A207982 -id:A207982 - cp's OEIS frontend

A208643 Least positive integer m such that those k*(k-1) mod m with k=1,...,n are pairwise distinct.

1, 3, 5, 7, 11, 11, 13, 16, 17, 19, 23, 23, 29, 29, 29, 31, 37, 37, 37, 41, 41, 43, 47, 47, 53, 53, 53, 59, 59, 59, 61, 64, 67, 67, 71, 71, 73, 79, 79, 79, 83, 83, 89, 89, 89, 97, 97, 97, 97, 101, 101, 103, 107, 107, 109, 113, 113, 127, 127, 127

Offset: 1

Zhi-Wei Sun, Feb 29 2012

nonn

On Feb. 29, 2012, Zhi-Wei Sun proved that a(n) = min{m>2n-2: m is a prime or a power of two}. He also showed that if we replace k(k-1) in the definition of a(n) by 2k(k-1) then a(n) is the least prime greater than 2n-2 for every n=2,3,4,....

Zhi-Wei Sun, Table of n, a(n) for n = 1..500
Zhi-Wei Sun, A function taking only prime values, a message to Number Theory List, Feb. 21, 2012.
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.

Cf. A000040, A207982, A208494.

R[n_,i_] := Union[Table[Mod[k(k-1),i], {k,1,n}]]; Do[Do[If[Length[R[n,i]]==n, Print[n," ",i]; Goto[aa]], {i,1,4n}]; Print[n]; Label[aa]; Continue, {n,1,1000}]

A208494 Least integer m>1 such that those k! mod m with k=1,...,n are pairwise distinct.

Original entry on oeis.org

2, 2, 3, 7, 10, 13, 13, 13, 31, 37, 37, 37, 61, 61, 61, 83, 83, 83, 127, 127, 127, 127, 127, 179, 179, 179, 179, 179, 193, 193, 193, 193, 193, 193, 193, 193, 277, 277, 277, 277, 277, 277, 383, 383, 383, 383, 383, 479, 479, 479, 479, 479, 479, 479, 541, 541, 541, 541, 541, 541, 541, 541, 541, 641, 641, 641, 641, 641, 641, 641, 641, 641, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 1013, 1013, 1013, 1013, 1013, 1013, 1013, 1013, 1013, 1013, 1013

Offset: 1

Zhi-Wei Sun, Feb 27 2012

On Feb 27 2012 Zhi-Wei Sun conjectured that a(n) is a prime with the only exception a(5)=10, and that a(n) does not exceed n^2/2 for all n=2,3,4,... He also conjectured that max{n>0: 1!,...,n! are pairwise incongruent mod p} is asymptotically equivalent to sqrt(p), where p is an odd prime.
He guessed that if we replace k! in the definition of a(n) by (-1)^k*k! then a(n) is a prime with the only exception a(3)=6. If we replace k! in the definition of a(n) by (2k)! or (-1)^k*(2k)!, then Zhi-Wei Sun conjectured that a(n) will take only prime values.
He also has similar conjectures involving (r*k)! or (-1)^k*(r*k)! with r>2.

			For n=5 we have a(5)=10 since 1!=1, 2!=2, 3!=6, 4!=24 and 5!=120 are pairwise incongruent mod 10 but not pairwise incongruent modulo any of 2,3,...,9.

Zhi-Wei Sun and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 400 terms from Sun)
Zhi-Wei Sun, A function taking only prime values, a message to Number Theory List, Feb 21 2012.
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.

Cf. A000142, A207982.

R[n_,i_]:=Union[Table[Mod[k!,i],{k,1,n}]]
Do[Do[If[Length[R[n,i]]==n,Print[n," ",i];Goto[aa]],{i,2,Max[n^2,2]}];
Print[n];Label[aa];Continue,{n,1,1000}]

has(n,m)=my(t=1); #Set(vector(n,i,t=(t*i)%m))==n
a(n,last=2)=while(!has(n,last), last++); last
t=2;vector(100,n,t=a(n,t)) \\ Charles R Greathouse IV, Jul 31 2016

A210144 a(n) = least integer m>1 such that the product of the first k primes for k=1,...,n are pairwise distinct modulo m.

Original entry on oeis.org

2, 3, 5, 11, 11, 23, 29, 37, 37, 41, 47, 47, 47, 47, 47, 73, 131, 131, 131, 131, 131, 151, 151, 151, 151, 199, 223, 223, 271, 271, 271, 281, 281, 281, 281, 281, 281, 281, 281, 281, 353, 353, 457, 457, 457, 457, 457, 457, 457, 457, 457, 641, 641, 641, 641, 641, 643, 643, 643, 643

Offset: 1

Zhi-Wei Sun, Mar 17 2012

nonn

Conjecture: all the terms are primes and a(n) 1.

			a(3)=5 because 2, 2*3=6, 2*3*5=30 are distinct modulo m=5 but not distinct modulo m=2,3,4.

Jason Yuen, Table of n, a(n) for n = 1..10000 (first 1172 terms from Zhi-Wei Sun)
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2023. - From _N. J. A. Sloane_, Jun 13 2012
Zhi-Wei Sun, A function taking only prime values, a message to Number Theory List, Feb. 21, 2012.
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.

Cf. A000040, A207982, A208494, A208643, A214196.

R[n_,m_]:=Union[Table[Mod[Product[Prime[j],{j,1,k}],m],{k,1,n}]]
Do[Do[If[Length[R[n,m]]==n,Print[n," ",m];Goto[aa]],{m,2,Max[2,n^2]}];
Print[n];Label[aa];Continue,{n,1,1000}]

A210186 a(n) = least integer m>1 such that m divides none of P_i + P_j with 0

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 23, 23, 23, 47, 59, 61, 71, 71, 71, 101, 101, 101, 101, 101, 101, 113, 113, 113, 113, 113, 113, 113, 113, 113, 223, 223, 223, 223, 223, 223, 223, 223, 223, 223, 223, 223, 223, 223, 223, 223, 223, 223, 487, 487, 661, 661, 661, 661, 661, 661, 661, 661, 661, 719, 719, 719, 719, 719, 719, 811, 811, 811, 811, 811, 811, 811, 811, 811, 811

Offset: 1

Zhi-Wei Sun, Mar 18 2012

nonn

Conjecture: all the terms are primes and a(n) < n^2 for all n > 1.

			We have a(3)=5 since 2+2*3, 2+2*3*5, 2*3+2*3*5 are pairwise distinct modulo m=5 but not pairwise distinct modulo m=2,3,4.

Zhi-Wei Sun, Table of n, a(n) for n = 1..258
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018. - From _N. J. A. Sloane_, Jun 13 2012
Zhi-Wei Sun, A function taking only prime values, message to Number Theory List, Feb. 21, 2012.
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory, Vol. 133, No. 8 (2013), pp. 2794-2812.

Cf. A000040, A210144, A208494, A208643, A207982.

P[n_]:=Product[Prime[k],{k,1,n}]
R[n_,m_]:=Product[If[Mod[P[k]+P[j],m]==0,0,1],{k,2,n},{j,1,k-1}]
Do[Do[If[R[n,m]==1,Print[n," ",m];Goto[aa]],{m,2,Max[2,n^2]}]; Print[n];Label[aa];Continue,{n,1,300}]

A210393 a(n) = least integer m>1 such that S_k! for k=1,...,n are pairwise distinct modulo m where S_k is the sum of the first k primes.

Original entry on oeis.org

2, 3, 7, 13, 19, 29, 43, 61, 79, 101, 131, 167, 199, 239, 293, 331, 389, 443, 503, 571, 641, 719, 797, 877, 971, 1063, 1163, 1277, 1373, 1481, 1601, 1721, 1861, 1997, 2131, 2281, 2437, 2591, 2753, 2927, 3089, 3271, 3457, 3659, 3847, 4049, 4231, 4441, 4663, 4889

Offset: 1

Zhi-Wei Sun, Mar 20 2012

nonn

When n>1, we have S_n!=S_{n-1}!=0 (mod m) for all m=1,...,S_{n-1} and hence a(n)>S_{n-1}. Zhi-Wei Sun conjectured that a(n) is always a prime not exceeding S_n.

			a(3)=7 since 2!,(2+3)!,(2+3+5)! are pairwise distinct modulo m=7 but not pairwise distinct modulo m=2,3,4,5,6.

Zhi-Wei Sun, Table of n, a(n) for n = 1..720
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.

Cf. A000040, A210394, A210186, A210144, A208494, A208643, A207982.

s[n_]:=s[n]=Sum[Prime[k],{k,1,n}]
f[n_]:=f[n]=s[n]!
R[n_,m_]:=Union[Table[Mod[f[k],m],{k,1,n}]]
Do[Do[If[Length[R[n,m]]==n,Print[n," ",m];Goto[aa]],{m,Max[2,s[n-1]],s[n]}];
   Print[n];Label[aa];Continue,{n,1,720}]

A210394 a(n) = least integer m>1 such that m divides none of S_i!+S_j! with 0

Original entry on oeis.org

2, 3, 7, 11, 19, 31, 43, 67, 79, 101, 131, 163, 199, 241, 283, 331, 383, 443, 503, 571, 641, 719, 797, 877, 967, 1061, 1163, 1277, 1373, 1481, 1597, 1721, 1871, 1999, 2129, 2281, 2437, 2593, 2749, 2927, 3089, 3271, 3457, 3643, 3833, 4057, 4229, 4441, 4673, 4889

Offset: 1

Zhi-Wei Sun, Mar 20 2012

nonn

When n>1, we have S_n!+S_{n-1}!=0 (mod m) for all m=1,...,S_{n-1} and hence a(n)>S_{n-1}. Zhi-Wei Sun conjectured that a(n) is always a prime not exceeding S_n.

			We have a(3)=7, since  m=7 divides none of 2!+(2+3)!,2!+(2+3+5)!,(2+3)!+(2+3+5)! but this fails for m=2,3,4,5,6.

Zhi-Wei Sun, Table of n, a(n) for n = 1..225
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.

Cf. A000040, A210393, A210186, A210144, A208494, A208643, A207982.

s[n_]:=s[n]=Sum[Prime[k],{k,1,n}]
f[n_]:=s[n]!
R[n_,m_]:=Product[If[Mod[f[k]+f[j],m]==0,0,1],{k,2,n},{j,1,k-1}]
Do[Do[If[R[n,m]==1,Print[n," ",m];Goto[aa]],{m,Max[2,s[n-1]],s[n]}];
   Print[n];Label[aa];Continue,{n,1,225}]

A181901 a(n) = least positive integer m such that 2(s_k)^2 for k=1,...,n are pairwise distinct modulo m where s_k = Sum_{j=1..k} (-1)^(k-j)*p_j and p_j is the j-th prime.

Original entry on oeis.org

1, 4, 7, 9, 13, 17, 19, 23, 25, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233

Offset: 1

Zhi-Wei Sun, Mar 31 2012

nonn

On Mar 28 2012, Zhi-Wei Sun conjectured that a(n) is the (n+1)-th prime p_{n+1} with the only exceptions being a(1)=1, a(2)=4, a(4)=9 and a(9)=25. He has shown that 2(s_k)^2 (k=1,...,n) are indeed pairwise distinct modulo p_{n+1} and hence a(n) does not exceed p_{n+1}.
Note that the sequence 0,s_1,s_2,s_3,... is A008347 introduced by N. J. A. Sloane and J. H. Conway.
Compare a(n) with the sequence A210640.
The conjecture was verified for n up to 2*10^5 by the author in 2018, and for n up to 3*10^5 by Chang Zhang (a student at Nanjing Univ.) in June 2020. - Zhi-Wei Sun, Jun 22 2020

			We have a(4)=9 since 2(s_1)^2=8, 2(s_2)^2=2, 2(s_3)^2=32, 2(s_4)^2=18 are pairwise distinct modulo 9 but not pairwise distinct modulo any of 1,...,8.

Zhi-Wei Sun, Table of n, a(n) for n = 1..600
Zhi-Wei Sun, An amazing recurrence for primes, a message to Number Theory List, March 31, 2012.
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.

Cf. A000040, A008347, A210640, A210393, A210394, A210186, A210144, A208494, A208643, A207982.

s[n_]:=Sum[(-1)^k*Prime[k],{k,1,n}]
f[n_]:=2*s[n]^2
R[n_,m_]:=Union[Table[Mod[f[k],m],{k,1,n}]]
Do[Do[If[Length[R[n,m]]==n,Print[n," ",m];Goto[aa]],{m,1,Prime[n+1]}];
   Print[n];Label[aa];Continue,{n,1,600}]

A210640 a(n) = least integer m > 1 such that 2S_k^2 (k=1,...,n) are pairwise distinct modulo m, where S_k is the sum of the first k primes.

Original entry on oeis.org

2, 4, 9, 13, 17, 28, 37, 37, 37, 37, 37, 61, 61, 61, 151, 151, 151, 151, 151, 151, 151, 227, 227, 227, 227, 227, 307, 307, 307, 337, 433, 433, 433, 433, 433, 433, 433, 433, 433, 433, 433, 509, 509, 509, 509, 509, 643, 727, 727, 761, 761, 761, 971, 971, 971

Offset: 1

Zhi-Wei Sun, Mar 26 2012

nonn

If a(n) is an odd prime p_k then a(n)|2(S_k^2-S_{k-1}^2) and hence k>n. Zhi-Wei Sun conjectured that a(n) is a prime smaller than n^2 unless n divides 6.

			We have a(3)=9, because 2*2^2=8, 2*(2+3)^2=50, 2(2+3+5)^2=200 are pairwise distinct modulo m=9 but not pairwise distinct modulo any of 2, 3, 4, 5, 6, 7, 8.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5258
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.
Zhi-Wei Sun, On sums of consecutive primes, a message to Number Theory List, March 23, 2012.

Cf. A000040, A210393, A210394, A210186, A210144, A208494, A208643, A207982.

s[n_] := s[n] = Sum[Prime[k], {k,1,n}]; f[n_] := f[n] = 2*s[n]^2; R[n_,m_] := Union[Table[Mod[f[k],m], {k,1,n}]]; Do[Do[If[Length[R[n,m]] == n, Print[n," ",m]; Goto[aa]], {m,2,Max[2,n^2]}]; Print[n]; Label[aa]; Continue, {n,1,5000}]

A182003 a(n) = least prime p_k such that n = p_k - p_{k-1} + ... + (-1)^{k-j}p_j for some 1 <= j <= k, with p_k the k-th prime.

Original entry on oeis.org

3, 2, 3, 5, 5, 11, 7, 11, 11, 17, 11, 17, 13, 23, 17, 23, 17, 31, 19, 41, 23, 41, 23, 47, 29, 47, 37, 59, 29, 59, 31, 59, 43, 67, 37, 67, 37, 67, 43, 73, 41, 83, 43, 83, 47, 101, 47, 97, 53, 97, 59, 97, 53, 103, 61, 109, 67, 127, 59, 131

Offset: 1

Zhi-Wei Sun, Apr 05 2012

a(n) is obviously at least n. In March 2012, Zhi-Wei Sun conjectured that a(n) always exists and does not exceed 2.2*n for n > 1; he even thought that 2.2*n might be replaced by 2*n + 2.5*sqrt(n) for n > 1.
IFF n=p, a(n)=1.
For odd n's, lim_{n->inf.} a(n)/n = 1; for even n's, lim_{n->inf.} a(n)/n = 2.

			We have a(4) = 5 since 4 = 5 - 3 + 2 and 3 - 2 < 4.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 5000 terms from Zhi-Wei Sun)
Zhi-Wei Sun, On functions taking only prime values, preprint, arxiv:1202.6589 [math.NT], 2012-2013.
Zhi-Wei Sun, A representation problem involving consecutive primes, a message to Number Theory List, Mar 31 2012.
Zhi-Wei Sun, An amazing recurrence for primes, a message to Number Theory List, Mar 31 2012.

Cf. A000040, A181901, A210640, A210393, A210394, A210186, A210144, A208494, A208643, A207982.

s[0_]:=0; s[n_]:=s[n]=Prime[n]-s[n-1]; Do[Do[If[s[j]-(-1)^(j-i)*s[i]==n, Print[n," ",Prime[j]]; Goto[aa]], {j, 1, PrimePi[3n]}, {i, 0, j-1}]; Print[n]; Label[aa]; Continue, {n, 1, 5000}]
f[n_] := Block[{j = PrimePi[n]}, While[ !MemberQ[ Accumulate@ Table[(-1)^(j - i) Prime[i], {i, j, 1, -1}], n], j++]; Prime[j]]; Array[f, 60] (* Robert G. Wilson v, Apr 06 2012 *)

A210642 a(n) = least integer m > 1 such that k! == n! (mod m) for no 0 < k < n.

Original entry on oeis.org

2, 2, 3, 4, 5, 9, 7, 13, 17, 17, 11, 13, 13, 19, 23, 17, 17, 29, 19, 23, 31, 31, 23, 41, 31, 29, 31, 37, 29, 31, 31, 37, 41, 41, 59, 37, 37, 59, 43, 41, 41, 59, 43, 67, 53, 53, 47, 53, 67, 59, 61, 53, 53, 79, 59, 59, 67, 73, 59, 67

Offset: 1

Zhi-Wei Sun, Mar 26 2012

nonn

Conjecture: a(n) is a prime not exceeding 2n with the only exceptions a(4)=4 and a(6)=9.
Note that a(n) is at least n and there is at least a prime in the interval [n,2n] by the Bertrand Postulate first confirmed by Chebyshev.
Compare this sequence with A208494.

			We have a(4)=4, because 4 divides none of 4!-1!=23, 4!-2!=22, 4!-3!=18, and both 2 and 3 divide 4!-3!=18.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2500
Oliver Gerard, Re: A new conjecture on primes, a message to Number Theory List, March 23, 2012.
Zhi-Wei Sun, A new conjecture on primes, a message to Number Theory List, March 20, 2012.
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.

Cf. A000040, A208494, A210640, A210393, A210394, A210186, A210144, A208643, A207982.

R[n_,m_]:=If[n==1,1,Product[If[Mod[n!-k!, m]==0, 0, 1], {k, 1, n-1}]] Do[Do[If[R[n,m]==1, Print[n, " ", m]; Goto[aa]], {m,Max[2,n],2n}]; Print[n]; Label[aa]; Continue,{n,1,2500}]

cp's OEIS Frontend

A208643 Least positive integer m such that those k*(k-1) mod m with k=1,...,n are pairwise distinct.

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A208494 Least integer m>1 such that those k! mod m with k=1,...,n are pairwise distinct.

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A210144 a(n) = least integer m>1 such that the product of the first k primes for k=1,...,n are pairwise distinct modulo m.

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A210186 a(n) = least integer m>1 such that m divides none of P_i + P_j with 0

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A210393 a(n) = least integer m>1 such that S_k! for k=1,...,n are pairwise distinct modulo m where S_k is the sum of the first k primes.

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A210394 a(n) = least integer m>1 such that m divides none of S_i!+S_j! with 0

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A181901 a(n) = least positive integer m such that 2(s_k)^2 for k=1,...,n are pairwise distinct modulo m where s_k = Sum_{j=1..k} (-1)^(k-j)*p_j and p_j is the j-th prime.

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A210640 a(n) = least integer m > 1 such that 2S_k^2 (k=1,...,n) are pairwise distinct modulo m, where S_k is the sum of the first k primes.

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