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.

Showing 1-4 of 4 results.

A166264 If the n-th prime is denoted by p(n) then a(j) = frequency with which each distinct value of (Sum of the quadratic non-residues of p(n) - Sum of the quadratic residues of p(n)) / p(n) occurs.

Original entry on oeis.org

174195, 6, 16, 25, 31, 34, 41, 37, 68, 45, 47, 85, 68, 95, 93, 83, 73, 101, 103, 85, 115, 109, 106, 154, 107, 132, 159, 114, 163, 179, 128, 132, 216, 164, 120, 209, 150, 119, 237, 216, 175, 228, 150, 221, 222, 192, 214, 262, 241, 185, 289, 196, 181, 379, 189
Offset: 1

Views

Author

Christopher Hunt Gribble, Oct 10 2009

Keywords

Comments

The table below shows a(j) for each distinct value of (Sum of the quadratic non-residues of p(n) - Sum of the quadratic residues of p(n)) / p(n) for 1 <= n <= 348513, with p(348513) = 4999999 (< 5*10^6).
a(1) appears to increase indefinitely, so the static sequence starts at a(2).
j (SQN-SQR)/p(n) a(j)
-- -------------- ------
1 0 174195
2 1 6
3 3 16
4 5 25
5 7 31
6 9 34
7 11 41
8 13 37
9 15 68
10 17 45
11 19 47
12 21 85
13 23 68
14 25 95
15 27 93
16 29 83
17 31 73
18 33 101
19 35 103
20 37 85
21 39 115
22 41 109
23 43 106
24 45 154
25 47 107
26 49 132
27 51 159
28 53 114
29 55 163
30 57 179
31 59 128
32 61 132
33 63 216
34 65 164
35 67 120
36 69 209
37 71 150
38 73 119
39 75 237
40 77 216
41 79 175
42 81 228
43 83 150
44 85 221
45 87 222
46 89 192
47 91 214
48 93 262
49 95 241
50 97 185
51 99 289
52 101 196
53 103 181
54 105 379
55 107 189
56 109 209
57 111 314
58 113 239

Links

Crossrefs

Cf. A165951, A165974, A004273.

A166408 a(n) = floor(A166407(n)/3).

Original entry on oeis.org

-1, 0, 0, 1, -3, 1, 0, 2, 0, 1, 0, 3, -10, 0, 0, 3, 0, 2, 0, 4, 0, 1, 0, 5, -21, 2, 0, 4, 0, 3, 0, 2, 0, 1, 0, 7, 0, 0, 0, 5, -27, 3, 0, 6, 0, 2, 0, 8, 0, 0, 0, 5, 0, 3, 0, 8, 0, 2, 0, 10, -55, 2, 0, 5, 0, 5, 0, 2, 0, 3, 0, 10, 0, 0, 0, 7, 0, 4, 0, 10, 0, 1, 0, 11, -78, 2, 0, 2, 0, 5, 0, 8, 0, 2, 0
Offset: 0

Views

Author

Antti Karttunen, Oct 21 2009

Keywords

Comments

See the conjecture in A166407. If true, then a(i) = A166406(i)/A005408(i), whenever i is not in A166101.

Crossrefs

A165951(n)=a(A102781(n)) for n>=2.

Programs

  • Python
    from sympy import floor, jacobi_symbol as J
def a(n):
    l=0
    m=0
    for i in range(1, 2*n + 2):
        if J(i, 2*n + 1)==-1: l+=i
        elif J(i, 2*n + 1)==1: m+=i
    return floor(3*((l - m)/(2*n + 1)))//3
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 12 2017

A166131 a(j) = minimum value of n for each distinct increasing value of (Sum of the quadratic non-residues of prime(n) - Sum of the quadratic residues of prime(n)) / prime(n) for each j.

Original entry on oeis.org

1, 4, 9, 15, 20, 46, 39, 43, 52, 76, 64, 83, 118, 92, 166, 154, 128, 146, 173, 236, 228, 190, 283, 215, 434, 240, 246, 395, 607, 377, 357, 536, 349, 492, 519, 444, 722, 430, 635, 814, 598, 512, 541, 562, 700, 821, 633, 708, 893, 729, 738
Offset: 1

Views

Author

Christopher Hunt Gribble, Oct 07 2009

Keywords

Examples

			The table below shows for each value of a(j) the corresponding values of prime(a(j)) and (Sum of the quadratic non-residues of prime(a(j)) - Sum of the quadratic residues of prime(a(j))) / prime(a(j))
.
   j      a(j)    prime(a(j))   (SQN-SQR)/prime(a(j))
  --      ----    -----------   ---------------------
   1         1          2          0
   2         4          7          1
   3         9         23          3
   4        15         47          5
   5        20         71          7
   6        46        199          9
   7        39        167         11
   8        43        191         13
   9        52        239         15
  10        76        383         17
  11        64        311         19
  12        83        431         21
  13       118        647         23
  14        92        479         25
  15       166        983         27
  16       154        887         29
  17       128        719         31
  18       146        839         33
  19       173       1031         35
  20       236       1487         37
  21       228       1439         39
  22       190       1151         41
  23       283       1847         43
  24       215       1319         45
  25       434       3023         47
  26       240       1511         49
  27       246       1559         51
  28       395       2711         53
  29       607       4463         55
  30       377       2591         57
  31       357       2399         59
  32       536       3863         61
  33       349       2351         63
  34       492       3527         65
  35       519       3719         67
  36       444       3119         69
  37       722       5471         71
  38       430       2999         73
  39       635       4703         75
  40       814       6263         77
  41       598       4391         79
  42       512       3671         81
  43       541       3911         83
  44       562       4079         85
  45       700       5279         87
  46       821       6311         89
  47       633       4679         91
  48       708       5351         93
  49       893       6959         95
  50       729       5519         97
  51       738       5591         99

Links

Crossrefs

Cf. A165951, A165974, A004273.

Extensions

Sequence corrected and comments added by Christopher Hunt Gribble, Oct 10 2009

A166263 a(j) = maximum value of n for each distinct increasing value of (Sum of the quadratic non-residues of prime(n) - Sum of the quadratic residues of prime(n)) / prime(n) for each j.

Original entry on oeis.org

348511, 38, 155, 389, 778, 1296, 1828, 2321, 3683, 3935, 4078, 6184, 8783, 9013, 9880, 15182, 12449, 19828, 18884, 14593, 22316, 25738, 26064, 26670, 31953, 33332, 45025, 35788, 37881, 50299, 39562, 49598, 77850, 56777, 53024, 70443, 71992
Offset: 1

Views

Author

Christopher Hunt Gribble, Oct 10 2009

Keywords

Comments

a(1) appears to increase indefinitely, so the static sequence starts from a(2).
The value of a(1) is the index of the largest prime p < 5*10^6 for which Sum of the quadratic non-residues of p = Sum of the quadratic residues of p.
The table below shows for each value of a(j) the corresponding values of p(a(j)) and (Sum of the quadratic non-residues of p(a(j)) - Sum of the quadratic residues of p(a(j))) / p(a(j)):
.
j a(j) prime(a(j)) (SQN-SQR)/prime(a(j))
-- ------ ----------- ---------------------
1 348511 4999961 0
2 38 163 1
3 155 907 3
4 389 2683 5
5 778 5923 7
6 1296 10627 9
7 1828 15667 11
8 2321 20563 13
9 3683 34483 15
10 3935 37123 17
11 4078 38707 19
12 6184 61483 21
13 8783 90787 23
14 9013 93307 25
15 9880 103387 27
16 15182 166147 29
17 12449 133387 31
18 19828 222643 33
19 18884 210907 35
20 14593 158923 37
21 22316 253507 39
22 25738 296587 41
23 26064 300787 43
24 26670 308323 45
25 31953 375523 47
26 33332 393187 49
27 45025 546067 51
28 35788 425107 53
29 37881 452083 55
30 50299 615883 57
31 39562 474307 59
32 49598 606643 61
33 77850 991027 63
34 56777 703123 65
35 53024 652723 67
36 70443 888427 69
37 71992 909547 71
38 70328 886867 73
39 72479 916507 75

Links

Crossrefs

Cf. A165951, A165974, A004273.
Showing 1-4 of 4 results.

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