A380494 a(1) = 1. For n > 1 a(n) is the smallest positive integer not yet in the sequence which is divisible by A007953(a(n-1)) + 1.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 16, 24, 14, 18, 20, 15, 21, 28, 11, 27, 30, 32, 36, 40, 25, 48, 13, 35, 45, 50, 42, 49, 56, 60, 63, 70, 64, 22, 55, 33, 77, 75, 26, 54, 80, 72, 90, 100, 34, 88, 17, 81, 110, 39, 52, 96, 112, 65, 84, 78, 128, 108, 120, 44, 99

Offset: 1

Author

David James Sycamore and Ali Sada, Jan 25 2025

Keywords

• nonn
• base
• easy

Comments

For n > 1 a(n) is the smallest novel multiple of digsum(a(n-1)) + 1. The "+1" is intended to avoid an infinite number of consecutive multiples of 9, which would otherwise occur after a(13). The only way a(n) = prime p can occur is if A007953(a(n-1)) is the first occasion of p-1. Sequence is conjectured to be a permutation of the natural numbers, with primes in order.

Examples

			a(1) = 1 has digsum = 1, therefore a(2) = 2, the smallest novel multiple of 1+1 = 2.
Likewise, a(n) = n for all n <= 10. a(11) = 12 because 1 + digsum(10) = 2 and 12 is the smallest number not already a term which is divisible by 2.
a(20) = 28, and digsum(28) + 1 = 11. Therefore a(21) = 11 because 28 is the first occasion of digsum(a(n-1)) = 10.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^6.

Crossrefs

Cf. A007953, A051855.

Programs

Mathematica

nn = 10^4; j = 1; u = 2; c[] := False; m[] := 1;
{j}~Join~Reap[Do[
  k = u; d = DigitSum[j] + 1;
  While[c[Set[k, m[d]*d]], m[d]++];
  Set[{j, c[k]}, {k, True}]; Sow[k];
  If[k == u, While[c[u], u++]],
  {n, 2, nn}] ][[-1, 1]]

Extensions

More terms from Michael De Vlieger, Jan 25 2025.