A334468 -id:A334468 - cp's OEIS frontend

A333518 a(n) = A000720(A006530(A334468(n))).

1, 2, 1, 2, 3, 1, 2, 2, 2, 3, 1, 2, 3, 3, 2, 2, 3, 4, 1, 4, 2, 3, 3, 2, 3, 2, 3, 4, 2, 3, 3, 1, 3, 4, 2, 3, 3, 2, 4, 4, 3, 4, 2, 3, 4, 2, 4, 3, 6, 3, 2, 3, 1, 3, 4, 2, 4, 3, 5, 6, 4, 3, 2, 4, 4, 4, 3, 7, 3, 4, 2, 4, 3, 3, 6, 4, 6, 2, 4, 4, 3, 5, 6, 8, 7, 3, 2, 5, 3, 4, 1, 4, 5, 3, 5, 4, 4, 2, 4, 5, 3, 5, 6, 3, 4

Offset: 1

Michael De Vlieger, May 05 2020

nonn

Indices of the greatest prime factor of A334468(n).
Consider A334468, a list of numbers m = n+j such that j > 0 is also the smallest number such that n+j has no prime factor > j for some n and j = A217287(n).
Since prime q always contributes a novel prime divisor (i.e., q itself) to the set of distinct primes that divide at least 1 number i the range n + i (1 <= i <= j), the numbers m in A334468 are composite, and given the above, m is a product of relatively small prime factors.

			Start with n = 1, the empty product. Incrementing n and storing the distinct prime factors each time, we encounter 2, which does not divide any previous number n. Therefore we proceed to n = 3, which is prime and its distinct prime divisor again does not divide any previous number. Finally, at 4, we have the distinct prime divisor 2, since 2 divides the product of the previous range {1, 2, 3}, we end the chain. Therefore 4 is the first term of this sequence.
We list row n of A217438 below, starting with n aligned in columns:
1  2  3
   2  3
      3  4  5
         4  5  6  7
            5  6  7
               6  7
                  7  8  9  10  11
                     8  9  10  11
                        9  10  11
                           10  11  12  13  14
                               11  12  13  14  15
                                   12  13  14  15
                                       13  14  15
                                           14  15
                                               ...
Adding 1 to the last numbers seen in all the rows, we generate the sequence A334468: {4, 6, 8, 12, 15, 16, ...}. Of these, we have greatest prime factors {2, 3, 2, 3, 5, 2, ...} with indices {1, 2, 1, 2, 3, 1, ...}.
Least indices of prime(k) in a(n):
   i  p(i)    n    a(n)
  ---------------------
   1    2     1      4
   2    3     2      6
   3    5     5     15
   4    7    18     63
   5   11    59    308
   6   13    49    234
   7   17    68    374
   8   19    84    475
   9   23   292   2392
  10   29   401   3625
  11   31   518   4991
  12   37   791   8547
  ...

Cf. A000720, A006530, A217287, A217438, A334468.

Block[{nn = 2^10, r}, r = Array[If[# == 1, 0, Total[2^(PrimePi /@ FactorInteger[#][[All, 1]] - 1)]] &, nn]; Map[PrimePi@ FactorInteger[#][[-1, 1]] &, #] &@ Union@ Array[Block[{k = # + 1, s = r[[#]]}, While[UnsameQ[s, Set[s, BitOr[s, r[[k]] ] ] ], k++]; k] &, nn - Ceiling@ Sqrt@ nn] ]

A334469 Indices of zero or positive first differences in A217287.

Original entry on oeis.org

1, 3, 4, 7, 10, 11, 15, 16, 22, 25, 26, 31, 34, 36, 41, 46, 52, 56, 57, 63, 64, 70, 71, 76, 79, 86, 94, 96, 99, 106, 116, 121, 127, 131, 134, 142, 146, 156, 160, 162, 169, 176, 183, 190, 196, 204, 214, 218, 221, 222, 236, 241, 246, 255, 266, 274, 286, 288, 296

Offset: 1

Michael De Vlieger, May 02 2020

nonn

Starting with i, we increment i to build a chain of consecutive numbers such that all distinct prime factors of ensuing numbers i + 1, i + 2, etc., divide at least one previous number in the chain. We store the chains in an irregular triangle T(i,j) described in A217438.

This sequence lists rows i such that the last term exceeds that of the previous row.

			We list numbers in row i of A217438 below, starting with i, aligned in columns:
1  2  3
   2  3
      3  4  5
         4  5  6  7
            5  6  7
               6  7
                  7  8  9  10  11
                     8  9  10  11
                        9  10  11
                           10  11  12  13  14
                               11  12  13  14  15
                                   12  13  14  15
                                       13  14  15
                                           14  15
1 is in the sequence since it is the first row.
2 is not in the sequence, since the last term (3) in row 2 of A217438 is equal to that of the previous row.
3 is in the sequence since its last term (5) exceeds that of the previous row (3).
Further, we observe the terms in row i breaking through resistance in the previous row at i = {1, 3, 4, 7, 10, 11, ...}

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Plot (x,y) of x in rows 1 <= y <= 4096 of A217438 in gray, accentuating rows m (in this sequence) in red where no term x in row (y - 1) exists. (Pixels attributed to A334468 appear in yellow).
Michael De Vlieger, Analysis of prime decompositions of terms in this sequence.

Cf. A217287, A217438, A334468.

Block[{nn = 2^9, r}, r = Array[If[# == 1, 0, Total[2^(PrimePi /@ FactorInteger[#][[All, 1]] - 1)]] &, nn]; Position[Prepend[#, 1], _?(# > 0 &)][[All, 1]] &@ Differences@ Array[Block[{k = # + 1, s = r[[#]]}, While[UnsameQ[s, Set[s, BitOr[s, r[[k]] ] ] ], k++]; k] &, nn - Ceiling@ Sqrt@ nn] ]

cp's OEIS Frontend

A333518 a(n) = A000720(A006530(A334468(n))).

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