A379525 a(1) = 1. For n > 1, if a(n-1) is a novel term, a(n) = A003132(a(n-1)). If a(n-1) has been seen k (>1) times already, a(n) = k*a(n-1).

1, 1, 2, 4, 16, 37, 58, 89, 145, 42, 20, 4, 8, 64, 52, 29, 85, 89, 178, 114, 18, 65, 61, 37, 74, 65, 130, 10, 1, 3, 9, 81, 65, 195, 107, 50, 25, 29, 58, 116, 38, 73, 58, 174, 66, 72, 53, 34, 25, 50, 100, 1, 4, 12, 5, 25, 75, 74, 148, 81, 162, 41, 17, 50, 150

Offset

1,3

Comments

2 and 3 appear only once since the only way they can be expressed as the sum of squares is as 2 = 1^2 + 1^2 and 3 = 1^2 + 1^2 + 1^2 (see Example). Maximum number of times prime(k) can appear is A379551(k).

Conjecture: Every positive integer appears in the sequence at least once.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Example

a(2) = 1^2 = 1 so a(3) = 2 since 1 has occurred twice. This is the only occasion of a(n) = 2 in the sequence. Because a(3) = 2 is a novel term, a(4) = 2^2 = 4, then a(5) = 4^2 = 16.
a(8) = a(18) = 89, so a(19) = 2*89 = 178.
a(28) = 10, a novel term, so a(29) = 1, the 3rd occurrence of 1, so a(30) = 3 (the only occasion of 3 in the sequence).

Mathematica

nn = 120; c[_] := 0; a[1] = j = 1;
Do[If[c[j] == 0,
    k = Total[IntegerDigits[j]^2]; c[j]++,
    k = ++c[j]*j ];
  Set[{a[n], j}, {k, k}], {n, 2, nn}];
Array[a, nn] (* Michael De Vlieger, Dec 25 2024 *)

Crossrefs

Cf. A003132, A379551.

Sequence in context: A247219 A265835 A185074 * A000216 A110998 A051861

Adjacent sequences: A379522 A379523 A379524 * A379526 A379527 A379528

Keyword

nonn,base

Author

David James Sycamore, Dec 24 2024

Status

approved