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 1-6 of 6 results.

A087112 Triangle in which the n-th row contains n distinct semiprimes not listed previously with all prime factors from among the first n primes.

Original entry on oeis.org

4, 6, 9, 10, 15, 25, 14, 21, 35, 49, 22, 33, 55, 77, 121, 26, 39, 65, 91, 143, 169, 34, 51, 85, 119, 187, 221, 289, 38, 57, 95, 133, 209, 247, 323, 361, 46, 69, 115, 161, 253, 299, 391, 437, 529, 58, 87, 145, 203, 319, 377, 493, 551, 667, 841, 62, 93, 155, 217, 341, 403, 527, 589, 713, 899, 961
Offset: 1

Views

Author

Ray Chandler, Aug 21 2003

Keywords

Comments

Terms through row n, sorted, will provide terms for A077553 through row n*(n+1)/2.

Examples

			Triangle begins:
   4;
   6,   9;
  10,  15,  25;
  14,  21,  35,  49;
  22,  33,  55,  77, 121;
  26,  39,  65,  91, 143, 169;

Links

Crossrefs

Cf. A100484 (left edge), A001248 (right edge), A143215 (row sums), A219603 (central terms of odd-indexed rows); A000040, A065342.
Cf. A276086, A370121.

Programs

  • Haskell
    a087112 n k = a087112_tabl !! (n-1) !! (k-1)
a087112_row n = map (* last ps) ps where ps = take n a000040_list
a087112_tabl = map a087112_row [1..]
-- Reinhard Zumkeller, Nov 25 2012
  • Maple
    T := (n, k) -> ithprime(n) * ithprime(k):
seq(print(seq(T(n, k), k = 1..n)), n = 1..11);  # Peter Luschny, Jun 25 2024
  • Mathematica
    Table[ Prime[j]*Prime[k], {j, 11}, {k, j}] // Flatten (* Robert G. Wilson v, Feb 06 2017 *)
  • PARI
    A087112(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2); (prime(1+c) * prime(1+(n-binomial(1+c, 2)))); }; \\ Antti Karttunen, Feb 29 2024

Formula

The n-th row consists of n terms, prime(n)*prime(i), i=1..n.
T(n, k) = A000040(n) * A000040(k).
For n >= 2, a(n) = A276086(A370121(n-1)). - Antti Karttunen, Feb 29 2024

A173786 Triangle read by rows: T(n,k) = 2^n + 2^k, 0 <= k <= n.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, 20, 24, 32, 33, 34, 36, 40, 48, 64, 65, 66, 68, 72, 80, 96, 128, 129, 130, 132, 136, 144, 160, 192, 256, 257, 258, 260, 264, 272, 288, 320, 384, 512, 513, 514, 516, 520, 528, 544, 576, 640, 768, 1024, 1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2048
Offset: 0

Views

Author

Reinhard Zumkeller, Feb 28 2010

Keywords

Comments

Essentially the same as A048645. - T. D. Noe, Mar 28 2011

Examples

			Triangle begins as:
     2;
     3,    4;
     5,    6,    8;
     9,   10,   12,   16;
    17,   18,   20,   24,   32;
    33,   34,   36,   40,   48,   64;
    65,   66,   68,   72,   80,   96,  128;
   129,  130,  132,  136,  144,  160,  192,  256;
   257,  258,  260,  264,  272,  288,  320,  384,  512;
   513,  514,  516,  520,  528,  544,  576,  640,  768, 1024;
  1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2048;

Links

Crossrefs

Cf. A048645, A118413, A118416.
Cf. also A087112, A370121.

Programs

  • Magma
    [2^n + 2^k: k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 07 2021
  • Mathematica
    Flatten[Table[2^n + 2^m, {n,0,10}, {m, 0, n}]] (* T. D. Noe, Jun 18 2013 *)
  • PARI
    A173786(n) = { my(c = (sqrtint(8*n + 1) - 1) \ 2); 1 << c + 1 << (n - binomial(c + 1, 2)); }; \\ Antti Karttunen, Feb 29 2024, after David A. Corneth's PARI-program in A048645
  • Python
    from math import isqrt, comb
def A173786(n):
    a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
    return (1<Chai Wah Wu, Jun 20 2025
  • Sage
    flatten([[2^n + 2^k for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 07 2021

Formula

1 <= A000120(T(n,k)) <= 2.
For n>0, 0<=kA048645(n+1,k+2) and T(n,n) = A048645(n+2,1).
Row sums give A006589(n).
Central terms give A161168(n).
T(2*n+1,n) = A007582(n+1).
T(2*n+1,n+1) = A028403(n+1).
T(n,k) = A140513(n,k) - A173787(n,k), 0<=k<=n.
T(n,k) = A059268(n+1,k+1) + A173787(n,k), 0
T(n,k) * A173787(n,k) = A173787(2*n,2*k), 0<=k<=n.
T(n,0) = A000051(n).
T(n,1) = A052548(n) for n>0.
T(n,2) = A140504(n) for n>1.
T(n,3) = A175161(n-3) for n>2.
T(n,4) = A175162(n-4) for n>3.
T(n,5) = A175163(n-5) for n>4.
T(n,n-4) = A110287(n-4) for n>3.
T(n,n-3) = A005010(n-3) for n>2.
T(n,n-2) = A020714(n-2) for n>1.
T(n,n-1) = A007283(n-1) for n>0.
T(n,n) = 2*A000079(n).

Extensions

Typo in first comment line fixed by Reinhard Zumkeller, Mar 07 2010

A370129 Triangle read by rows: T(n,k) = A003415(A002110(n)+A002110(k)), 0 <= k <= n; arithmetic derivatives of the sums of two primorial numbers.

Original entry on oeis.org

1, 1, 4, 1, 12, 16, 1, 80, 60, 92, 1, 216, 540, 608, 704, 1, 3740, 3100, 4548, 6324, 8164, 568, 60080, 40060, 56292, 116208, 61768, 110752, 33975, 1021040, 1041768, 794468, 2415104, 1091004, 1357128, 1942844, 28300, 9789116, 29099520, 19722884, 18576860, 35347200, 35779644, 26575580, 37935056, 704080, 335024060
Offset: 0

Views

Author

Antti Karttunen, Feb 29 2024

Keywords

Comments

Apart from those positions (A014545) at the left edge where a(n) = 1, a(n) <= A087112(1+n) only at n=2, 4 and 5, i.e., never after the third row.

Examples

			Triangle begins as:
      1;
      1,       4;
      1,      12,       16;
      1,      80,       60,       92;
      1,     216,      540,      608,      704;
      1,    3740,     3100,     4548,     6324,     8164;
    568,   60080,    40060,    56292,   116208,    61768,   110752;
  33975, 1021040,  1041768,   794468,  2415104,  1091004,  1357128,  1942844;
  28300, 9789116, 29099520, 19722884, 18576860, 35347200, 35779644, 26575580, 37935056;

Links

Crossrefs

Cf. A002110, A003415, A024451, A370121, A370135, A370136.
Cf. A014545 (positions of 1's at the left edge), A087112.
Cf. also A024451 (arithmetic derivatives of primorials).

Programs

Formula

a(n) = A003415(A370121(n)).
For n, k >= 1, T(n,k) = A002110(k)*A370136(n,k) + A024451(k)*A370135(n,k).

A370134 Triangle read by rows: T(n,k) = A002110(n) + A002110(k), 1 <= k <= n; sums of two primorials > 1, not necessarily distinct.

Original entry on oeis.org

4, 8, 12, 32, 36, 60, 212, 216, 240, 420, 2312, 2316, 2340, 2520, 4620, 30032, 30036, 30060, 30240, 32340, 60060, 510512, 510516, 510540, 510720, 512820, 540540, 1021020, 9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200, 19399380, 223092872, 223092876, 223092900, 223093080, 223095180, 223122900, 223603380
Offset: 1

Views

Author

Antti Karttunen, Mar 07 2024

Keywords

Examples

			Triangle begins as:
        4;
        8,      12;
       32,      36,      60;
      212,     216,     240,     420;
     2312,    2316,    2340,    2520,    4620;
    30032,   30036,   30060,   30240,   32340,   60060;
   510512,  510516,  510540,  510720,  512820,  540540, 1021020;
  9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200, 19399380;

Links

Crossrefs

A370121 without its leftmost column. Subsequence of A370132.
Cf. A002110, A049345, A087112, A276150, A370135.
Cf. A088860 (right edge).

Programs

  • Mathematica
    nn = 20; MapIndexed[Set[P[First[#2] - 1], #1] &, FoldList[Times, 1, Prime@ Range[nn + 1]]]; Table[(P[n] + P[k]), {n, nn}, {k, n}] (* Michael De Vlieger, Mar 08 2024 *)
  • PARI
    A002110(n) = prod(i=1,n,prime(i));
A370134(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2); (A002110(1+c) + A002110(1+n - binomial(c + 1, 2))); };

Formula

For n >= 1, A276150(a(n)) = 2.

A373844 Triangle read by rows: T(n,k) = A276086(1 + A002110(n) + A002110(k)), 1 <= k <= n, where A276086 is the primorial base exp-function.

Original entry on oeis.org

18, 30, 50, 42, 70, 98, 66, 110, 154, 242, 78, 130, 182, 286, 338, 102, 170, 238, 374, 442, 578, 114, 190, 266, 418, 494, 646, 722, 138, 230, 322, 506, 598, 782, 874, 1058, 174, 290, 406, 638, 754, 986, 1102, 1334, 1682, 186, 310, 434, 682, 806, 1054, 1178, 1426, 1798, 1922, 222, 370, 518, 814, 962, 1258, 1406, 1702, 2146, 2294, 2738
Offset: 1

Views

Author

Antti Karttunen, Jun 21 2024

Keywords

Comments

Triangle giving all products of three primes, of which one is even (2) and two are odd (not necessarily distinct), so that the product is of the form 4m+2.
The only terms such that T(n, k) > A373845(n, k) > 1 are 30, 42, 110 at positions T(2,1), T(3,1), T(4,2), and the corresponding terms in A373845 are 6, 14, 38.

Examples

			Triangle begins as:
   18,
   30,  50,
   42,  70,  98,
   66, 110, 154, 242,
   78, 130, 182, 286, 338,
  102, 170, 238, 374, 442,  578,
  114, 190, 266, 418, 494,  646,  722,
  138, 230, 322, 506, 598,  782,  874, 1058,
  174, 290, 406, 638, 754,  986, 1102, 1334, 1682,
  186, 310, 434, 682, 806, 1054, 1178, 1426, 1798, 1922,
  222, 370, 518, 814, 962, 1258, 1406, 1702, 2146, 2294, 2738,
etc.

Links

Crossrefs

Cf. A002110, A276086, A370121, A373845.
Cf. also A087112, A369979, A373848.

Programs

  • PARI
    A002110(n) = prod(i=1,n,prime(i));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A373844(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2, x=A002110(1+n - binomial(c + 1, 2))); A276086(1+(A002110(1+c)+x)); };

Formula

For n, k >= 1, T(n, k) = A276086(1+A370121(n, k)).
For n, k >= 1, T(n, k) = 2*A087112(n+1, k+1).

A373845 Triangle read by rows: T(n,k) = arithmetic derivative of (1 + A002110(n) + A002110(k)), 1 <= k <= n, where A002110(n) is the n-th primorial number.

Original entry on oeis.org

1, 6, 1, 14, 1, 1, 74, 38, 1, 1, 1551, 338, 1, 1, 1, 21084, 8631, 1330, 1, 1, 3550, 172655, 72938, 1970, 3410, 1, 1, 5822, 3233234, 4157356, 421750, 228491, 10190, 13610, 537398, 289610, 297753138, 32805527, 5188250, 8698439, 761710, 1, 18344100, 1, 6954431, 2156564414, 929540471, 68769335, 335525472, 4283242, 21900155, 348965439, 109820278, 185002, 32593310
Offset: 1

Views

Author

Antti Karttunen, Jun 21 2024

Keywords

Comments

Arithmetic derivatives of the sums of three primorials, of which one is 1 [= A002110(0)], and two are > 1.
Ones occur in positions where 1 + A002110(n) + A002110(k) is a prime.
See also comments in A373844, and in A373848.

Examples

			Triangle begins as:
        1,
        6,        1,
       14,        1,       1,
       74,       38,       1,       1,
     1551,      338,       1,       1,      1,
    21084,     8631,    1330,       1,      1,  3550,
   172655,    72938,    1970,    3410,      1,     1,     5822,
  3233234,  4157356,  421750,  228491,  10190, 13610,   537398, 289610,
297753138, 32805527, 5188250, 8698439, 761710,     1, 18344100,      1, 6954431,
etc.

Links

Crossrefs

Cf. A002110, A003415, A370121, A373844, A373848.
Cf. also A024451, A370129, A370138 (arithmetic derivative applied to the sums of a constant number of primorials).

Programs

  • PARI
    A002110(n) = prod(i=1,n,prime(i));
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A373845(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2, x=A002110(1+n - binomial(c + 1, 2))); A003415(1+(A002110(1+c)+x)); };

Formula

For n, k >= 1, T(n, k) = A003415(1+A370121(n, k)).
Showing 1-6 of 6 results.

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