A165974 -id:A165974 - cp's OEIS frontend

A166347 Irregular triangle read by rows of the frequency with which each value v of floor (j^2/p) occurs within A165974 for each prime p, taken over 1 <= j <= p - 1.

1, 0, 1, 1, 0, 2, 1, 0, 1, 0, 2, 1, 1, 1, 0, 1, 0, 3, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 3, 2, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 4, 1, 2, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 4, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0

Offset: 1

Christopher Hunt Gribble, Oct 12 2009

Part of the irregular triangle from which the sequence is constructed
is shown below.
....v..0..1..2..3..4..5..6..7..8..9.10.11.12.13.14.15.16.17.18.19.20.21.22
..p
..2....1..0
..3....1..1..0
..5....2..1..0..1..0
..7....2..1..1..1..0..1..0
.11....3..1..1..1..1..1..0..1..0..1..0
.13....3..2..1..1..1..0..1..1..0..1..0..1..0
.17....4..1..2..1..1..1..0..1..1..1..0..1..0..1..0..1..0
.19....4..2..1..1..1..1..1..1..1..0..1..1..0..1..0..1..0..1..0
.23....4..2..2..1..1..1..1..1..1..1..0..1..1..0..1..1..0..1..0..1..0..1..0

C. H. Gribble, Flattened irregular triangle read by rows of primes < 300

Cf. A165974

Minor edits by Christopher Hunt Gribble, Oct 13 2009

A166348 Irregular triangle read by rows giving the frequency with which each nonzero value v of floor (j^2/p) occurs within A165974 for each j, taken over all primes p.

Original entry on oeis.org

1, 2, 0, 2, 1, 1, 4, 1, 1, 0, 4, 2, 1, 0, 1, 6, 3, 1, 1, 0, 0, 7, 3, 2, 1, 1, 0, 0, 10, 3, 1, 2, 0, 1, 1, 0, 10, 4, 2, 1, 2, 0, 1, 0, 1, 13, 5, 2, 1, 1, 1, 1, 0, 1, 0, 14, 5, 4, 2, 0, 1, 1, 1, 0, 0, 1, 16, 7, 3, 2, 2, 0, 1, 1, 1, 0, 0, 0, 19, 7, 3, 3, 1, 2, 0, 1, 0, 1, 1, 0, 0

Offset: 1

Christopher Hunt Gribble, Oct 12 2009

			Part of the irregular triangle from which the sequence is constructed is shown below.
....v...1..2..3..4..5..6..7..8..9.10.11.12.13
..j
..1.....
..2.....1
..3.....2..0
..4.....2..1..1
..5.....4..1..1..0
..6.....4..2..1..0..1
..7.....6..3..1..1..0..0
..8.....7..3..2..1..1..0..0
..9....10..3..1..2..0..1..1..0
.10....10..4..2..1..2..0..1..0..1
.11....13..5..2..1..1..1..1..0..1..0
.12....14..5..4..2..0..1..1..1..0..0..1
.13....16..7..3..2..2..0..1..1..1..0..0..0
.14....19..7..3..3..1..2..0..1..0..1..1..0..0

C. H. Gribble, Flattened triangle read by rows as table of n, a(n) for n = 1:495510 (j = 1:995).

Cf. A165974

Minor corrections by Christopher Hunt Gribble, Dec 13 2009

A117490 Number of primes between n and n^2 (with n and n^2 excluded).

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 11, 14, 18, 21, 25, 29, 33, 38, 42, 48, 54, 59, 64, 70, 77, 84, 90, 96, 105, 113, 120, 128, 136, 144, 151, 161, 170, 180, 189, 199, 207, 216, 228, 239, 250, 261, 269, 281, 292, 305, 314, 327, 342, 352, 363, 378, 393, 405, 418, 429, 441, 458, 470

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Mar 22 2006

A famous Japanese mathematics book states that this sequence is nonzero (for n>1) if the Riemann Hypothesis is true, but this statement seems to be false.

If the n-th prime is denoted by p(n) then a(j) = number of nonzero values of floor (j^2/p(n)), over all n >= 1, (derived from A165974). - Christopher Hunt Gribble, Oct 03 2009

			For n = 5: between 5+1 = 6 and 5^2-1 = 24 there are the following six primes: 7, 11, 13, 17, 19, 23.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000720, A010051, A014085, A038107, A060715, A073882, A117491, A165974.

P:=proc(n) local i,j,np; for i from 1 by 1 to n do np:=0; for j from i+1 by 1 to i^2-1 do if isprime(j) then np:=np+1; fi; od; print(np); od; end: P(100);

a[n_] := PrimePi[n^2 - 1] - PrimePi[n]; Array[a, 59] (* Robert G. Wilson v, Apr 06 2006 *)

a(n) = pi(n^2) - pi(n), cf. A000720.

a(n) = A038107(n) - A000720(n) = A073882(n) - A010051(n). - Reinhard Zumkeller, May 20 2010

A166373 Triangle read by rows for floor(j^2 / n) with n >= 2 and 1<=j

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 0, 1, 3, 0, 0, 1, 2, 4, 0, 0, 1, 2, 3, 5, 0, 0, 1, 2, 3, 4, 6, 0, 0, 1, 1, 2, 4, 5, 7, 0, 0, 0, 1, 2, 3, 4, 6, 8, 0, 0, 0, 1, 2, 3, 4, 5, 7, 9, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 10, 0, 0, 0, 1, 1, 2, 3, 4, 6, 7, 9, 11, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 0, 0, 0, 1, 1, 2, 3, 4, 5, 6, 8, 9

Offset: 2

Christopher Hunt Gribble, Oct 13 2009. Offset corrected Oct 18 2009

			Part of the triangle from which the sequence is constructed is shown below.
.....j..1..2..3..4..5..6..7..8..9.10.11.12.13.14
...n
...2....0
...3....0..1
...4....0..1..2
...5....0..0..1..3
...6....0..0..1..2..4
...7....0..0..1..2..3..5
...8....0..0..1..2..3..4..6
...9....0..0..1..1..2..4..5..7
..10....0..0..0..1..2..3..4..6..8
..11....0..0..0..1..2..3..4..5..7..9
..12....0..0..0..1..2..3..4..5..6..8.10
..13....0..0..0..1..1..2..3..4..6..7..9.11
..14....0..0..0..1..1..2..3..4..5..7..8.10.12
..15....0..0..0..1..1..2..3..4..5..6..8..9.11.13

A165974 is contained in this sequence.

Cf. A166381 (column sums), A166375 (row sums).

T(n,j)=j^2\n \\ Charles R Greathouse IV, Jan 16 2012

A165993 a(n) = sum_{j=1..prime(n)-1} floor (j^2/prime(n)).

Original entry on oeis.org

0, 1, 4, 11, 31, 44, 80, 103, 157, 252, 293, 420, 520, 575, 695, 884, 1105, 1180, 1431, 1617, 1704, 2007, 2217, 2552, 3040, 3300, 3439, 3713, 3852, 4144, 5255, 5595, 6120, 6305, 7252, 7457, 8060, 8695, 9141, 9804, 10507, 10740, 11983, 12224, 12740

Offset: 1

Christopher Hunt Gribble, Oct 03 2009

nonn

C. H. Gribble, Table of n, a(n) for n=1..78498 (i.e. for primes < 10^6).

Cf. A166375, A165974, A014817.

Table[Sum[Floor[j^2/n],{j,n-1}],{n,Prime[Range[50]]}] (* Harvey P. Dale, Aug 10 2014 *)

a(n) = sum(j=1, prime(n)-1, floor (j^2/prime(n))) \\ Michel Marcus, Jun 20 2013

a(n)=my(p=prime(n));sum(j=1,p-1,j^2\p) \\ Charles R Greathouse IV, Jun 20 2013

a(n) = A166375(prime(n)-1). - Charles R Greathouse IV, Jun 28 2013

Definition rephrased by R. J. Mathar, Oct 09 2009

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

Christopher Hunt Gribble, Oct 10 2009

nonn

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)
  --   --------------  ------
        0        174195
        1             6
        3            16
        5            25
        7            31
        9            34
       11            41
       13            37
       15            68
       17            45
       19            47
       21            85
       23            68
       25            95
       27            93
       29            83
       31            73
       33           101
       35           103
       37            85
       39           115
       41           109
       43           106
       45           154
       47           107
       49           132
       51           159
       53           114
       55           163
       57           179
       59           128
       61           132
       63           216
       65           164
       67           120
       69           209
       71           150
       73           119
       75           237
       77           216
       79           175
       81           228
       83           150
       85           221
       87           222
       89           192
       91           214
       93           262
       95           241
       97           185
       99           289
      101           196
      103           181
      105           379
      107           189
      109           209
      111           314
      113           239

Christopher Hunt Gribble, Table of n, a(n) for n = 1..1973.

Cf. A165951, A165974, A004273.

A165994 a(n) is the number of nonzero values of floor (j^2/prime(n)), over 1 <= j < prime(n).

Original entry on oeis.org

0, 1, 2, 4, 7, 9, 12, 14, 18, 23, 25, 30, 34, 36, 40, 45, 51, 53, 58, 62, 64, 70, 73, 79, 87, 90, 92, 96, 98, 102, 115, 119, 125, 127, 136, 138, 144, 150, 154, 159, 165, 167, 177, 179, 182, 184, 196, 208, 211, 213, 217, 223, 225, 235, 240, 246, 252, 254, 260

Offset: 1

Christopher Hunt Gribble, Oct 03 2009

nonn

C. H. Gribble, Table of n, a(n) for n = 1..78498 (i.e., for primes < 10^6).

Cf. A165974.

[Floor(NthPrime(n) - Sqrt(NthPrime(n))): n in [1..60]]; // Vincenzo Librandi, Nov 13 2018

Table[Floor[Prime[n] - Sqrt[Prime[n]]], {n, 60}] (* Vincenzo Librandi, Nov 13 2018 *)

a(n) = floor(A000040(n) - sqrt(A000040(n))). - Jon Maiga, Nov 13 2018

Definition rephrased by R. J. Mathar, Oct 09 2009

A165995 a(n) = Sum_{p > n} floor(n^2/p), for primes p.

Original entry on oeis.org

0, 1, 2, 7, 9, 16, 19, 28, 40, 54, 60, 76, 81, 100, 121, 143, 151, 177, 184, 210, 241, 272, 281, 314, 349, 386, 424, 465, 480, 522, 538, 582, 628, 677, 728, 782, 800, 856, 910, 970, 991, 1051, 1072, 1133, 1198, 1263, 1285, 1353, 1424, 1497, 1571

Offset: 1

Christopher Hunt Gribble, Oct 03 2009

nonn

Also the number of integers between n and n^2 with a prime factor > n. - Orson R. L. Peters, Dec 04 2017

C. H. Gribble, Table of j, a(j) for j=1..2236.

Cf. A165974.

Block[{nn = 51, s}, s = Array[FactorInteger[#][[All, 1]] &, nn^2]; Table[Count[s[[#]] & /@ Range[n, n^2], ?(AnyTrue[#, # > n &] &)], {n, nn}]] (* _Michael De Vlieger, Dec 04 2017 *)

Definition corrected by Orson R. L. Peters, Dec 04 2017

A166127 Minimum value of j such that floor(j^2 / prime(n)) > 0.

Original entry on oeis.org

0, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20

Offset: 1

Christopher Hunt Gribble, Oct 07 2009

nonn

Essentially the same as A104103. - R. J. Mathar, Jul 21 2015

C. H. Gribble, Table of n, a(n) for n=1,..., 78498 (i.e. for primes < 10^6).

Cf. A165974

Join[{0},Table[Ceiling[Sqrt[p]],{p,Prime[Range[2,80]]}]] (* Harvey P. Dale, Oct 10 2020 *)

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

Christopher Hunt Gribble, Oct 07 2009

nonn

			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

Christopher Hunt Gribble, Table of n, a(n) for n = 1..1973.

Cf. A165951, A165974, A004273.

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

cp's OEIS Frontend

A166347 Irregular triangle read by rows of the frequency with which each value v of floor (j^2/p) occurs within A165974 for each prime p, taken over 1 <= j <= p - 1.

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A166348 Irregular triangle read by rows giving the frequency with which each nonzero value v of floor (j^2/p) occurs within A165974 for each j, taken over all primes p.

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A117490 Number of primes between n and n^2 (with n and n^2 excluded).

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Maple

Mathematica

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A166373 Triangle read by rows for floor(j^2 / n) with n >= 2 and 1<=j

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PARI

A165993 a(n) = sum_{j=1..prime(n)-1} floor (j^2/prime(n)).

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PARI

PARI

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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.

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A165994 a(n) is the number of nonzero values of floor (j^2/prime(n)), over 1 <= j < prime(n).

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Magma

Mathematica

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Extensions

A165995 a(n) = Sum_{p > n} floor(n^2/p), for primes p.

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Mathematica

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A166127 Minimum value of j such that floor(j^2 / prime(n)) > 0.

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