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 A365048 a(n) is the number of steps required for the n-th odd prime number to reach 3 when iterating the following hailstone map: If P+1 == 0 (mod 6), then the next number = smallest prime >= P + (P-1)/2; otherwise the next number = largest prime <= (P+1)/2. 2
 0, 2, 1, 6, 2, 5, 2, 4, 4, 3, 3, 5, 3, 8, 5, 13, 4, 4, 7, 4, 4, 6, 12, 9, 6, 9, 6, 6, 14, 5, 8, 11, 5, 8, 5, 5, 5, 16, 13, 13, 13, 13, 10, 7, 10, 10, 7, 15, 15, 15, 12, 15, 15, 12, 12, 12, 9, 6, 12, 6, 12, 6, 17, 6, 14, 6, 17, 14, 14, 11, 11, 14, 14, 14, 8, 11, 11, 14, 11, 8, 11, 16 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjecture: This hailstone operation on odd prime numbers will always reach 3. If the condition "(P + (P-1)/2)" is changed to "(P + (P+1)/2)" then some prime numbers will go into a loop. For example, 449 will loop through 2609. If the condition "(P+1)/2" is changed to "(P+3)/2" then some prime numbers will go into a loop. For example, 5 will go into the loop 5,7,5,7,.... LINKS Paolo Xausa, Table of n, a(n) for n = 1..10000 EXAMPLE Case 3: 0 steps required. Case 5: 2 steps required: 5,7,3. Case 7: 1 step required: 7,3. Case 11: 6 steps required: 11,17,29,43,19,7,3. case 17: 5 steps required: 17,29,43,19,7,3. MATHEMATICA A365048[n_]:=Length[NestWhileList[If[Divisible[#+1, 6], NextPrime[#+(#-1)/2-1], NextPrime[(#+1)/2+1, -1]]&, Prime[n+1], #>3&]]-1; Array[A365048, 100] (* Paolo Xausa, Nov 13 2023 *) PROG (Python) from sympy import nextprime, prevprime def hailstone(prime): if (prime + 1) % 6 == 0: jump = prime + ((prime - 1) / 2) jump = nextprime(jump - 1) else: jump = ((prime + 1) / 2) jump = prevprime(jump + 1) return jump q = 2 lst = [] while q < 3000: count = 0 p = nextprime(q) q = p while p != 3: p = hailstone(p) count = count + 1 lst.append(count) CROSSREFS Cf. A007528, A065091 (odd primes). Sequence in context: A198870 A359273 A372909 * A050457 A195441 A338025 Adjacent sequences: A365045 A365046 A365047 * A365049 A365050 A365051 KEYWORD nonn AUTHOR Najeem Ziauddin, Oct 21 2023 STATUS approved

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Last modified September 15 12:32 EDT 2024. Contains 375938 sequences. (Running on oeis4.)