Brilliant

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325 A.D.

 

This year, Christians celebrate the 1700th anniversary of the Council of Nicaea (present day İznik, Turkey). The council was convened by the Roman Emperor, Constantine I. This ecumenical gathering representing all Christendom was represented by over 200 bishops. The council sought to establish common understanding of the technical aspects of Christology.

Establishing the date of Easter and the proclamation of faith, the Nicene Creed, were two major products of this council.

A389240 - A contribution to the On-Line Encyclopedia of Integer Sequences (OEIS)

This new sequence was just published today. I was happy to collaborate with my friend and fellow Illinois Tech alumnus and swimmer, Michael De Vlieger.

A389240

Start the sequence S with S(1) = n and extend S with S(m+1) = S(m)/2 when S(m) is even, otherwise with S(m) + the smallest odd number not yet added. a(n) is the number of steps to reach 1, or -1 if 1 is never reached.

0

0, 1, 3, 2, 8, 4, 4, 3, 6, 9, 9, 5, 33, 5, 5, 4, 9, 7, 7, 10, -1, 10, 10, 6, 7, 34, 34, 6, 13, 6, 6, 5, 34, 10, 10, 8, 9, 8, 8, 11, 10, -1, -1, 11, 11, 11, 11, 7, -1, 8, 8, 35, 34, 35, 35, 7, 8, 14, 14, 7, 96, 7, 7, 6, 14, 35, 35, 11, -1, 11, 11, 9, 35, 10, 10

(list; graph; refs; listen; history; edit; text; internal format)

OFFSET

1,3

COMMENTS

For n = 1, S begins 1, 2, 1, 4, 2. The subsequent terms are A066070.

LINKS

James C. McMahon, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Scatterplot of first 1000000 terms

FORMULA

a(2^k) = k.

a(2^k-1) = k+1 for k > 1.

a(2^k-2) = k+1 for k > 2.

For odd m and a(m) = h, a(2^k*m) = h+k.

EXAMPLE

For n = 5, S begins 5, 6, 3, 6, 3, 8, 4, 2, 1, thus 8 steps to reach 1, so a(5) = 8.

For n = 21, S begins 21, 22, 11, 14, 7, 12, 6, 3, 10, 5, 14, 7, 18, 9. Starting with the 7th step, 3, alternating terms of S are the odd numbers 3, 5, 7, 9..., so the sequence never reaches 1; a(21) = -1.

MATHEMATICA

Table[m = -1;

If[#[[-1]] == 1, -1 + Length[#], -1] &@

If[n < 5,

NestWhileList[If[EvenQ[#], #/2, # + (m += 2)] &, n, # > 1 &],

NestWhileList[If[EvenQ[#], #/2, # + (m += 2)] &, n,

And[#4 > 1, Nand[#1 == #3 - 2, #2 == #4 - 4]] &, 4]], {n, 75}]

CROSSREFS

Cf. A066070, A386819, A389391.

Sequence in context: A019666 A110938 A135852 * A191731 A143515 A082333

Adjacent sequences: A389235 A389236 A389237 * A389241 A389242 A389244

KEYWORD

sign,new

AUTHOR

James C. McMahon and Michael De Vlieger, Sep 26 2025

STATUS

approved