A210144 -id:A210144 - cp's OEIS frontend

A214196 Unique terms in sequence A210144.

2, 3, 5, 11, 23, 29, 37, 41, 47, 73, 131, 151, 199, 223, 271, 281, 353, 457, 641, 643, 659, 1259, 1531, 1747, 1951, 2671, 2953, 4259, 4967, 5419, 5839, 7013, 7963, 11261, 12653, 15733, 16189, 18367, 19237, 29129, 32381, 33161, 33247, 57653, 61723, 63823, 66739

Offset: 1

N. J. A. Sloane, Jul 07 2012

nonn

The sequence is the set of numbers m which are the minimum m for some triple 1 <= i < j <= k such that m divides none of the differences A002110(i)-A002110(j). - R. J. Mathar, Jul 08 2012
In Sun (2012), these numbers are called "Primes of the first kind".
Conjecture: all the terms are prime. See Conjecture 1.5(i) in Sun 2012. - Jason Yuen, Feb 25 2024

Jason Yuen, Table of n, a(n) for n = 1..128
Zhi-Wei Sun, On functions taking only prime values, arXiv preprint arXiv:1202.6589 [math.NT], 2012; see p. 5-6.

Cf. A210144, A214197.

A214196 := proc(n)
        local m ,i,j,ddvs;
        for m from 2 do
                ddvs := false ;
                for i from 1 to n-1 do
                        for j from i+1 to n do
                                if (A002110(j)-A002110(i)) mod m = 0 then
                                        ddvs := true;
                                        break;
                                end if;
                        end do:
                        if ddvs then
                                break;
                        end if;
                end do:
                if ddvs = false then
                        return m;
                end if;
        end do:
end proc:
# loop generates m multiples times (pipe through 'uniq')
for n from 1 do
        printf("%d,\n",A214196(n)) ;
end do: # R. J. Mathar, Jul 08 2012

primorial[n_] := primorial[n] = Product[Prime[i], {i, 1, n}];
p[0] = 1; p[n_] := p[n] = Module[{m, i, j, ddvs}, For[m = 2, True, m++, ddvs = False ; For[i = 1, i <= n - 1, i++, For[j = i + 1, j <= n, j++, If[Mod[primorial[j] - primorial[i], m] == 0, ddvs = True; Break[]]]; If[ddvs, Break[]]]; If[ddvs == False, Return[m]]]];
A214196 = Reap[n = k = 1; While[n <= 400, If[p[n] != p[n - 1], a[k] = p[n]; Print[n, " a(", k, ") = ", a[k]]; Sow[a[k]]; k++]; n++]][[2, 1]] (* Jean-François Alcover, Jan 20 2018, after R. J. Mathar *)

a(28)-a(34) from Jean-François Alcover, Jan 20 2018

Definition simplified and more terms from Jason Yuen, Feb 24 2024

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

A214196 Unique terms in sequence A210144.

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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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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.

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

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