A057522 Start with 73, to get the next term divide by the smallest prime factor < 13 if it exists, else multiply by 13 and add 1.

73, 950, 475, 95, 19, 248, 124, 62, 31, 404, 202, 101, 1314, 657, 219, 73, 950, 475, 95, 19, 248, 124, 62, 31, 404, 202, 101, 1314, 657, 219, 73, 950, 475, 95, 19, 248, 124, 62, 31, 404, 202, 101, 1314, 657, 219, 73, 950, 475, 95, 19, 248, 124, 62, 31, 404, 202

Offset

0,1

Comments

Original name: "a(n+1) = a(n)/2 if 2|a(n), a(n)/3 if 3|a(n), a(n)/5 if 5|a(n), a(n)/7 if 7|a(n), a(n)/11 if 11|a(n), otherwise 13*a(n)+1."

This is the '13x+1' map. The 'Px+1 map': if x is divisible by any prime < P then divide out these primes one at a time starting with the smallest; otherwise multiply x by P and add 1.

Links

Table of n, a(n) for n=0..55.

Eric Weisstein's World of Mathematics, Collatz problem

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Formula

For n >= 15, a(n) = a(n-15). - Harvey P. Dale, May 02 2011

Example

73 -> 13 * 73 + 1 = 950, 950 = 2 * 5^2 * 19 -> 950/2 = 475, etc.

Mathematica

nxt[n_]:=Which[Divisible[n, 2], n/2, Divisible[n, 3], n/3, Divisible[n, 5], n/5, Divisible[n, 7], n/7, Divisible[n, 11], n/11, True, 13n+1]; NestList[nxt, 73, 60] (* Harvey P. Dale, May 02 2011 *)

Prog

(PARI) a(n)=[73, 950, 475, 95, 19, 248, 124, 62, 31, 404, 202, 101, 1314, 657, 219][n%15+1] \\ Charles R Greathouse IV, Apr 29 2026

Crossrefs

Cf. A057446 (short version), A057216, A057534, A057614.

Sequence in context: A296024 A254136 A123811 * A320205 A305549 A320214

Adjacent sequences: A057519 A057520 A057521 * A057523 A057524 A057525

Keyword

nonn,easy

Author

Murad A. AlDamen (Divisibility(AT)yahoo.com), Oct 17 2000

Extensions

Name corrected and clarified by M. F. Hasler, Feb 11 2026

Status

approved