A045535 -id:A045535 - cp's OEIS frontend

A196942 a(n) is the prime order of sequence A196941.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 3, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 1, 1, 1, 2, 1, 3, 2, 4, 1, 2, 2, 2, 2, 3, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 2

Lei Zhou, Oct 07 2011

Assuming 1 is the 0th prime, as what in Mathematica: PrimePi[1] = 0.

So far the first occurrence of this sequence agree with A062241 and A045535. Is this a coincidence or a theorem?

			A196941(3) = 2, which is the first prime number, so a(3) = 1;

Cf. A196941, A062241.

FactorSet[seed_] := Module[{fset2, a, l, i}, a = FactorInteger[seed]; l = Length[a]; fset2 = {}; Do[fset2 = Union[fset2, {a[[i]][[1]]}], {i, 1, l}]; fset2]; Table[min = n; Do[r = n - k; s = Union[FactorSet[k], FactorSet[r]]; If[a = s[[Length[s]]]; a < min, min = a], {k, 1, IntegerPart[n/2]}]; PrimePi[min], {n, 2, 88}]

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

A323051 Numbers that cannot be written as a sum of two or fewer 11-smooth numbers (A051038).

Original entry on oeis.org

479, 958, 1151, 1319, 1437, 1559, 1679, 1916, 2302, 2351, 2395, 2638, 2874, 2999, 3013, 3071, 3118, 3353, 3358, 3453, 3671, 3737, 3769, 3832, 3911, 3957, 4199, 4309, 4311, 4604, 4677, 4702, 4703, 4751, 4790, 4919, 5037, 5057, 5269, 5276, 5389, 5443, 5519, 5597, 5683

Offset: 1

Carlos Alves, Jan 03 2019

nonn

Similar to A323046 (3-smooth), A323049 (5-smooth) or A323050 (7-smooth).
This sequence is a subsequence of A323046, A323049, and A323050.
Notice that A045535(4) = a(1) = 479.

See A323046 (3-smooth), A323049 (5-smooth) or A323050 (7-smooth). Cf. A051038, A045535 (or A062241).

f[n_] := Union@Flatten@Table[2^a*3^b*5^c*7^d, {a, 0, Log2[n]}, {b, 0, Log[3, n/2^a]}, {c, 0, Log[5, n/(2^a*3^b)]}, {d, 0, Log[7, n/(2^a*3^b*5^c)]}];
b = Block[{nn = 3000, s}, s = f[nn]; {0, 1}~Join~
    Select[Union@Flatten@Outer[Plus, s, s], # <= nn &]];
Complement[Range[3000], b]

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A196942 a(n) is the prime order of sequence A196941.

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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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A323051 Numbers that cannot be written as a sum of two or fewer 11-smooth numbers (A051038).

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Mathematica