A232932 -id:A232932 - cp's OEIS frontend

A306538 The least prime q such that Kronecker(D/q) = 1 where D runs through all negative fundamental discriminants (-A003657).

7, 5, 2, 3, 3, 2, 5, 3, 2, 5, 2, 3, 2, 7, 11, 2, 5, 7, 2, 3, 3, 17, 3, 2, 2, 3, 5, 2, 13, 5, 2, 2, 3, 3, 2, 7, 3, 2, 11, 11, 2, 3, 7, 5, 5, 2, 19, 2, 3, 3, 2, 41, 3, 2, 13, 3, 2, 5, 7, 2, 7, 2, 3, 5, 3, 2, 5, 2, 3, 11, 2, 31, 13, 2, 5, 2, 3, 3, 2, 5, 3, 2, 5, 23, 2, 5, 17, 7, 2, 5, 7, 2, 3, 3

Offset: 1

Jianing Song, Feb 22 2019

nonn

a(n) is the least prime that decomposes in the imaginary quadratic field with discriminant D, D = -A003657(n).
For most n, a(n) is relatively small. There are only 472 n's among [1, 3043] (there are 3043 terms in A003657 below 10000) that violate a(n) < log(A003657(n)).
Also a(n) is the smallest prime p such that the imaginary quadratic field with discriminant D = -A003657(n) can be embedded into the p-adic field Q_p. - Jianing Song, Feb 14 2021

			Let K = Q[sqrt(-177)] with D = -708 = -A003657(218), we have: 2 and 3 divides 708, (-708/5) = (-708/7) = ... = (-708/29) = -1 and (-708/31) = +1, so 2 and 3 ramify in K, 5, 7, ..., 29 remain inert in K and 31 decomposes in K, so a(218) = 31.

Cf. A003657.

Similar sequences: A232931, A232932 (the least prime that remains inert); A306537, this sequence (the least prime that decomposes); A306541, A306542 (the least prime that decomposes or ramifies).

b(D)=forprime(p=2, oo, if(kronecker(D, p)>0, return(p)))
for(n=1, 300, if(isfundamental(-n), print1(b(-n), ", ")))

A306537 The least prime q such that Kronecker(D/q) = 1 where D runs through all positive fundamental discriminants (A003658).

Original entry on oeis.org

2, 11, 7, 11, 3, 2, 5, 5, 3, 5, 2, 3, 3, 2, 5, 7, 5, 2, 7, 3, 2, 5, 2, 3, 13, 3, 3, 2, 7, 7, 2, 5, 5, 2, 3, 2, 7, 3, 2, 3, 3, 2, 13, 5, 2, 5, 11, 5, 3, 2, 7, 11, 3, 13, 2, 3, 3, 2, 11, 2, 7, 2, 5, 3, 2, 11, 2, 3, 5, 3, 3, 2, 5, 13, 2, 13, 2, 3, 2, 5, 2, 3, 5, 2

Offset: 1

Jianing Song, Feb 22 2019

nonn

a(n) is the least prime that decomposes in the real quadratic field with discriminant D, D = A003658(n).
For most n, a(n) is relatively small. There are only 459 n's among [1, 3044] (there are 3044 terms in A003658 below 10000) that violate a(n) < log(A003658(n)).
Also a(n) is the smallest prime p such that the real quadratic field with discriminant D = A003658(n) can be embedded into the p-adic field Q_p. - Jianing Song, Feb 14 2021

			Let K = Q[sqrt(635)] with D = 2540 = A003658(774), we have: 2 and 5 divides 2540, (2540/3) = (2540/7) = ... = (2540/37) = -1 and (2540/41) = +1, so 2 and 5 ramify in K, 3, 7, ..., 37 remain inert in K and 41 decomposes in K, so a(774) = 41.

Cf. A003658.

Similar sequences: A232931, A232932 (the least prime that remains inert); this sequence, A306538 (the least prime that decomposes); A306541, A306542 (the least prime that decomposes or ramifies).

b(D)=forprime(p=2, oo, if(kronecker(D, p)>0, return(p)))
for(n=1, 300, if(isfundamental(n), print1(b(n), ", ")))

A306541 The least prime q such that Kronecker(D/q) >= 0 where D runs through all positive fundamental discriminants (A003658).

Original entry on oeis.org

2, 5, 2, 2, 3, 2, 3, 2, 2, 5, 2, 3, 2, 2, 2, 7, 2, 2, 2, 3, 2, 3, 2, 2, 7, 3, 2, 2, 2, 3, 2, 5, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 5, 2, 2, 3, 2, 3, 2, 2, 13, 2, 3, 2, 2, 2, 2, 7, 2, 2, 3, 2, 3, 2, 2, 5, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 5, 2, 3, 2, 2, 2

Offset: 1

Jianing Song, Feb 22 2019

nonn

a(n) is the least prime that either decomposes or ramifies in the real quadratic field with discriminant D, D = A003658(n).

For most n, a(n) is relatively small. There are only 86 n's among [1, 3044] (there are 3044 terms in A003658 below 10000) that violate a(n) < log(A003658(n)).

			Let K = Q[sqrt(293)] with D = 293 = A003658(90), we have: (293/2) = (293/3) = ... = (293/13) = -1 and (293/17) = +1, so 2, 3, 5, 7, 11 and 13 remain inert in K and 17 decomposes in K, so a(90) = 17.

Cf. A003658.

Similar sequences: A232931, A232932 (the least prime that remains inert); A306537, A306538 (the least prime that decomposes); this sequence, A306542 (the least prime that decomposes or ramifies).

b(D)=forprime(p=2, oo, if(kronecker(D, p)>=0, return(p)))
for(n=1, 300, if(isfundamental(n), print1(b(n), ", ")))

A306542 The least prime q such that Kronecker(D/q) >= 0 where D runs through all negative fundamental discriminants (-A003657).

Original entry on oeis.org

3, 2, 2, 2, 3, 2, 5, 2, 2, 2, 2, 3, 2, 2, 11, 2, 3, 2, 2, 2, 3, 17, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 5, 2, 2, 2, 3, 2, 3, 2, 2, 5, 2, 2, 2, 2, 3, 2, 41, 2, 2, 2, 3, 2, 2, 7, 2, 3, 2, 3, 5, 2, 2, 3, 2, 3, 2, 2, 2, 5, 2, 2, 2, 2, 3, 2, 5, 2, 2, 2, 3, 2, 2, 2, 7

Offset: 1

Jianing Song, Feb 22 2019

nonn

a(n) is the least prime that either decomposes or ramifies in the imaginary quadratic field with discriminant D, D = -A003657(n).
The quadratic field with discriminant D = -A003657(n) has class number 1 if and only if a(n) >= (1/4)*A003657(n). If the quadratic field with discriminant D = -A003657(n) has class number 3 then a(n)^2 < (1/4)*A003657(n) < a(n)^3.
For most n, a(n) is relatively small. There are only 86 n's among [1, 3043] (there are 3043 terms in A003657 below 10000) that violate a(n) < log(A003657(n)). In fact, if we ignore the first term, the only terms among the first 3043 ones that seem unusually large are a(15) = 11 (with A003657(15) = 43), a(22) = 17 (with A003657(22) = 67), a(52) = 41 (with A003657(52) = 163), a(1147) = 23 (with A003657(1147) = 3763), a(2677) = 23 (with A003657(2677) = 8803) and a(2758) = 23 (with A003657(2758) = 9067).

			Let K = Q[sqrt(-3763)] with D = -3763 = -A003657(1147), we have: (-3763/2) = (-3763/3) = ... = (-3763/19) = -1 and (-3763/23) = +1, so 2, 3, 5, 7, 11, 13, 17 and 19 remain inert in K and 23 decomposes in K, so a(1147) = 23.

Cf. A003657.

Similar sequences: A232931, A232932 (the least prime that remains inert); A306537, A306538 (the least prime that decomposes); A306541, this sequence (the least prime that decomposes or ramifies).

b(D)=forprime(p=2, oo, if(kronecker(D, p)>=0, return(p)))
for(n=1, 300, if(isfundamental(-n), print1(b(-n), ", ")))

A306218 Fundamental discriminant D < 0 with the least absolute value such that the first n primes p have (D/p) >= 0, negated.

Original entry on oeis.org

4, 8, 15, 20, 24, 231, 264, 831, 920, 1364, 1364, 9044, 67044, 67044, 67044, 67044, 268719, 268719, 3604695, 4588724, 5053620, 5053620, 5053620, 5053620, 60369855, 364461096, 532735220, 715236599, 1093026360, 2710139064, 2710139064, 3356929784, 3356929784

Offset: 1

Jianing Song, Jan 29 2019

nonn

a(n) is the negated fundamental discriminant D < 0 with the least absolute value such that the first n primes either decompose or ramify in the imaginary quadratic field with discriminant D. See A241482 for the real quadratic field case.

			(-231/2) = 1, (-231/3) = 0, (-231/5) = 1, (-231/7) = 0, (-231/11) = 0, (-231/13) = 1, so 2, 5, 13 decompose in Q[sqrt(-231)] and 3, 7, 11 ramify in Q[sqrt(-231)]. For other fundamental discriminants -231 < D < 0, at least one of 2, 3, 5, 7, 11, 13 is inert in the imaginary quadratic field with discriminant D, so a(6) = 231.

Cf. A003657, A232932, A241482 (the real quadratic field case).

A045535 and A094841 are similar sequences.

a(n) = my(i=1); while(!isfundamental(-i)||sum(j=1, n, kronecker(-i,prime(j))==-1)!=0, i++); i

a(n) = A003657(k), where k is the smallest number such that A232932(k) >= prime(n+1).

a(26)-a(33) from Jinyuan Wang, Apr 06 2019

A249272 Decimal expansion of a constant associated with fundamental discriminants and Dirichlet characters.

Original entry on oeis.org

4, 9, 8, 0, 9, 4, 7, 3, 3, 9, 6, 1, 4, 9, 3, 4, 1, 5, 0, 7, 9, 1, 3, 2, 5, 3, 2, 5, 8, 8, 0, 7, 7, 5, 2, 8, 1, 2, 3, 7, 7, 3, 2, 6, 9, 6, 5, 8, 5, 2, 0, 4, 7, 9, 5, 4, 6, 2, 3, 3, 1, 2, 7, 1, 8, 6, 7, 3, 3, 2, 6, 3, 8, 1, 9, 6, 8, 0, 0, 3, 8, 1, 5, 2, 0, 9, 0, 4, 7, 7, 4, 9, 0, 0, 6, 1, 7, 6, 1, 6, 2, 1, 2

Offset: 1

Jean-François Alcover, Oct 24 2014

			4.9809473396149341507913253258807752812377326965852...

Peter J. Cho, Henry H. Kim, The average of the smallest prime in a conjugacy class, arXiv:1601.03012 [math.NT], 2016.
Steven R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 250.
P. Pollack, The average least quadratic nonresidue modulo m and other variations on a theme of Erdős, J. Number Theory 132 (2012) 1185-1202.

Cf. A232929, A232930, A232931, A232932.

digits = 103; Clear[s, P]; P[j_] := P[j] = Product[(Prime[k] + 2)/(2*(Prime[k] + 1)), {k, 1, j - 1}] // N[#, digits + 100]&; s[m_] := s[m] = Sum[Prime[j]^2/(2*(Prime[j] + 1))*P[j], {j, 1, m}]; s[10]; s[m = 20]; While[RealDigits[s[m]] != RealDigits[s[m/2]], Print[m, " ", N[s[m]]]; m = 2*m]; RealDigits[s[m], 10, digits] // First

suminf(k=1, prime(k)^2/(2*(prime(k)+1))*prod(i=1, k-1, (prime(i)+2)/(2*(prime(i)+1)))); \\ Michel Marcus, Apr 15 2017

sum_{q} q^2/(2(q+1)) prod_{p

A249273 Decimal expansion of a constant associated with the set of all complex nonprincipal Dirichlet characters.

Original entry on oeis.org

2, 5, 3, 5, 0, 5, 4, 1, 8, 0, 3, 6, 0, 4, 3, 8, 8, 3, 0, 1, 6, 5, 5, 3, 0, 0, 0, 7, 1, 8, 5, 9, 0, 8, 3, 5, 0, 8, 6, 1, 1, 7, 8, 0, 1, 3, 8, 5, 3, 7, 0, 1, 6, 4, 5, 3, 7, 7, 5, 1, 2, 6, 4, 9, 4, 3, 6, 4, 1, 4, 7, 5, 3, 8, 2, 9, 6, 7, 8, 5, 4, 7, 0, 1, 7, 0, 3, 3, 6, 6, 5, 1, 7, 9, 1, 0, 9, 0, 3, 4, 2, 4, 5

Offset: 1

Jean-François Alcover, Oct 24 2014

			2.5350541803604388301655300071859083508611780138537...

Steven R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]
G. Martin and P. Pollack, The average least character non-residue and further variations on a theme of Erdős, J. London Math. Soc. 87 (2013) 22-42.

Cf. A232929, A232930, A232931, A232932.

digits = 103; Clear[s]; s[m_] := s[m] = Sum[Prime[k]^2/Product[Prime[j] + 1, {j, 1, k}] , {k, 1, m}] // N[#, digits + 100]&; s[10]; s[m = 20]; While[RealDigits[s[m]] != RealDigits[s[m/2]], Print[m, " ", N[s[m]]]; m = 2*m]; RealDigits[s[m], 10, digits] // First

sum_{k >= 1} p_k^2/((p_1 + 1)(p_2 + 1)...(p_k + 1)), where p_k is the k-th prime number.

A249274 Decimal expansion of a constant associated with the set of all complex primitive Dirichlet characters.

Original entry on oeis.org

2, 1, 5, 1, 4, 3, 5, 1, 0, 5, 6, 8, 6, 1, 4, 6, 5, 4, 8, 6, 2, 4, 2, 8, 1, 0, 0, 5, 0, 9, 6, 5, 8, 4, 0, 5, 3, 2, 6, 3, 3, 0, 4, 5, 7, 1, 8, 5, 8, 4, 5, 7, 8, 9, 5, 8, 8, 9, 7, 3, 3, 3, 9, 1, 0, 7, 8, 1, 8, 4, 2, 8, 7, 3, 2, 5, 7, 4, 6, 4, 5, 2, 0, 7, 1, 8, 4, 6, 3, 0, 4, 2, 4, 4, 6, 9, 1, 7, 9, 3, 2

Offset: 1

Jean-François Alcover, Oct 24 2014

			2.1514351056861465486242810050965840532633...

Steven R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 251.
G. Martin and P. Pollack, The average least character non-residue and further variations on a theme of Erdős, J. London Math. Soc. 87 (2013) 22-42.

Cf. A232929, A232930, A232931, A232932.

digits = 101; Clear[s, P]; P[j_] := P[j] = Product[(Prime[k]^2 - Prime[k] - 1)/((Prime[k] + 1)^2*(Prime[k] - 1)), {k, 1, j - 1}] // N[#, digits + 100]&; s[m_] := s[m] = Sum[Prime[j]^4/((Prime[j] + 1)^2*(Prime[j] - 1))*P[j], {j, 1, m}]; s[10]; s[m = 20]; While[ RealDigits[s[m]] != RealDigits[s[m/2]], Print[m, " ", N[s[m]]]; m = 2*m]; RealDigits[s[m], 10, digits] // First

sum_{q} q^4/((q+1)^2 (q-1)) prod_{p

Showing 1-8 of 8 results.

cp's OEIS Frontend

A306538 The least prime q such that Kronecker(D/q) = 1 where D runs through all negative fundamental discriminants (-A003657).

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A306537 The least prime q such that Kronecker(D/q) = 1 where D runs through all positive fundamental discriminants (A003658).

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A306541 The least prime q such that Kronecker(D/q) >= 0 where D runs through all positive fundamental discriminants (A003658).

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A306542 The least prime q such that Kronecker(D/q) >= 0 where D runs through all negative fundamental discriminants (-A003657).

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A306218 Fundamental discriminant D < 0 with the least absolute value such that the first n primes p have (D/p) >= 0, negated.

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A249272 Decimal expansion of a constant associated with fundamental discriminants and Dirichlet characters.

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A249273 Decimal expansion of a constant associated with the set of all complex nonprincipal Dirichlet characters.

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A249274 Decimal expansion of a constant associated with the set of all complex primitive Dirichlet characters.

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