A340708 Maximum denominator of resistances obtained by an electrical network with n unit resistors.

1, 2, 3, 5, 8, 13, 24, 40, 69, 130, 231, 408, 689, 1272, 2153, 3960, 6993, 12560

Offset

1,2

Comments

a(n) is taken from the set of resistance values counted by A337517(n). These sets can be computed by the PARI program of Andrew Howroyd in A180414.

Also the maximum numerator of these electrical networks for small n.

Maximum numerator and maximum denominator coincide for planar networks: for every resistance R in a planar network with n resistors there is always another planar network with n resistors and resistance 1/R. For nonplanar networks this is not necessarily so, as can be seen in A338573.

The asymmetry is illustrated by the example a(15) = 2153.

The author conjectures that this asymmetry will increase with n, and eventually the maxima will differ.

Conjecture: a(19) = 22233, a(20) = 39918. It would be very desirable to know at which value of n > 18 the maximum values of numerators and denominators differ for the first time. - Hugo Pfoertner, Apr 19 2021

Links

Table of n, a(n) for n=1..18.

Index to sequences related to resistances.

Example

Denominators for numerator a(15) = 2153 in electrical networks with 15 resistors:
  1025,1049,1051,1058,1089,1104,1145,1184,1185,1193,1208,
  1212,1219,1248,1254,1337,1382,1403,1526,1527,1529,1530,
  1545,1547,1555,1579,1586,1632,1642,1647,1687,1699,1719.
Numerators for denominator a(15) = 2153 in electrical networks with 15 resistors:
   899, 905, 934, 941, 945, 960, 968, 969,1008,1049,1064,
  1095,1102,1104,1128,1137,1143,1147,1164,1182,1207,1296,
  1359,1367,1387,1400,1447,1543.

Crossrefs

Cf. A180414, A337517, A338601, A339548, A339808, A340726.

Sequence in context: A257418 A370001 A125730 * A074030 A336604 A024318

Adjacent sequences: A340705 A340706 A340707 * A340709 A340710 A340711

Keyword

nonn,hard,more

Author

Rainer Rosenthal, Jan 16 2021

Extensions

a(18) from Hugo Pfoertner, Apr 11 2021

Status

approved