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.

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A053650 Cototient function of n^2.

Original entry on oeis.org

0, 2, 3, 8, 5, 24, 7, 32, 27, 60, 11, 96, 13, 112, 105, 128, 17, 216, 19, 240, 189, 264, 23, 384, 125, 364, 243, 448, 29, 660, 31, 512, 429, 612, 385, 864, 37, 760, 585, 960, 41, 1260, 43, 1056, 945, 1104, 47, 1536, 343, 1500, 969, 1456, 53, 1944, 825, 1792, 1197
Offset: 1

Views

Author

Labos Elemer, Feb 18 2000

Keywords

Comments

Seems to be invertible like n*Phi(n). Compare with A002618, A038040.

Links

Crossrefs

Cf. A000005, A038040.
Cf. A001248, A002618, A053650, A053192, A053193, A036689, A229099.

Programs

Formula

a(n) = n*(n - phi(n)) = n^2 - n*phi(n) = Cototient(n^2) = A051953(A000290(n)).
a(n) = n^2 - A002618(n).
For p prime, Cototient(p)=1 and a(p)=p.
a(n) = n*cototient(n) = n*A051953(n). - Omar E. Pol, Nov 22 2012
Dirichlet g.f.: zeta(s-2)*(1 - 1/zeta(s-1)). - Ilya Gutkovskiy, Jul 26 2016
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 1 - 6/Pi^2 (A229099). - Amiram Eldar, Dec 15 2023

A127918 Half of product of three numbers: n-th prime, previous and following number.

Original entry on oeis.org

3, 12, 60, 168, 660, 1092, 2448, 3420, 6072, 12180, 14880, 25308, 34440, 39732, 51888, 74412, 102660, 113460, 150348, 178920, 194472, 246480, 285852, 352440, 456288, 515100, 546312, 612468, 647460, 721392, 1024128, 1123980, 1285608
Offset: 1

Views

Author

Artur Jasinski, Feb 06 2007

Keywords

Comments

Apart from the first term, the same as A117762. - R. J. Mathar, Jun 14 2008
Except the first term, a(n) is the area of the integer-sided isosceles triangle ABC with AB=AC such that the altitude AH is of prime(n) length.
The couples (a(n), altitude) are (12,3), (60,5), (168,7), (660,11), (1092,13), ... and the sequence of the ratio a(n)/prime(n) is {4, 12, 24, 60, 84, 144, 180, ...} - see A084921. - Michel Lagneau, Oct 23 2013
a(n) is also equal to the number of reducible quadratic polynomials in the field of size prime(n). - James East, Apr 26 2024

Links

Crossrefs

Cf. A036689, A034953, A127917, A127920.

Programs

  • Magma
    [(NthPrime(n)+1)*NthPrime(n)*(NthPrime(n)-1)/2: n in [1..40]]; // Vincenzo Librandi, Apr 09 2017
  • Mathematica
    Table[(Prime[n] + 1) Prime[n](Prime[n] - 1)/2, {n, 1, 100}]
  • PARI
    forprime(p=2,1e3,print1(3*binomial(p+1,3)", ")) \\ Charles R Greathouse IV, Jun 16 2011

A127919 1/3 of product of three numbers: the n-th prime, the previous number and the following number.

Original entry on oeis.org

2, 8, 40, 112, 440, 728, 1632, 2280, 4048, 8120, 9920, 16872, 22960, 26488, 34592, 49608, 68440, 75640, 100232, 119280, 129648, 164320, 190568, 234960, 304192, 343400, 364208, 408312, 431640, 480928, 682752, 749320, 857072, 895160, 1102600
Offset: 1

Views

Author

Artur Jasinski, Feb 06 2007

Keywords

Comments

Number of irreducible monic cubic polynomials over GF(prime(n)). - Robert Israel, Jan 06 2015

Links

Crossrefs

Cf. A036689, A034953, A127917, A127918, A127920.

Programs

  • Magma
    [(p^3 - p) div 3: p in PrimesUpTo(150)]; // Vincenzo Librandi, Jan 08 2015
  • Maple
    seq((ithprime(n)^3 - ithprime(n))/3, n=1..100); # Robert Israel, Jan 06 2015
  • Mathematica
    Table[(Prime[n] + 1) Prime[n] (Prime[n] - 1)/3, {n, 100}]
  • PARI
    forprime(p=2,1e3,print1(2*binomial(p+1,3)", ")) \\ Charles R Greathouse IV, Jun 16 2011

Formula

a(n) = (prime(n)^3 - prime(n))/3. - Wesley Ivan Hurt, Oct 15 2023

A138402 a(n) = (n-th prime)^4-(n-th prime)^2.

Original entry on oeis.org

12, 72, 600, 2352, 14520, 28392, 83232, 129960, 279312, 706440, 922560, 1872792, 2824080, 3416952, 4877472, 7887672, 12113880, 13842120, 20146632, 25406640, 28392912, 38943840, 47451432, 62734320, 88519872, 104050200, 112540272
Offset: 1

Views

Author

Artur Jasinski, Mar 19 2008

Keywords

Links

Crossrefs

Cf. A001248, A036689, A036690, A065471, A084920.

Programs

  • Magma
    [NthPrime((n))^4 - NthPrime((n))^2: n in [1..30] ]; // Vincenzo Librandi, Jun 17 2011
  • Mathematica
    a = {}; Do[p = Prime[n]; AppendTo[a, p^4 - p^2], {n, 1, 50}]; a
#^4-#^2&/@Prime[Range[30]] (* Harvey P. Dale, Sep 19 2018 *)
  • PARI
    forprime(p=2,1e3,print1(p^4-p^2", ")) \\ Charles R Greathouse IV, Jun 16 2011
  • PARI
    apply(p->p^4-p^2, primes(100)) \\ Charles R Greathouse IV, Apr 17 2015

Formula

Product_{n>=1} (1 - 1/a(n)) = A065471.
From Amiram Eldar, Nov 22 2022: (Start)
a(n) = A001248(n) * A084920(n).
a(n) = A036689(n) * A036690(n). (End)

A074791 Numbers k such that k does not divide the denominator of the k-th harmonic number.

Original entry on oeis.org

6, 18, 20, 21, 33, 42, 54, 63, 66, 77, 100, 110, 120, 156, 162, 189, 198, 272, 294, 336, 342, 363, 377, 435, 486, 500, 506, 559, 567, 594, 600, 610, 629, 685, 703, 812, 847, 880, 924, 930, 957, 1067, 1166, 1210, 1243, 1247, 1287, 1320, 1332, 1458, 1590, 1640
Offset: 1

Views

Author

Benoit Cloitre, Sep 07 2002

Keywords

Comments

k such that A064169(k) is different from A027612(k).
Also k such that A096617(k) is different from A001008(k). - Alexander Adamchuk, Jun 26 2006
This sequence contains A036689(k) for all k > 1. - Wouter van Doorn, Nov 06 2024

Links

Crossrefs

Cf. A027612, A064169.
Cf. A096617, A001008.

Programs

  • Mathematica
    Select[ Range[1700], Mod[ Denominator[ HarmonicNumber[ # ]], # ] != 0 &] (* Robert G. Wilson v, Sep 28 2005 *)
seq = {}; s = 0; Do[s += 1/n; If[! Divisible[Denominator[s], n], AppendTo[seq, n]], {n, 1, 2000}]; seq (* Amiram Eldar, Dec 01 2020 *)

Formula

Is a(n) asymptotic to c*n^2 0.5
a(n) < 2*n^2*log(n)^2 for all n > 2. This follows from the fact that for all k > 1 there exists an n such that A036689(k) is equal to A074791(n). - Wouter van Doorn, Nov 06 2024

Extensions

Better description and more terms from Robert G. Wilson v, Sep 28 2005

A257251 Square array A(row,col) = A083221(row,col+1) - A083221(row,col): the first differences of each row of array constructed from the sieve of Eratosthenes.

Original entry on oeis.org

2, 2, 6, 2, 6, 20, 2, 6, 10, 42, 2, 6, 20, 28, 110, 2, 6, 10, 14, 22, 156, 2, 6, 20, 28, 44, 52, 272, 2, 6, 10, 14, 22, 26, 34, 342, 2, 6, 20, 28, 44, 52, 68, 76, 506, 2, 6, 10, 42, 66, 78, 102, 114, 138, 812, 2, 6, 20, 14, 22, 26, 34, 38, 46, 58, 930, 2, 6, 10, 42, 66, 78, 102, 114, 138, 174, 186, 1332
Offset: 1

Views

Author

Antti Karttunen, Apr 19 2015

Keywords

Comments

The array is read by downwards antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Examples

			The top left corner of the array:
     2,   2,   2,   2,   2,   2,   2,   2,   2,   2,   2,   2,   2,   2,   2
     6,   6,   6,   6,   6,   6,   6,   6,   6,   6,   6,   6,   6,   6,   6
    20,  10,  20,  10,  20,  10,  20,  10,  20,  10,  20,  10,  20,  10,  20
    42,  28,  14,  28,  14,  28,  42,  14,  42,  28,  14,  28,  14,  28,  42
   110,  22,  44,  22,  44,  66,  22,  66,  44,  22,  44,  66,  66,  22,  66
   156,  52,  26,  52,  78,  26,  78,  52,  26,  52,  78,  78,  26,  78,  52
   272,  34,  68, 102,  34, 102,  68,  34,  68, 102, 102,  34, 102,  68,  34
   342,  76, 114,  38, 114,  76,  38,  76, 114, 114,  38, 114,  76,  38, 114
   506, 138,  46, 138,  92,  46,  92, 138, 138,  46, 138,  92,  46, 138,  92
   812,  58, 174, 116,  58, 116, 174, 174,  58, 174, 116,  58, 174, 116, 174
   930, 186, 124,  62, 124, 186, 186,  62, 186, 124,  62, 186, 124, 186, 248
  1332, 148,  74, 148, 222, 222,  74, 222, 148,  74, 222, 148, 222, 296, 148
  1640,  82, 164, 246, 246,  82, 246, 164,  82, 246, 164, 246, 328, 164,  82
  1806, 172, 258, 258,  86, 258, 172,  86, 258, 172, 258, 344, 172,  86, 172
  2162, 282, 282,  94, 282, 188,  94, 282, 188, 282, 376, 188,  94, 188,  94
  2756, 318, 106, 318, 212, 106, 318, 212, 318, 424, 212, 106, 212, 106, 212
  ...

Links

Crossrefs

Transpose: A257252.
Column 1: A036689.
Row 4: 7 * A145011.
Cf. A083221, A257253 (same array but with terms divided by 2).
Cf. arrays A257255 and A257257, also A257513.

Programs

Formula

A(row,col) = A083221(row,col+1) - A083221(row,col).
A(row,col) = 2*A257253(row,col).

A379010 Square array A(n, k) = phi(A246278(n, k)), read by falling antidiagonals; Euler totient function applied to the prime shift array.

Original entry on oeis.org

1, 2, 2, 2, 6, 4, 4, 8, 20, 6, 4, 18, 24, 42, 10, 4, 12, 100, 60, 110, 12, 6, 24, 40, 294, 120, 156, 16, 8, 20, 120, 72, 1210, 192, 272, 18, 6, 54, 48, 420, 160, 2028, 288, 342, 22, 8, 40, 500, 96, 1320, 216, 4624, 396, 506, 28, 10, 36, 168, 2058, 180, 2496, 352, 6498, 616, 812, 30, 8, 24, 200, 660, 13310, 264, 4896, 504, 11638, 840, 930, 36
Offset: 1

Views

Author

Antti Karttunen, Dec 14 2024

Keywords

Comments

Each column is strictly increasing.

Examples

			The top left corner of the array:
k=  |  1     2     3      4     5      6     7        8      9     10
2k= |  2     4     6      8    10     12    14       16     18     20
----+-------------------------------------------------------------------
1   |  1,    2,    2,     4,    4,     4,    6,       8,     6,     8,
2   |  2,    6,    8,    18,   12,    24,   20,      54,    40,    36,
3   |  4,   20,   24,   100,   40,   120,   48,     500,   168,   200,
4   |  6,   42,   60,   294,   72,   420,   96,    2058,   660,   504,
5   | 10,  110,  120,  1210,  160,  1320,  180,   13310,  1560,  1760,
6   | 12,  156,  192,  2028,  216,  2496,  264,   26364,  3264,  2808,
7   | 16,  272,  288,  4624,  352,  4896,  448,   78608,  5472,  5984,
8   | 18,  342,  396,  6498,  504,  7524,  540,  123462,  9108,  9576,
9   | 22,  506,  616, 11638,  660, 14168,  792,  267674, 17864, 15180,
10  | 28,  812,  840, 23548, 1008, 24360, 1120,  682892, 26040, 29232,
11  | 30,  930, 1080, 28830, 1200, 33480, 1260,  893730, 39960, 37200,
12  | 36, 1332, 1440, 49284, 1512, 53280, 1656, 1823508, 59040, 55944,

Links

Crossrefs

Cf. A000010, A246278, A379011.
Cf. A062570 (row 1), A006093 (column 1), A036689 (column 2), A083553 (column 3), A135177 (column 4).

Programs

  • PARI
    up_to = 11325; \\ = binomial(150+1,2)
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A379010sq(row,col) = eulerphi(A246278sq(row,col));
A379010list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A379010sq(col,(a-(col-1))))); (v); };
v379010 = A379010list(up_to);
A379010(n) = v379010[n];

A127920 1/6 of product of three numbers: n-th prime, previous and following number.

Original entry on oeis.org

1, 4, 20, 56, 220, 364, 816, 1140, 2024, 4060, 4960, 8436, 11480, 13244, 17296, 24804, 34220, 37820, 50116, 59640, 64824, 82160, 95284, 117480, 152096, 171700, 182104, 204156, 215820, 240464, 341376, 374660, 428536, 447580, 551300, 573800, 644956
Offset: 1

Views

Author

Artur Jasinski, Feb 06 2007

Keywords

Links

Crossrefs

Cf. A036689, A034953, A127917, A127918, A127919.

Programs

  • Magma
    [(NthPrime(n) + 1)*NthPrime(n)*(NthPrime(n) - 1)/6: n in [1..40]]; // Vincenzo Librandi, Apr 09 2017
  • Mathematica
    Table[(Prime[n] + 1) Prime[n](Prime[n] - 1)/6, {n, 1, 100}]
((#-1)#(#+1))/6&/@Prime[Range[40]] (* Harvey P. Dale, Dec 23 2019 *)
  • PARI
    forprime(p=2,1e3,print1(binomial(p+1,3)", ")) \\ Charles R Greathouse IV, Jun 16 2011
  • Python
    from sympy import prime
print([(prime(n) - 1)*prime(n)*(prime(n) + 1)//6 for n in range(1, 101)]) # Indranil Ghosh, Apr 09 2017

Formula

a(n) = A127918(n)/3. - Michel Marcus, Apr 09 2017

A139224 M(M-1)/2, where M is Mersenne prime A000668(n).

Original entry on oeis.org

3, 21, 465, 8001, 33542145, 8589737985, 137438167041, 2305843005992468481, 2658455991569831742348849606740148225, 191561942608236107294793377465333618488307184098607105
Offset: 1

Views

Author

Omar E. Pol, May 10 2008

Keywords

Comments

Perfect number A000396(n) minus Mersenne prime A000668(n).

Links

Crossrefs

Cf. A000217, A000396, A000668, A036689, A139096, A139115, A139116, A139223, A139225, A139226, A139256, A139306.

Formula

a(n) = A000668(n)*(A000668(n)-1)/2.
a(n) = A000396(n)-A000668(n).

Extensions

More terms from Max Alekseyev, Mar 09 2009

A119959 p^2-p+1 central polygonal numbers with prime indices A002061(prime(n)).

Original entry on oeis.org

3, 7, 21, 43, 111, 157, 273, 343, 507, 813, 931, 1333, 1641, 1807, 2163, 2757, 3423, 3661, 4423, 4971, 5257, 6163, 6807, 7833, 9313, 10101, 10507, 11343, 11773, 12657, 16003, 17031, 18633, 19183, 22053, 22651, 24493, 26407, 27723, 29757, 31863
Offset: 1

Views

Author

Alexander Adamchuk, Aug 02 2006

Keywords

Comments

Prime terms belong to A074268, which is a subset of A002383, A087126, A034915, A085104.
In every interval of prime(n)^2 integers, a(n) is the number that are not divisible by prime(n) plus the number that are divisible by prime(n)^2. - Peter Munn, Dec 12 2020

Crossrefs

Cf. A002061, A074268, A002383, A087126, A034915, A068468, A085104, A083558.

Programs

  • Mathematica
    Table[Prime[n]^2-Prime[n]+1,{n,1,100}]
  • PARI
    a(n) = {my(p = prime(n)); p^2 - p + 1; } \\ Amiram Eldar, Nov 07 2022

Formula

a(n) = prime(n)^2 - prime(n) + 1.
a(n) = A036689(n)+1. - R. J. Mathar, Aug 13 2019
Product_{n>=1} (1 - 1/a(n)) = zeta(6)/(zeta(2)*zeta(3)) (A068468). - Amiram Eldar, Nov 07 2022
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