A217438 -id:A217438 - cp's OEIS frontend

A217287 Length of chain of consecutive integers starting with n, where each new integer in the chain has a prime factor which no previous member in the chain has.

3, 2, 3, 4, 3, 2, 5, 4, 3, 5, 5, 4, 3, 2, 3, 8, 7, 6, 5, 4, 3, 5, 4, 3, 5, 6, 5, 4, 3, 2, 5, 4, 3, 6, 5, 9, 8, 7, 6, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 3, 8, 7, 6, 5, 7, 7, 6, 5, 4, 3, 2, 7, 8, 7, 6, 5, 4, 3, 5, 9, 8, 7, 6, 5, 5, 4, 3, 11, 10, 9, 8, 7, 6, 5, 10, 9, 8, 7, 6, 5, 4, 3, 6, 5, 9, 8, 7, 9, 8

Offset: 1

Lior Manor, Sep 30 2012

nonn

a(n) >= 2. If n < 2 is prime or prime power, a(n) >= 3. For any n > 1, k > 1, a(n^k - n) <= n.

a(n) is also the smallest k>0 such that n+k is k-smooth (i.e. has no prime factor > k). - N. J. A. Sloane, Apr 25 2020

			Example: a(7)=5 since 7 starts a chain of 5 integers 7-11 with the following property: 7 is divisible by 7, 8 is divisible by 2, 9 is divisible by 3, 10 is divisible by 5, 11 is divisible by 11. And the next integer 12 is divisible by 2 and 3, both of them are prime factors of prior members in the chain.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Lior Manor, First 1000 entries with the associated chains (For n=1, the chain 1,2,3 should be added. - _N. J. A. Sloane_, Apr 25 2020)
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.

Cf. A006530, A217288 and A217289 (records), A217438.

A006530 := n->max(1, op(numtheory[factorset](n)));
a:=[]; M:=120;
for n from 1 to M do
   for k from 1 to 3*n do
   if A006530(n+k) <= k then a:=[op(a),k]; break; fi;
   od;
od:
a; # N. J. A. Sloane, Apr 25 2020

Block[{nn = 111, r}, r = Prime@ Range[PrimePi@ nn]; r = Table[FromDigits[#, 2] &@ Map[Boole[Mod[n, #] == 0] &, r], {n, nn}]; Array[Block[{k = # + 1, s = r[[#]]}, While[UnsameQ[s, Set[s, BitOr[s, r[[k]] ] ] ], k++]; k - #] &, nn - Ceiling@ Sqrt@ nn] ] (* Michael De Vlieger, Apr 30 2020 *)

a(1) = 3 added by N. J. A. Sloane, Apr 25 2020

A217288 Records in A217287.

Original entry on oeis.org

3, 4, 5, 8, 9, 11, 14, 18, 19, 20, 22, 25, 30, 32, 35, 38, 41, 45, 46, 49, 52, 54, 62, 68, 72, 73, 74, 85, 86, 88, 90, 105, 110, 112, 126, 128, 143, 144, 149, 154, 161, 166, 182

Offset: 1

Lior Manor, Sep 30 2012

nonn

Cf. A217287, A217289, A217438.

Beginning changed because of new term a(1)=3 in A217287. - N. J. A. Sloane, Apr 25 2020

A217289 Where records occur in A217287.

Original entry on oeis.org

1, 4, 7, 16, 36, 79, 106, 222, 456, 460, 650, 1046, 1090, 2110, 2205, 2302, 3455, 4946, 6072, 6671, 10388, 12000, 12174, 24772, 26790, 29490, 36862, 39767, 72361, 79240, 86810, 87580, 135014, 172256, 210917, 223312, 293452, 538746, 550786, 585886, 617819, 838074, 879410

Offset: 1

Lior Manor, Sep 30 2012

nonn

Cf. A217287, A217288, A217438.

Beginning changed because of new term a(1)=3 in A217287. - N. J. A. Sloane, Apr 25 2020

A334468 List of distinct values of n + A217287(n).

Original entry on oeis.org

4, 6, 8, 12, 15, 16, 18, 24, 27, 30, 32, 36, 40, 45, 48, 54, 60, 63, 64, 70, 72, 75, 80, 81, 90, 96, 100, 105, 108, 120, 125, 128, 135, 140, 144, 150, 160, 162, 168, 175, 180, 189, 192, 200, 210, 216, 224, 225, 234, 240, 243, 250, 256, 270, 280, 288, 294, 300

Offset: 1

Michael De Vlieger, May 02 2020

nonn

This sequence is a list of primitive least m > n whose distinct prime factors p are not a subset of those prime factors p found in the range n..(m - 1), i.e., the smallest A217287(n)-smooth number m > n. These numbers serve as "obstructions" that end or break the chains described at A217287.

The number (a(n) - 1) can be found in at least one row of A217438. In other words, this sequence includes any number T(n, A217287(n)) + 1 where T(n, k) is the irregular triangle described at A217438.

			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 and considering each value only once, we generate the sequence {4, 6, 8, 12, 15, 16, ...}

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, with the single pixel m (in this sequence) that exceeds the largest term of A217438 in red.
Michael De Vlieger, Analysis of prime decompositions of terms in this sequence.

Cf. A000918, A217287, A217438, A334469.

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

a(n) > n + 2 for all n.

a(m) = m + 2 for m = 2^k - 2 and k > 1, since m is even and 2^k only has the distinct prime divisor 2. Therefore 2^k for k > 1 is in this sequence.

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] ]

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

Original entry on oeis.org

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] ]

cp's OEIS Frontend

A217287 Length of chain of consecutive integers starting with n, where each new integer in the chain has a prime factor which no previous member in the chain has.

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A217288 Records in A217287.

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A217289 Where records occur in A217287.

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A334468 List of distinct values of n + A217287(n).

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A334469 Indices of zero or positive first differences in A217287.

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

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