A370135 -id:A370135 - cp's OEIS frontend

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.

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

Antti Karttunen, Feb 29 2024

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.

			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;

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).

A002110(n) = prod(i=1,n,prime(i));
A370121(n) = { my(c = (sqrtint(8*n + 1) - 1) \ 2); (A002110(c) + A002110(n - binomial(c + 1, 2))); };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A370129(n) = A003415(A370121(n));

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

Antti Karttunen, Mar 07 2024

			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;

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

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 *)

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))); };

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

A370136 Triangle read by rows: T(n,k) = arithmetic derivative of ((A002110(n) + A002110(k)) / A002110(k)), 1 <= k <= n.

Original entry on oeis.org

1, 4, 1, 32, 5, 1, 55, 60, 12, 1, 1292, 195, 71, 16, 1, 22532, 2505, 841, 384, 9, 1, 382892, 102723, 8897, 8640, 191, 21, 1, 2469635, 3502740, 323328, 34133, 9980, 756, 24, 1, 111738812, 18755325, 10308201, 1568312, 50621, 5211, 371, 44, 1, 4853127108, 2003156919, 107924801, 178347008, 2376149, 251367, 6339, 672, 31, 1

Offset: 1

Antti Karttunen, Mar 07 2024

			Triangle begins as:
          1;
          4,        1;
         32,        5,        1;
         55,       60,       12,       1;
       1292,      195,       71,      16,     1;
      22532,     2505,      841,     384,     9,    1;
     382892,   102723,     8897,    8640,   191,   21,   1;
    2469635,  3502740,   323328,   34133,  9980,  756,  24,  1;
  111738812, 18755325, 10308201, 1568312, 50621, 5211, 371, 44, 1;

Cf. A002110, A003415, A370129, A370135.

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]));
A370135(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2, x=A002110(1+n - binomial(c + 1, 2))); ((A002110(1+c)+x)/x); };
A370136(n) = A003415(A370135(n));

a(n) = A003415(A370135(n)).

cp's OEIS Frontend

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.

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

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A370136 Triangle read by rows: T(n,k) = arithmetic derivative of ((A002110(n) + A002110(k)) / A002110(k)), 1 <= k <= n.

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