A374445 -id:A374445 - cp's OEIS frontend

A374351 Lexicographically earliest infinite sequence of distinct positive integers such that for any triple i,j,k of consecutive terms, gcd(i,k) = 1 and A007947(ijk) is a term in A002110.

1, 2, 3, 5, 4, 6, 9, 25, 8, 12, 15, 7, 14, 30, 11, 49, 60, 10, 77, 21, 20, 16, 27, 35, 22, 18, 105, 55, 26, 42, 165, 13, 28, 330, 33, 91, 40, 66, 63, 65, 44, 84, 45, 121, 56, 75, 81, 32, 50, 135, 147, 64, 80, 189, 99, 100, 70, 231, 39, 130, 154, 51, 195, 308, 17, 585, 462, 34, 325, 693

Offset: 1

David James Sycamore, Jul 09 2024

nonn

In other words a(n) is least k such that (k, a(n-2)) = 1 and rad(a(n-2)*a(n-1)*k) is a primorial number (alternatively i*j*k is in A055932).

Conjectures: A permutation of the positive integers, a(n) = prime p (> 2) iff p is least unused odd term, 2|i, and rad(i*j) is a primorial number divisible by all primes < p but not by p; primes appear in natural order.

			a(1,2,3) = 1,2,3 the lexicographically earliest triple of numbers satisfying the definition: 3 and 1 are coprime whilst 1*2*3 = 6 = A002110(2).
a(4) = 5 because (2,5) = 1 and 2*3*5 = 30 = A002110(3).
a(9,10,11) = 8,12,15 and 7 is the smallest novel number prime to 12 and rad(12*15*7) = 210 = A002110(4).

Dominic McCarty, Table of n, a(n) for n = 1..10000
Dominic McCarty, Java program for A374351
Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^4, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple represents powerful numbers that are not prime powers.
Michael De Vlieger, Plot prime(i)^m | n at (x,y) = (n,i), n = 1..2048, 8X vertical exaggeration, with a color function showing m = 1 in black, m = 2 in red, m = 3 in orange, ..., largest m in the dataset in magenta.

Cf. A002110, A007947, A055932, A374445.

Java
```
See Dominic McCarty link.
```

nn = 120; c[_] := False;
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; i = a[1]; j = a[2]; u = 3;
f[x_] := f[x] = Or[IntegerQ@ Log2[x], And[EvenQ[x], Union@ Differences@ PrimePi@ FactorInteger[x][[All, 1]] == {1}]];
Do[k = u; While[Or[! CoprimeQ[i, k], c[k], ! f[i*j*k]], k++];
  Set[{a[n], c[k], i, j}, {k, True, j, k}];
  If[k == u, While[c[u], u++]], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Jun 27 2025 *)

More terms from Dominic McCarty, Aug 19 2024

A374916 Lexicographically earliest sequence of distinct positive integers in which any three consecutive terms are pairwise coprime whereas the squarefree kernel of their product is a primorial number (A002110).

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 25, 8, 21, 55, 16, 63, 125, 22, 147, 65, 44, 189, 325, 88, 357, 845, 176, 441, 625, 32, 27, 35, 64, 33, 175, 26, 99, 245, 52, 297, 595, 104, 363, 875, 128, 81, 385, 208, 51, 1925, 338, 153, 2695, 416, 243, 4235, 256, 39, 6545, 38, 117, 32725

Offset: 1

David James Sycamore and Michael De Vlieger, Jul 23 2024

nonn

Variation on a theme of A374445. Let i = a(n-2), j = a(n-1) and k = a(n). Then rad(i*j*k) is a term in A002110 whereas (i, j) = (i, k) = (j, k) = 1.
 If i is even, both j and k must be odd (by combination of coprimality and primorial conditions). Furthermore two odd terms must be followed by an even term (by the primorial condition). Consequently, starting with a(2) = 2 the sequence displays parity pattern {even, odd, odd} throughout. After a(1) = 1 the sequence continues a(2,3,4) = 2,3,5 implying a(5) = 4 and a(6) = 9, establishing the pattern that if the Lpf of the term preceding even a(n) is 5 then the term following a(n) must have Lpf = 3, which forces the next term to have Lpf = 5, and so on. This pattern is locked in at a(2,3,4) and persists throughout the sequence so that starting from any a(3*k+2), (k >= 0) the Lpf pattern {2,3,5} is continued in successive consecutive triples of adjacent terms.
6 cannot be a term in this sequence since if so a(i) = 6 would require Lpf(j) >= 5; contradiction. By similar arguments 10,15 and 30 are also not  terms. Consequently no multiples of 6, 10, 15 or 30 are terms. 2,3,5 are the only prime terms and 1,2 are the only primorial terms. All terms are numbers divisible by one and only one of {2,3,5}, and the sequence is conjectured to be a permutation of such numbers, namely those congruent to (+ -){2,3,4,5,8,9,14} (mod 30).
Outside 1 and powers of 2, the sequence is a proper subset of A080259.
Conjecture: As n approaches infinity all primorial numbers will eventually appear as rad(i*j*k).

			The sequence must start with a(1,2,3) = 1,2,3 since this is the earliest triple of distinct terms which satisfy the definition.
a(4) = 5 since 2*3*5 = 30 = A002110(3), the coprime conditions are satisfied and no smaller distinct number is possible.
a(20) = 88 and a(21) = 357.  88 = 2^*11, 357 = 3*7*17 means that a(22) must be the least unused number divisible by both 5 and 13. Thus a(22) = 845 = 5*13^2. (because a(16) = 65 = 5*13 and a(19) = 325 = 5^2*13 have both occurred already).

Cf. A002110, A016789, A020639, A055932, A083720, A374445.

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A374351 Lexicographically earliest infinite sequence of distinct positive integers such that for any triple i,j,k of consecutive terms, gcd(i,k) = 1 and A007947(ijk) is a term in A002110.

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A374916 Lexicographically earliest sequence of distinct positive integers in which any three consecutive terms are pairwise coprime whereas the squarefree kernel of their product is a primorial number (A002110).

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cp's OEIS Frontend

A374351 Lexicographically earliest infinite sequence of distinct positive integers such that for any triple i,j,k of consecutive terms, gcd(i,k) = 1 and A007947(i*j*k) is a term in A002110.

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Programs

Java

Mathematica

Extensions

A374916 Lexicographically earliest sequence of distinct positive integers in which any three consecutive terms are pairwise coprime whereas the squarefree kernel of their product is a primorial number (A002110).

Views

Author

Keywords

Comments

Examples

Crossrefs

A374351 Lexicographically earliest infinite sequence of distinct positive integers such that for any triple i,j,k of consecutive terms, gcd(i,k) = 1 and A007947(ijk) is a term in A002110.