A392505 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused number such that a(n) shares a factor with a(n-1) while rad(a(n)*a(n-1)) has not previously occurred as the rad value of the product of adjacent terms, where rad = A007947.

1, 2, 6, 10, 4, 14, 12, 22, 8, 26, 18, 34, 16, 38, 20, 28, 30, 33, 3, 9, 15, 21, 7, 35, 55, 5, 25, 65, 39, 13, 91, 42, 44, 40, 46, 23, 69, 24, 57, 19, 95, 45, 51, 17, 85, 50, 52, 56, 58, 29, 87, 36, 62, 31, 93, 60, 68, 66, 70, 74, 32, 82, 48, 86, 43, 129, 63, 77

Offset

1,2

Comments

The distribution of terms in the sequence is very similar to the EKG sequence, see the attached image, although some terms are outliers and do not fall on the three dominant lines - these are due to the occurrence of semiprimes containing a large prime.

The vast majority of primes occur in their natural order, especially for large n, and, like the EKG sequence, they are preceded and followed by terms that are 2 and 3 times their value. Those that are not in natural order are preceded and followed by larger multiples.

In the first 500K terms there are no fixed points other than 1 and 2. The sequence is conjectured to be a permutation of the positive numbers.

See A393471 for the rad value of the product of adjacent terms.

The general pattern 2*p -> p -> 3*p is disturbed when 2*p follows 2^k for some k in this sequence. Entry of a(n) = m*p sometimes occurs when m | a(n-1), especially when m is prime. Examples: a(13) = 16, a(14) = 2*19, thus, a(39..41) = {3*19, 19, 5*19}; a(140) = 2^7, a(141) = 2*83 and a(235) = 3*83 following a product of 3, so a(386..388) = {5*83, 83, 7*83}. - Michael De Vlieger, Mar 24 2026

Links

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, proper prime powers in gold, squarefree composites in green, powerful numbers that are not prime powers in purple, and numbers neither squarefree nor powerful in blue.

Scott R. Shannon, Image of the first 100000 terms. The colors are graduated across the spectrum from red to violet to show the number of prime factors of each term, counted with multiplicity. The thin green line is a(n) = n.

Example

a(3) = 6 as the rad value for the product of previous adjacent terms is 2, so a(3) cannot be 4, as rad(2*4) = 2, while 6 shares a factor with 2 and rad(2*6) = 6.
a(4) = 10 as the rad values for the products of previous adjacent terms are 2 and 6, so a(4) cannot be 3, 4 or 8, while 10 shares a factor with 6 and rad(6*10) = 30.

Mathematica

Block[{c, q}, nn = 120; c[_] := False; q[_] := False; Set[{i, j}, {1, 2}]; u = 3; MapIndexed[Set[{c[#], q[#]}, {True, True}] &, {1, 2}]; rad[x_] := Times @@ FactorInteger[x][[All, 1]]; {i, j}~Join~Reap[Do[If[PrimePowerQ[j], p = FactorInteger[j][[1, 1]]; k = 1; While[Or[c[#], q[rad[j*#] ] ] &[p*k], k++]; k *= p, k = u; While[Or[c[k], q[rad[j*k] ], CoprimeQ[j, k] ], k++] ]; Sow[k]; Set[{q[rad[j*k] ], c[k], i, j}, {True, True, j, k}]; If[k == u, While[c[u], u++]], {n, 3, nn}] ][[-1, 1]] ] (* Michael De Vlieger, Mar 24 2026 *)

Crossrefs

Cf. A393471 (rad values), A007947, A064413, A372043.

Sequence in context: A095105 A220338 A052194 * A320383 A073662 A086553

Adjacent sequences: A392502 A392503 A392504 * A392506 A392507 A392508

Keyword

nonn

Author

Scott R. Shannon, Mar 23 2026

Status

approved