A104656 -id:A104656 - cp's OEIS frontend

A104655 Let c(i) = A018252(i) be the i-th nonprime; then the final entry in row n of A101513 is c(a(n)) (see A104656), for n >= 3.

4, 8, 11, 17, 22, 30, 37, 46, 55, 66, 77, 90, 103, 117, 132, 148, 166, 183, 201, 222, 242, 264, 287, 310, 334, 360, 387, 413, 442, 470, 500, 533, 564, 598, 631, 665, 701, 738, 775, 813, 853, 893, 936, 979, 1022, 1065, 1110, 1154, 1202, 1251, 1299, 1349, 1401

Offset: 3

N. J. A. Sloane, Apr 22 2005

The entries up to 66 have been checked, but the remaining entries are based on my hypothetical formula (see A101513). (It would be easy to check them.)

I have checked that these are correct entries from the triangle, but not that they match the hypothetical formula (from A101513). - Joshua Zucker, May 20 2006

Cf. A101513, A104656, A073425.

More terms from Joshua Zucker, May 20 2006

A101513 a(1) = 1, a(2) = 2, a(3) = 3; triangle where n-th row has lowest n positive integers not yet in the sequence such that each integer has a prime divisor in common with at least one element of the (n-1)th row.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 5, 7, 15, 16, 18, 20, 21, 22, 24, 25, 26, 11, 13, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 17, 19, 45, 46, 48, 49, 50, 51, 52, 23, 54, 55, 56, 57, 58, 60, 62, 63, 64, 29, 31, 65, 66, 68, 69, 70, 72, 74, 75, 76, 37, 77, 78, 80, 81, 82

Offset: 1

Leroy Quet, Jan 25 2005

Is this a permutation of the positive integers?
Conjectures from N. J. A. Sloane, Apr 22 2005: (Start)
"Call a number "postponed" if it cannot be placed right away, that is, if it is relatively prime to the numbers in the previous row. Then I conjecture that:
"(1) a number n >= 4 is postponed iff n is prime,
"(2) every number appears,
"(3) the primes appear in order,
"(4) 2p (p prime) will appear in one row and p will appear in the next row,
"(5) let c(i) = A018252(i) be the i-th nonprime and define a sequence k(n) [see A104655], n >= 3, by k(3) = 4 and for n >= 4, n*(n+1)/2 = pi( floor( c(k(n-1))/2 ) ) + k(n). Then the final term in row n, for n >= 3, is c(k(n)) [A104656]." (End)

			Triangle begins:
   1;
   2,  3;
   4,  6,  8;
   9, 10, 12, 14;
   5,  7, 15, 16, 18;
  20, 21, 22, 24, 25, 26;
  11, 13, 27, 28, 30, 32, 33;
  ...
7 is in the 5th row because it does not occur earlier and 14 is in the 4th row.

Michael De Vlieger, Table of n, a(n) for n = 1..11628 (rows 1 <= n <= 150).

Cf. the EKG sequence A064413. See also A104654, A104655, A104656.

f[w_List] := Block[{k = 4, m = {}}, Do[While[Nand[FreeQ[Join[w, m], k], AnyTrue[Last@ w, GCD[k, #] > 1 &]], k++]; AppendTo[m, k], {i, Length@ w + 1}]; m]; Nest[Append[#, f@ #] &, Table[n + k - 1, {n, 2}, {k, n}], 10] // Flatten (* Michael De Vlieger, Sep 25 2017 *)

More terms from Joshua Zucker, May 20 2006

A104654 Leading entry in row n of triangle A101513.

Original entry on oeis.org

1, 2, 4, 9, 5, 20, 11, 34, 17, 23, 29, 37, 41, 47, 53, 61, 71, 79, 89, 97, 107, 127, 131, 149, 157, 167, 181, 197, 211, 227, 241, 257, 277, 293, 307, 331, 347, 367, 389, 409, 431, 449, 467, 491, 521, 541, 563, 587, 613, 641, 673, 691, 719, 751, 773, 809, 839, 863

Offset: 1

N. J. A. Sloane, Apr 22 2005

Cf. A101513, A104655, A104656.

More terms from Joshua Zucker, May 20 2006

cp's OEIS Frontend

A104655 Let c(i) = A018252(i) be the i-th nonprime; then the final entry in row n of A101513 is c(a(n)) (see A104656), for n >= 3.

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A101513 a(1) = 1, a(2) = 2, a(3) = 3; triangle where n-th row has lowest n positive integers not yet in the sequence such that each integer has a prime divisor in common with at least one element of the (n-1)th row.

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A104654 Leading entry in row n of triangle A101513.

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