A062241 -id:A062241 - cp's OEIS frontend

A045535 Least negative pseudosquare modulo the first n odd primes.

7, 23, 71, 311, 479, 1559, 5711, 10559, 18191, 31391, 118271, 366791, 366791, 2155919, 2155919, 2155919, 6077111, 6077111, 98538359, 120293879, 131486759, 131486759, 508095719, 2570169839, 2570169839, 2570169839, 2570169839, 2570169839, 2570169839, 328878692999, 328878692999, 513928659191, 844276851239

Offset: 0

N. J. A. Sloane

a(n) is the smallest positive integer m such that m == 7 (mod 8) and for the first n odd primes p, -m is a (nonzero) quadratic residue mod p.

N. D. Bronson and D. A. Buell, Congruential sieves on FPGA computers, pp. 547-551 of Mathematics of Computation 1943-1993 (Vancouver, 1993), Proc. Symp. Appl. Math., Vol. 48, Amer. Math. Soc. 1994.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Bert Dobbelaere, Table of n, a(n) for n = 0..50
D. H. Lehmer, E. Lehmer and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp., 24 (1970), 433-451.
D. H. Lehmer, E. Lehmer and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp., 24 (1970), 433-451 [Annotated scanned copy]
OEIS Index entries for sequences related to pseudo-squares

Cf. A002189, A062241.

{A045535 = (n,m=7)->until(!m+=8,for(i=2,n+1,m%prime(i)||next(2);issquare(Mod(-m,prime(i)))||next(2));return(m))} \\ Starting value (e.g., a(n-1); must be in 7+8Z) may be given as 2nd arg. - M. F. Hasler, Oct 24 2013

The Bronson-Buell reference gives terms through 227. The Math. Comp. version is erroneous.
Edited by Don Reble, Nov 14 2006
Corrected link to OEIS index, following a remark by Don Reble. Values a(0..21) double-checked. - M. F. Hasler, Oct 24 2013
a(27)-a(28) from Jinyuan Wang, Mar 24 2020
More terms from Bert Dobbelaere, Feb 28 2021

A112084 Column 2 of A112070.

Original entry on oeis.org

5, 9, 25, 71, 311, 479, 1559, 5711, 10559, 18191, 31391, 118271, 701399, 366791, 2704679, 2954591, 2155919, 13845841, 6077111, 25411681, 28398241

Offset: 1

Antti Karttunen, Aug 28 2005

Note the subsequence equal to the portion of A045535 / A062241 and also the non-monotone drops present, like the one from a(13)=701399 to a(14)=366791.

An independent recomputation with another software, e.g. Mathematica, would be welcome.

Row 2 of A112071. a(n) = A005408(A112083(n)).

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

Original entry on oeis.org

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}]

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]

cp's OEIS Frontend

A045535 Least negative pseudosquare modulo the first n odd primes.

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A112084 Column 2 of A112070.

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

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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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Author

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Programs

Mathematica