = a(30) = a(31) = a(33) = a(40) = 1, a(44) = a(48) = a(49) = 2, a(42) = 3, a(14) = 5, a(24) = a(46) = 7, a(12) = 8, a(13) = 17, a(47) = 19, a(25) = 49, a(38) = 54, a(37) = 179, a(16) = 207, a(22) = 676, a(18) = 690, a(43) = 880, a(17) = 1068, a(34) = 1199. - Chai Wah Wu, Nov 20 2018
a(15) = 2590, a(23) = 3734. - Chai Wah Wu, Nov 21 2018

Cf. A320077.

a[n_] := Do[s = HarmonicNumber[n, r]; If[PrimeQ[Numerator[s] - Denominator[s]], Return[r]], {r, 1, Infinity}]; Table[a[n], {n, 3, 10}] (* Vaclav Kotesovec, Nov 14 2018 *)

a(n)={for(i=1, +oo, s=sum(j=1, n, j^(-i)); p=numerator(s); q=denominator(s); if(ispseudoprime(p-q), return(i)))};

A320077 a(n) is smallest positive integer i such that sum of numerator and denominator of sum of j^(-i), when j=1..n, is prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 34, 1, 1, 5

Offset: 1

Dmitry Ezhov, Oct 05 2018

a(17) > 7000, a(32) = 2015.

a(18) = a(25) = a(31) = 6, a(19) = a(22) = a(37) = 5, a(20) = a(27) = a(35) = a(36) = a(39) = a(48) = 1, a(23) = a(38) = a(49) = 2, a(24) = a(29) = a(42) = 3, a(26) = 12, a(28) = 75, a(30) = 8, a(33) = 7, a(41) = 121, a(44) = 1052, a(46) = 125, a(47) = 1527. - Chai Wah Wu, Nov 19 2018

dxdy Blog, MSE Crusher Questioner - will anyone try? (in Russian).
Math StackExchange, Is there are similar conjecture like this??, Sep 2018.

Cf. A320076.

a[n_] := Do[s = HarmonicNumber[n, r]; If[PrimeQ[Numerator[s] + Denominator[s]], Return[r]], {r, 1, Infinity}]; Table[a[n], {n, 1, 16}] (* Vaclav Kotesovec, Nov 14 2018 *)

a(n)={for(i=1, +oo, s=sum(j=1, n, j^(-i)); p=numerator(s); q=denominator(s); if(ispseudoprime(p+q), return(i)))};

A282669 Prime numbers p > 3 such that 2^p - 9 is prime.

Original entry on oeis.org

5, 11, 17, 251, 563, 21011

Offset: 1

Dmitry Ezhov, Mar 07 2017

Let W = 2^p - 9 and s = (W+7)/(2*p), then 3^s == 4 (mod W) for terms 1..6.

a(7) > 3480081 using A059610. - Michael S. Branicky, Jan 27 2025

Prime terms of A059610.

Select[Prime[Range[3,565]],PrimeQ[2^#-9]&] (* The program generates the first five terms of the sequence. *) (* Harvey P. Dale, Aug 24 2024 *)

forprime(p=5, 10^5, W= 2^p-9; if(ispseudoprime(W), print1(p, ", ")))

A283266 Prime numbers p such that 2^p - 3 is prime.

Original entry on oeis.org

3, 5, 29, 233, 42689, 69337

Offset: 1

Dmitry Ezhov, Mar 04 2017

Let W = 2^p - 3 and s = (W+1)/(2*p), then 3^s == -2 (mod W) for terms 1..6.

a(7) > 2086750 using A050414. - Michael S. Branicky, Jan 27 2025

Prime terms in A050414.

forprime(p=2, 10^5, W= 2^p-3; if(ispseudoprime(W), print1(p, ", ")))

A283191 Prime numbers p such that (2^p - 5)/3 is prime.

Original entry on oeis.org

7, 13, 19, 31, 373, 811, 1117, 5059, 12601

Offset: 1

Dmitry Ezhov, Mar 02 2017

Let W = (2^p - 5)/3 and s = (W+1)/(2*p), then 5^s == 2 (mod W) for terms 1..9.
Subsequence of 7, 13, 19, 31, 51, 55, 85, 111, 319, 373,.. which are numbers m such that (2^m-5)/3 is prime. - R. J. Mathar, Mar 05 2017
a(10) > 3*10^5. - Michael S. Branicky, Jan 30 2025

Cf. A000978.

Select[Prime@ Range[2, 1000], PrimeQ[(2^# - 5)/3] &] (* Michael De Vlieger, Mar 03 2017 *)

forprime(p=3, 30000, W= (2^p-5)/3; if(ispseudoprime(W), print1(p, ", ")))

A278792 a(n) is the positive integer x such that 3^((M-1)/(2*p)) == -2^x (mod M), where p > 2 is prime, M=2^p-1 is the n-th Mersenne prime and x < p.

Original entry on oeis.org

2, 2, 1, 6, 16, 4, 5, 25, 18, 20, 45, 61, 91, 939, 817, 336, 862, 2533, 3404, 2822, 3136, 1554, 9371, 10712, 21311, 44296, 68185, 66909, 31147, 25648

Offset: 2

Dmitry Ezhov, Nov 28 2016

Let s=(M-1)/(2*p) and z is multiplicative order of 3 modulo M, then M|2^(x+p*i)+3^(s+z*j), where integer i,j>=0.

Cf. A000043, A000668.

A000043=[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657];
for(n=2, #A000043, p= A000043[n]; M=2^p-1; s= (M-1)/2/p; x= valuation(lift(-Mod(3,M)^s), 2); print1(n,": ",x,", "));

A277630 Positive integers n such that 3^n == 8 (mod n).

Original entry on oeis.org

1, 5, 2352527, 193841707, 17126009179703, 380211619942943

Offset: 1

Dmitry Ezhov, Oct 24 2016

No other terms below 10^15. - Max Alekseyev, Sep 13 2017

Solutions to 3^n == k (mod n): A277340 (k=-11), A277289 (k=-7), A277288 (k=-5), A015973 (k=-2), A015949 (k=-1), A067945 (k=1), A276671 (k=2), A276740 (k=5), A277628 (k=6), A277126 (k=7), this sequence (k=8), A277274 (k=11).

PARI
```
isok(n) = Mod(3, n)^n == Mod(8, n);
```

a(5)-a(6) established by Max Alekseyev, Sep 13 2017

A277628 Positive integers n such that 3^n == 6 (mod n).

Original entry on oeis.org

1, 3, 21, 936340943, 10460353197, 9374251222371, 23326283250291, 615790788171551

Offset: 1

Dmitry Ezhov, Oct 24 2016

No other terms below 10^15. - Max Alekseyev, Sep 12 2017

Solutions to 3^n == k (mod n): A277340 (k=-11), A277289 (k=-7), A277288 (k=-5), A015973 (k=-2), A015949 (k=-1), A067945 (k=1), A276671 (k=2), A276740 (k=5), this sequence (k=6), A277126 (k=7), A277630 (k=8), A277274 (k=11).

PARI
```
isok(n) = Mod(3, n)^n == Mod(6, n);
```

a(6)-a(8) from Max Alekseyev, Sep 12 2017

A276740 Numbers n such that 3^n == 5 (mod n).

Original entry on oeis.org

1, 2, 4, 76, 418, 1102, 4687, 7637, 139183, 2543923, 1614895738, 9083990938, 23149317409, 497240757797, 4447730232523, 16000967516764, 65262766108619, 141644055557882

Offset: 1

Dmitry Ezhov, Sep 16 2016

No other terms below 10^15. Some larger terms: 194995887252090239, 2185052151122686482926861593785262. - Max Alekseyev, Oct 13 2016

			3 == 5 (mod 1), so 1 is a term;
9 == 5 (mod 2), so 2 is a term.

Cf. A066601.

Solutions to 3^n == k (mod n): A277340 (k=-11), A277289 (k=-7), A277288 (k=-5), A015973 (k=-2), A015949 (k=-1), A067945 (k=1), A276671 (k=2), this sequence (k=5), A277628 (k=6), A277126 (k=7), A277630 (k=8), A277274 (k=11).

Select[Range[10^7], PowerMod[3, #, #] == Mod[5, #] &] (* Michael De Vlieger, Sep 26 2016 *)

isok(n) = Mod(3, n)^n == Mod(5, n); \\ Michel Marcus, Sep 17 2016

A276740_list = [1,2,4]+[n for n in range(5,10**6) if pow(3,n,n) == 5] # Chai Wah Wu, Oct 04 2016

a(11)-a(13) from Chai Wah Wu, Oct 05 2016
a(14) from Lars Blomberg, Oct 12 2016
a(15)-a(18) from Max Alekseyev, Oct 13 2016
a(12) was missing Robert G. Wilson v, Oct 19 2016

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User: Dmitry Ezhov

A306666 Positive integers x such that x*(7-5*x+x^2)*(6-4*x+x^2) is a square.

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A320076 a(n) is smallest positive integer i such that difference of numerator and denominator of sum of j^(-i), when j=1..n and n > 2, is prime.

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A320077 a(n) is smallest positive integer i such that sum of numerator and denominator of sum of j^(-i), when j=1..n, is prime.

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A282669 Prime numbers p > 3 such that 2^p - 9 is prime.

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A283266 Prime numbers p such that 2^p - 3 is prime.

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A283191 Prime numbers p such that (2^p - 5)/3 is prime.

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A278792 a(n) is the positive integer x such that 3^((M-1)/(2*p)) == -2^x (mod M), where p > 2 is prime, M=2^p-1 is the n-th Mersenne prime and x < p.

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A277630 Positive integers n such that 3^n == 8 (mod n).

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A277628 Positive integers n such that 3^n == 6 (mod n).

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A306666 Positive integers x such that x(7-5x+x^2)(6-4x+x^2) is a square.