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

Author

Rainer Rosenthal, Jan 16 2021

Keywords

• nonn
• hard
• more

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

Examples

			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.
		

Links

• Index to sequences related to resistances.

Crossrefs

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

Extensions

a(18) from Hugo Pfoertner, Apr 11 2021