A338197 -id:A338197 - cp's OEIS frontend

A051389 Number of resistance values that can be constructed using exactly n 1-ohm resistors in series or parallel but not with fewer resistors.

1, 2, 4, 8, 20, 42, 102, 250, 610, 1486, 3710, 9228, 23050, 57718, 145288, 365820, 922194, 2327914, 5885800, 14890796, 37701452, 95550472, 242325118, 614869792, 1561228066, 3966071764, 10080113232, 25630109268, 65194419268, 165890640468

Offset: 1

Hugo van der Sanden

If x and y require xn and yn resistors respectively, then (x+y) and 1/(1/x + 1/y) require no more than (xn+yn). Inspired by a sci.math posting by Miguel A. Lerma (lerma(AT)math.nwu.edu).
Let A(n) be the set of resistances equivalent to a network of n 1-ohm resistors using only series and parallel combinations. Then A048211(n) = card(A(n)). Let L(n) be the set of resistances that first appear in A(n), i.e. L(n) = A(n) \ (A(1) U ... U A(n-1)). Then a(n) = card(L(n)). - Antoine Mathys, Nov 22 2024
If a resistance is equivalent to a n-resistor circuit, then it is equivalent to a 4n-resistor circuit. There is therefore no upper bound on the size of the networks to which it is equivalent. - Antoine Mathys, Nov 22 2024

			The a(1) = 1 resistance value is 1 ohm.
The a(2) = 2 resistance values are {1/2, 2}.
The a(3) = 4 resistance values are {1/3, 2/3, 3/2, 3}.
The a(4) = 8 resistance values are {1/4, 2/5, 3/5, 3/4, 4/3, 5/3, 5/2, 4}.
The a(5) = 20 resistance values are {1/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5, 5/6, 6/7, 7/6, 6/5, 5/4, 7/5, 8/5, 7/4, 7/3, 8/3, 7/2, 5}.
E.g. 6/5 is made from two resistors in series in parallel with three resistors in series, since 6/5 = 1/(1/2 + 1/3). It cannot be obtained using fewer resistors.

Miguel A. Lerma, resistors, post in the newsgroup sci.math, Nov 5 1999.
Index to sequences related to resistances.

Cf. A048211, A046825, A153588, A180414, A338197.

a(n) = A153588(n) - A153588(n-1) for n > 1. - Hugo Pfoertner, Nov 04 2020

a(15)-a(21) from Jon E. Schoenfield, Aug 28 2006
Definition corrected by Jon E. Schoenfield, Aug 27 2006
a(22)-a(23) from Graeme McRae, Aug 18 2007
a(24)-a(25) from Antoine Mathys, Mar 20 2017
Definition changed to say "exactly". - N. J. A. Sloane, Nov 07 2020
Definition clarified by Antoine Mathys, Nov 22 2024
a(26)-a(30) from Antoine Mathys, Dec 05 2024

A338607 Resistance values R < 1 ohm, multiplied by a common denominator 232792560 (= A338600(7)), that can be obtained from a network of exactly 7 one-ohm resistors, but not from any network with fewer than 7 one-ohm resistors.

Original entry on oeis.org

33256080, 42325920, 49884120, 53721360, 62078016, 64664600, 68468400, 71628480, 72747675, 73513440, 81477396, 82162080, 83140200, 85765680, 88682880, 90530440, 95855760, 98017920, 101846745, 106696590, 110270160, 110853600, 121938960, 122522400, 126095970

Offset: 1

Hugo Pfoertner, Nov 05 2020

The list of the A338197(7)/2 = 57 resistance values < 1 ohm is A338587(n)/A338597(n). a(n) = 232792560 * [1/7, 2/11, 3/14, 3/13, 4/15, 5/18, 5/17, ..., 19/21, 11/12, 12/13, 13/14, 14/15, 15/16, 18/19].

Hugo Pfoertner, Table of n, a(n) for n = 1..57

Cf. A180414, A338197, A338587, A338597, A338600, A338605, A338606, A338608, A338609.

A338608 Resistance values R < 1 ohm, multiplied by a common denominator 5342931457063200 (= A338600(8)), that can be obtained from a network of exactly 8 one-ohm resistors, but not from any network with fewer than 8 one-ohm resistors.

Original entry on oeis.org

667866432132900, 821989454932800, 942870257128800, 1001799648199350, 1124827675171200, 1161506838492000, 1214302603878000, 1257160342838400, 1272126537396000, 1282303549695168, 1385204451831200, 1393808206190400, 1406034593964000, 1438481546132400, 1473912126086400

Offset: 1

Hugo Pfoertner, Nov 06 2020

			The list of the A338197(8)/2 = 156 resistance values < 1 ohm is A338580(n)/A338598(n). a(n) = 5342931457063200 * [1/8, 2/13, 3/17, 3/16, 4/19, 5/23, 5/22, ..., 23/24, 24/25, 25/26, 26/27, 27/28, 30/31, 34/35].

Hugo Pfoertner, Table of n, a(n) for n = 1..156

Cf. A180414, A338197, A338580, A338598, A338600.

Cf. A338605, A338606, A338607, A338609 (similar for n = 5..9).

A338609 Resistance values R < 1 ohm, multiplied by a common denominator 591133442051411133755680800 (= A338600(9)), that can be obtained from a network of exactly 9 one-ohm resistors, but not from any network with fewer than 9 one-ohm resistors.

Original entry on oeis.org

65681493561267903750631200, 78817792273521484500757440, 88670016307711670063352120, 93336859271275442171949600, 102805816008941066740118400, 105559543223466273884943000, 109469155935446506251052000, 112596846105030692143939200, 113679508086809833414554000

Offset: 1

Hugo Pfoertner, Nov 06 2020

			The list of the A338197(9)/2 = 447 resistance values < 1 ohm is A338580(n)/A338599(n). a(n) = 591133442051411133755680800 * [1/9, 2/15, 3/20, 3/19, 4/23, 5/28, ..., 43/44, 45/46, 46/47, 48/49, 50/51, 55/56].

Hugo Pfoertner, Table of n, a(n) for n = 1..447

Cf. A180414, A338197, A338580, A338599, A338600.

Cf. A338605, A338606, A338607, A338608 (similar for n = 5..8).

A338583 Number of unlabeled 3-connected nonplanar graphs with n edges.

Original entry on oeis.org

1, 2, 3, 10, 29, 94, 343, 1291, 5206, 22061, 96908, 439837, 2053916, 9841412, 48319944, 242857491, 1248629027, 6563581656, 35258560001, 193463945790

Offset: 9

Hugo Pfoertner, Nov 21 2020

Hugo Pfoertner, Illustrations of terms a(9) - a(12).

Cf. A002840, A006290, A337517, A338197, A338511, A338573, A338584, A338593, A338594, A338604.

a(n) = A338511(n) - A002840(n).

a(n) <= A338593(n). The difference A338584(n) = A338593(n)-a(n) are the counts of nonplanar connected graphs with minimum degree 3 at each node that are not 3-connected.

A338606 Resistance values R < 1 ohm, multiplied by a common denominator 360360 (= A338600(6)), that can be obtained from a network of exactly 6 one-ohm resistors, but not from any network with fewer than 6 one-ohm resistors.

Original entry on oeis.org

60060, 80080, 98280, 108108, 131040, 138600, 150150, 160160, 163800, 166320, 194040, 196560, 200200, 210210, 221760, 229320, 252252, 262080, 280280, 304920, 324324, 327600

Offset: 1

Hugo Pfoertner, Nov 05 2020

			The list of resistance values < 1 ohm is A338580(n)/A338596(n). a(n) = 360360 * [1/6, 2/9, 3/11, 3/10, 4/11, 5/13, 5/12, 4/9, 5/11, 6/13, 7/13, 6/11, 5/9, 7/12, 8/13, 7/11, 7/10, 8/11, 7/9, 11/13, 9/10, 10/11].

Cf. A180414, A338197, A338580, A338596, A338600, A338605, A338607, A338608, A338609.

A338587 Numerators of resistance values < 1 ohm that can be obtained from a network of exactly 7 one-ohm resistors, but not from any network with fewer than 7 one-ohm resistors. Denominators are in A338597.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 4, 5, 6, 7, 6, 5, 7, 8, 7, 7, 8, 7, 11, 9, 10, 11, 10, 13, 9, 11, 10, 11, 13, 12, 9, 11, 13, 13, 11, 9, 12, 13, 11, 14, 10, 11, 15, 9, 14, 16, 17, 7, 8, 19, 11, 12, 13, 14, 15, 18

Offset: 1

Hugo Pfoertner, Nov 05 2020

			The list of the 114 = A338197(7) resistance values, sorted by increasing size of R = a(n)/A338597(n) = A338607(n)/A338600(7), is the union of [1/7, 2/11, 3/14, ..., 14/15, 15/16, 18/19] and of the corresponding reciprocal resistances > 1 ohm [19/18, 16/15, 15/14, ..., 14/3, 11/2, 7].

Cf. A338197, A338597, A338607.

Cf. A338595, A338596, A338598, A338599, A338590 (similar for n = 5..10).

cp's OEIS Frontend

A051389 Number of resistance values that can be constructed using exactly n 1-ohm resistors in series or parallel but not with fewer resistors.

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A338607 Resistance values R < 1 ohm, multiplied by a common denominator 232792560 (= A338600(7)), that can be obtained from a network of exactly 7 one-ohm resistors, but not from any network with fewer than 7 one-ohm resistors.

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A338608 Resistance values R < 1 ohm, multiplied by a common denominator 5342931457063200 (= A338600(8)), that can be obtained from a network of exactly 8 one-ohm resistors, but not from any network with fewer than 8 one-ohm resistors.

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A338609 Resistance values R < 1 ohm, multiplied by a common denominator 591133442051411133755680800 (= A338600(9)), that can be obtained from a network of exactly 9 one-ohm resistors, but not from any network with fewer than 9 one-ohm resistors.

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A338583 Number of unlabeled 3-connected nonplanar graphs with n edges.

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A338606 Resistance values R < 1 ohm, multiplied by a common denominator 360360 (= A338600(6)), that can be obtained from a network of exactly 6 one-ohm resistors, but not from any network with fewer than 6 one-ohm resistors.

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A338587 Numerators of resistance values < 1 ohm that can be obtained from a network of exactly 7 one-ohm resistors, but not from any network with fewer than 7 one-ohm resistors. Denominators are in A338597.

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