A358354 a(n) = n for n <= 3. Thereafter a(n) is the least m such that rad(m) = rad(rad(a(n-3)) + rad(a(n-1))) where rad is A007947.

1, 2, 3, 4, 8, 5, 7, 9, 16, 27, 6, 32, 25, 11, 13, 12, 17, 30, 18, 23, 53, 59, 82, 15, 74, 78, 93, 167, 35, 64, 169, 24, 128, 45, 21, 529, 38, 3481, 164, 60, 89, 57, 87, 22, 79, 166, 94, 173, 339, 433, 606, 105, 538, 286, 391, 929, 75, 406, 1335, 90, 218, 1553

Offset: 1

Author

David James Sycamore, Nov 11 2022

Keywords

• nonn

Comments

Similar sequence to A358093, though here the definition works between a(n-1) and a(n-3) (skipping a(n-2)) so there is no evident parity pattern, and adjacent pairs are not always coprime.

Conjecture: for n > 60, a(n) is squarefree. - Michael De Vlieger, Jul 01 2025

Examples

			To find a(4): a(1)=1, a(3)=3 so a(4) is the least m such that rad(n) = rad(rad(1) + rad(3)) = rad(4) = 2. Since 2 has occurred already, a(4) = 4.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1000
• Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..256, 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.

Crossrefs

Cf. A007947, A358093.

Programs

Mathematica

Block[{a, c, f, i, j, k, m, p, q, s, nn}, nn = 62; c[] = False; p[] = q[] = 1; f[n] := Times @@ FactorInteger[n][[All, 1]]; Array[Set[{a[#], c[#]}, {#, True}] &, 3]; Array[(q[#]++; p[#]++) &[f[a[#]] ] &, 3]; Set[{i, j, k}, Array[f[a[#]] &, 3]]; Do[s = f[i + k]; If[PrimeQ[s], m = s^p[s]; p[s]++, m = q[s]; While[Nand[! c[m s], PowerMod[s, s, m] == 0], m++]; m *= s]; q[s]++; Set[{a[n], c[m], i, j, k}, {m, True, j, k, s}], {n, 4, nn}]; Array[a, nn] ] (* Michael De Vlieger, Nov 12 2022 *)