cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Zachary Russ

Zachary Russ's wiki page.

Zachary Russ has authored 3 sequences.

A333997 Number of electrons per subshell in element Z=n expressed as a 56-bit unsigned integer.

Original entry on oeis.org

1, 2, 6, 10, 26, 42, 58, 74, 90, 106, 234, 362, 874, 1386, 1898, 2410, 2922, 3434, 68970, 134506, 138602, 142698, 146794, 89450, 154986, 159082, 163178, 167274, 109930, 175466, 437610, 699754, 961898, 1224042, 1486186, 1748330
Offset: 1

Views

Author

Zachary Russ, Sep 05 2020

Keywords

Comments

This is a memory-efficient way of encoding the number of electrons per subshell of all known elements.

Examples

			  |-----|----------------------------------------------------------------------------|
  |     |                                a(n) [bin]                                  |
  |  n  |                          Electrons per Subshell                            |
  |     |  7p|7s| 6d | 6p|6s| 5f | 5d | 5p|5s| 4f | 4d | 4p|4s| 3d | 3p|3s| 2p|2s|1s |
  |-----|----|--|----|---|--|----|----|---|--|----|----|---|--|----|---|--|---|--|---|
  |   1 | 000 00 0000 000 00 0000 0000 000 00 0000 0000 000 00 0000 000 00 000 00 01 |
  |   2 | 000 00 0000 000 00 0000 0000 000 00 0000 0000 000 00 0000 000 00 000 00 10 |
  |   3 | 000 00 0000 000 00 0000 0000 000 00 0000 0000 000 00 0000 000 00 000 01 10 |
  |   4 | 000 00 0000 000 00 0000 0000 000 00 0000 0000 000 00 0000 000 00 000 10 10 |
  |   5 | 000 00 0000 000 00 0000 0000 000 00 0000 0000 000 00 0000 000 00 001 10 10 |
  |   6 | 000 00 0000 000 00 0000 0000 000 00 0000 0000 000 00 0000 000 00 010 10 10 |
  |  .  |                                     .                                      |
  |  .  |                                     .                                      |
  |  .  |                                     .                                      |
  | 113 | 001 10 1010 110 10 1110 1010 110 10 1110 1010 110 10 1010 110 10 110 10 10 |
  | 114 | 010 10 1010 110 10 1110 1010 110 10 1110 1010 110 10 1010 110 10 110 10 10 |
  | 115 | 011 10 1010 110 10 1110 1010 110 10 1110 1010 110 10 1010 110 10 110 10 10 |
  | 116 | 100 10 1010 110 10 1110 1010 110 10 1110 1010 110 10 1010 110 10 110 10 10 |
  | 117 | 101 10 1010 110 10 1110 1010 110 10 1110 1010 110 10 1010 110 10 110 10 10 |
  | 118 | 110 10 1010 011 10 1110 1010 110 10 1110 1010 110 10 1010 110 10 110 10 10 |
  |-----|----------------------------------------------------------------------------|

Links

Crossrefs

Cf. A333662, A168208, A093907.

A333662 Number of electrons per shell in element Z=n expressed as a 32-bit unsigned integer.

Original entry on oeis.org

1, 2, 6, 10, 14, 18, 22, 26, 30, 34, 98, 162, 226, 290, 354, 418, 482, 546, 2594, 4642, 4706, 4770, 4834, 2914, 4962, 5026, 5090, 5154, 3234, 5282, 7330, 9378, 11426, 13474, 15522, 17570, 148642, 279714, 281762, 283810
Offset: 1

Views

Author

Zachary Russ, Sep 02 2020

Keywords

Comments

This is a memory-efficient way of encoding the number of electrons per shell of all known elements.

Examples

			  |-----|------------|------------|----------------------------------------|
  |     |            |            |               a(n) [bin]               |
  |  n  | a(n) [dec] | a(n) [hex] |           Electrons per Shell          |
  |     |            |            |  Q  |  P  |   O  |  N   |  M  |  L | K |
  |-----|------------|------------|-----|-----|------|------|-----|----|---|
  |   1 |          1 | 0x00000001 | 0000 00000 000000 000000 00000 0000 01 |
  |   2 |          2 | 0x00000002 | 0000 00000 000000 000000 00000 0000 10 |
  |   3 |          6 | 0x00000006 | 0000 00000 000000 000000 00000 0001 10 |
  |   4 |         10 | 0x0000000a | 0000 00000 000000 000000 00000 0010 10 |
  |   5 |         14 | 0x0000000e | 0000 00000 000000 000000 00000 0011 10 |
  |   6 |         18 | 0x00000012 | 0000 00000 000000 000000 00000 0100 10 |
  |  .  |     .      |      .     |                    .                   |
  |  .  |     .      |      .     |                    .                   |
  |  .  |     .      |      .     |                    .                   |
  | 113 |  960562338 | 0x394104a2 | 0011 10010 100000 100000 10010 1000 10 |
  | 114 | 1228997794 | 0x494104a2 | 0100 10010 100000 100000 10010 1000 10 |
  | 115 | 1497433250 | 0x594104a2 | 0101 10010 100000 100000 10010 1000 10 |
  | 116 | 1765868706 | 0x694104a2 | 0110 10010 100000 100000 10010 1000 10 |
  | 117 | 2034304162 | 0x794104a2 | 0111 10010 100000 100000 10010 1000 10 |
  | 118 | 2302739618 | 0x894104a2 | 1000 10010 100000 100000 10010 1000 10 |
  |-----|------------|------------|----------------------------------------|

Links

Crossrefs

Cf. A333997, A168208, A093907.

A322557 Smallest k such that floor(N*sqrt(Sum_{m=1..k} 6/m^2)) = floor(N*Pi), where N = 10^n.

Original entry on oeis.org

7, 23, 600, 1611, 10307, 359863, 1461054, 17819245, 266012440, 1619092245, 10634761313, 97509078554, 1203836807622, 10241799698090, 294871290395291, 4004525174270251, 24827457879988026, 112840588371964574, 2064072875704476882, 15243903003939891921
Offset: 0

Views

Author

Zachary Russ, Aug 28 2019

Keywords

Comments

6*A007406(k)/A007407(k) = Sum_{m=1..k} 6/m^2.
It seems nearly certain that, for all n >= 0, a(n) = ceiling(z - 1/2 - 1/(12*z)) where z = 6/(Pi^2 - (floor(Pi*10^n)/10^n)^2). - Jon E. Schoenfield, Aug 31 2019

Examples

			floor((10^0)*sqrt(Sum_{m=1..7} 6/m^2)) = 3.
floor((10^1)*sqrt(Sum_{m=1..23} 6/m^2)) = 31.
floor((10^2)*sqrt(Sum_{m=1..600} 6/m^2)) = 314.
floor((10^3)*sqrt(Sum_{m=1..1611} 6/m^2)) = 3141.
floor((10^4)*sqrt(Sum_{m=1..10307} 6/m^2)) = 31415.
floor((10^5)*sqrt(Sum_{m=1..359863} 6/m^2)) = 314159.

Links

Crossrefs

Cf. A000796, A013661, A013679, A002388, A274982, A007406, A007407.
Cf. A011545 (floor(Pi*10^n)).

Programs

  • PARI
    a(n) = {my(k = 1); t = floor(10^(n)*Pi); while(floor(10^(n)*sqrt(sum(m = 1, k, 6/m^2))) != t, k++); k; } \\ Jinyuan Wang, Aug 30 2019

Extensions

a(6)-a(19) from Jon E. Schoenfield, Aug 31 2019

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.