A341515 -id:A341515 - cp's OEIS frontend

A352892 Next even term in the trajectory of map x -> A341515(x), when starting from x=n; a(1) = 1. Here A341515 is the Collatz or 3x+1 map (A006370) conjugated by unary-binary-encoding (A156552).

1, 2, 2, 6, 2, 2, 2, 12, 4, 8, 2, 14, 2, 18, 6, 24, 2, 6, 2, 54, 10, 50, 2, 28, 4, 98, 8, 150, 2, 2, 2, 48, 14, 242, 6, 70, 2, 338, 22, 108, 2, 8, 2, 294, 12, 578, 2, 56, 4, 20, 26, 726, 2, 12, 10, 300, 34, 722, 2, 26, 2, 1058, 20, 96, 14, 18, 2, 1014, 38, 32, 2, 140, 2, 1682, 18, 1734, 6, 50, 2, 216, 16, 1922, 2, 686

Offset: 1

Antti Karttunen, Apr 08 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A005940, A064989, A156552, A329603, A341515, A348717.
Cf. also A139391, A352893, A352894, A352896, A352897, A352898, A352899, A353267.
Coincides with A353268 on even n, and with A348717 on odd n.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A348717(n) = { my(f=factor(n)); if(#f~>0, my(pi1=primepi(f[1, 1])); for(k=1, #f~, f[k, 1] = prime(primepi(f[k, 1])-pi1+1))); factorback(f); }; \\ From A348717
A352892(n) = A348717(A341515(n));

A352892(n) = if(1==n, n, n = A341515(n); while(n%2, n = A341515(n)); (n)); \\ A slower alternative.

a(n) = A348717(A341515(n)).
For all n >= 1, a(2n) = A353268(2n), a(2n-1) = A348717(2n-1).
a(p) = 2 for all primes p.
For n > 1, a(n) = A005940(1+A139391(A156552(n))).

A347115 Möbius transform of A341515.

Original entry on oeis.org

1, 4, 1, 10, 2, 5, 4, 30, 2, 118, 6, 12, 10, 236, 2, 90, 12, 64, 16, 240, 4, 594, 18, 36, 6, 830, 4, 480, 22, -116, 28, 270, 6, 1428, 8, 132, 30, 1784, 10, 720, 36, 1076, 40, 1200, 4, 2622, 42, 108, 20, 144, 12, 1680, 46, 458, 12, 1440, 16, 4178, 52, -228, 58, 4772, 8, 810, 20, 1242, 60, 2880, 18, 2752, 66, 396

Offset: 1

Antti Karttunen, Aug 20 2021

sign

Cf. A008683, A064989, A329603, A341515.
Cf. A285702 (odd bisection), A347116 (even bisection).
Cf. also A332463, A347117, A347118, A348045.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A347115(n) = sumdiv(n,d,moebius(n/d)*A341515(d));

a(n) = A008683(n/d) * A341515(d).

A352890 Number of iterations of map x -> A341515(x) needed to reach x <= 2 when starting from x=n, or -1 if such number is never reached. Here A341515 is the Collatz or 3x+1 map (A006370) conjugated by unary-binary-encoding (A156552).

Original entry on oeis.org

0, 0, 1, 7, 2, 5, 3, 16, 8, 19, 4, 14, 5, 12, 6, 17, 6, 9, 7, 20, 20, 26, 8, 15, 9, 27, 17, 13, 9, 7, 10, 106, 13, 121, 7, 111, 11, 122, 27, 34, 12, 21, 13, 27, 15, 35, 14, 104, 10, 23, 28, 28, 15, 18, 21, 102, 122, 36, 16, 29, 17, 156, 21, 107, 14, 14, 18, 122, 123, 109, 19, 112, 20, 113, 10, 123, 8, 28, 21, 35

Offset: 1

Antti Karttunen, Apr 08 2022

The unbroken ray in the scatter plot corresponds to primes.

Cf. A000040, A005940, A006577, A064989, A156552, A329603, A341515.

Cf. also A352891, A352892, A352893, A352894.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A352890(n) = { my(k=0); while(n>2, n = A341515(n); k++); (k); };

If n <= 2, a(n) = 0, otherwise a(n) = 1 + a(A341515(n)).
For n > 1, a(n) = A006577(A156552(n)).
For n >= 1, a(A000040(n)) = n-1.
For n >= 1, a(n) >= A352891(n).
For n >= 1, a(n) >= A352893(n).

A352891 Number of iterations of map x -> A341515(x) needed to reach x < n when starting from x=n, or 0 if such number is never reached. Here A341515 is the Collatz or 3x+1 map (A006370) conjugated by unary-binary-encoding (A156552).

Original entry on oeis.org

0, 0, 1, 6, 1, 3, 1, 11, 1, 3, 1, 9, 1, 7, 1, 11, 1, 3, 1, 6, 1, 6, 1, 9, 1, 11, 1, 7, 1, 1, 1, 91, 1, 106, 1, 5, 1, 16, 1, 14, 1, 4, 1, 7, 1, 20, 1, 89, 1, 3, 1, 7, 1, 3, 1, 87, 1, 21, 1, 1, 1, 50, 1, 92, 1, 5, 1, 18, 1, 3, 1, 8, 1, 98, 1, 14, 1, 5, 1, 14, 1, 34, 1, 6, 1, 35, 1, 12, 1, 2, 1, 21, 1, 71, 1, 90, 1, 3

Offset: 1

Antti Karttunen, Apr 08 2022

nonn

This is one possible analog for A102419 ("Dropping time" sequence) when computed for A341515. See also A352894.

Cf. A000040, A005940, A006577, A064989, A156552, A329603, A352890, A352894.

Cf. also A102419.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A352891(n) = if(n<=2, 0, my(k=0,x=n); while(x>=n, x = A341515(x); k++); (k));

For n >= 1, a(2n+1) = 1.

For n >= 1, A352894(n) <= a(n) <= A352890(n).

A329603 a(n) = A005940(1+(1+(3A156552(n)))) = (1/2) A005940(1+(3A156552(2n))).

Original entry on oeis.org

2, 5, 8, 15, 18, 11, 50, 45, 20, 125, 98, 33, 242, 245, 32, 135, 338, 77, 578, 375, 72, 605, 722, 99, 42, 845, 60, 735, 1058, 17, 1682, 405, 200, 1445, 162, 231, 1922, 1805, 392, 1125, 2738, 1331, 3362, 1815, 44, 2645, 3698, 297, 110, 275, 968, 2535, 4418, 539, 450, 2205, 1352, 4205, 5618, 51, 6962, 4805, 500, 1215, 882, 1859, 7442, 4335, 2312

Offset: 1

Antti Karttunen, Nov 21 2019

nonn

Function n -> 3n+1 (A016777) conjugated by A156552. - Antti Karttunen, Aug 21 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Permutation of A329604.
Cf. A000035, A005940, A016777, A064989, A156552, A329903, A332449, A332461, A332814, A332823, A341354, A341510, A341515, A341516, A347117 (Möbius transform).
A skewed diagonal of A341510.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A156552(n) = { my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = ((1/2)*A005940(1+(3*A156552(2*n))));

from math import prod
from itertools import accumulate
from collections import Counter
from sympy import prime, primepi, factorint
def A329603(n): return prod(prime(len(a)+1)**b for a, b in Counter(accumulate(bin(1+3*sum((1<Chai Wah Wu, Mar 11 2023

a(n) = (1/2) * A005940(1+(3*A156552(2*n))).
From Antti Karttunen, Feb 14 2021: (Start)
A156552(2*a(n)) = 3*A156552(2*n) = 3*(1+2*A156552(n)) = 3 + 6*A156552(n).
a(n) = A341510(n,2n) = A005940(1+A156552(n)+A156552(2n)) = A005940(1+(1+(3*A156552(n)))).
a(n) = A005940(1+A016777(A156552(n))).
For all n >= 1, A329903(a(n)) = A332814(a(n)) = A332823(A332461(a(n))) = 1.
For all n >= 1, A341354(a(n)) > 0.
For all n >= 1, A000035(a(n)) = 1 - A000035(n). [Flips the parity of n]
(End)
a(n) = A332449(2*n)/2, a(n) = Sum_{d|n} A347117(d). - Antti Karttunen, Aug 21 2021

New primary definition added by Antti Karttunen, Feb 14 2021

A341510 Symmetric square array A(n,k) = A005940(1+A156552(n)+A156552(k)), read by antidiagonals starting with A(1,1).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 9, 9, 9, 9, 9, 7, 8, 10, 8, 8, 8, 8, 10, 8, 9, 7, 15, 7, 7, 7, 15, 7, 9, 10, 8, 10, 12, 10, 10, 12, 10, 8, 10, 11, 15, 7, 15, 25, 15, 25, 15, 7, 15, 11, 12, 14, 12, 10, 12, 18, 18, 12, 10, 12, 14, 12, 13, 25, 21, 25, 15, 25, 11, 25, 15, 25, 21, 25, 13

Offset: 1

Antti Karttunen, Feb 13 2021

Considered as a binary operation on the positive integers, A(x, y) returns the term of the Doudna-sequence from the position that is the sum of the positions of x and y in the same sequence. (This is based on giving the Doudna-sequence an offset of 0, rather than 1 as used in A005940.) - Peter Munn, Feb 14 2021

			The top left 16x16 corner of the array:
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10,  11,  12,  13,  14,  15, 16,
   2,  3,  4,  5,  6,  9, 10,  7,  8, 15,  14,  25,  22,  21,  12, 11,
   3,  4,  5,  6,  9,  8, 15, 10,  7, 12,  21,  18,  33,  20,  25, 14,
   4,  5,  6,  9,  8,  7, 12, 15, 10, 25,  20,  27,  28,  35,  18, 21,
   5,  6,  9,  8,  7, 10, 25, 12, 15, 18,  35,  16,  55,  30,  27, 20,
   6,  9,  8,  7, 10, 15, 18, 25, 12, 27,  30,  11,  42,  45,  16, 35,
   7, 10, 15, 12, 25, 18, 11, 16, 27, 14,  49,  20,  77,  50,  21, 24,
   8,  7, 10, 15, 12, 25, 16, 27, 18, 11,  24,  21,  40,  49,  14, 45,
   9,  8,  7, 10, 15, 12, 27, 18, 25, 16,  45,  14,  63,  24,  11, 30,
  10, 15, 12, 25, 18, 27, 14, 11, 16, 21,  50,  35,  70,  75,  20, 49,
  11, 14, 21, 20, 35, 30, 49, 24, 45, 50,  13,  36, 121,  22,  75, 32,
  12, 25, 18, 27, 16, 11, 20, 21, 14, 35,  36,  45,  60, 125,  30, 75,
  13, 22, 33, 28, 55, 42, 77, 40, 63, 70, 121,  60,  17,  98, 105, 48,
  14, 21, 20, 35, 30, 45, 50, 49, 24, 75,  22, 125,  98,  33,  36, 13,
  15, 12, 25, 18, 27, 16, 21, 14, 11, 20,  75,  30, 105,  36,  35, 50,
  16, 11, 14, 21, 20, 35, 24, 45, 30, 49,  32,  75,  48,  13,  50, 81,

Cf. A341511 (the lower triangular section).
Cf. A003961 (main diagonal), A329603 (skewed diagonal).
Cf. A297165 (row 2 and column 2, when started from its term a(1)).
Cf. also A003056, A268715, A341515, A341520, A348041.

up_to = 105;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A341510sq(n,k) = A005940(1+A156552(n)+A156552(k));
A341510list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A341510sq(col,(a-(col-1))))); (v); };
v341510 = A341510list(up_to);
A341510(n) = v341510[n];

A(n, k) = A(k, n) = A005940(1 + A156552(n) + A156552(k)).
A(n, n) = A003961(n).
A(n, 2*n) = A(2*n, n) = A329603(n).
A(n, 2) = A(2, n) = A297165(n).

A352893 Number of iterations of map x -> A352892(x) needed to reach x <= 2 when starting from x=n, or -1 if such number is never reached. Here A352892 is the next odd term in the Collatz or 3x+1 map (A139391) conjugated by unary-binary-encoding (A156552).

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 1, 5, 3, 6, 1, 4, 1, 3, 2, 5, 1, 2, 1, 6, 7, 8, 1, 4, 3, 8, 6, 3, 1, 1, 1, 39, 4, 44, 2, 41, 1, 44, 9, 11, 1, 6, 1, 8, 5, 10, 1, 38, 3, 7, 9, 8, 1, 5, 7, 37, 45, 10, 1, 9, 1, 56, 7, 39, 4, 3, 1, 44, 45, 40, 1, 41, 1, 39, 3, 44, 2, 8, 1, 11, 6, 15, 1, 3, 9, 15, 11, 13, 1, 4, 7, 10, 11, 32, 9, 38

Offset: 1

Antti Karttunen, Apr 08 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A005940, A064989, A139391, A156552, A286380, A329603, A341515, A352890, A352892, A352894.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A348717(n) = { my(f=factor(n)); if(#f~>0, my(pi1=primepi(f[1, 1])); for(k=1, #f~, f[k, 1] = prime(primepi(f[k, 1])-pi1+1))); factorback(f); }; \\ From A348717
A352892(n) = A348717(A341515(n));
A352893(n) = { my(k=0); while(n>2, n = A352892(n); k++); (k); };

\\ Much faster than above program:
A139391(n) = my(x = if(n%2, 3*n+1, n/2)); x/2^valuation(x, 2); \\ From A139391
A286380(n) = { my(k=0); while(n>1, n = A139391(n); k++); (k); };
A352893(n) = if(1==n,0,A286380(A156552(n)));

If n <= 2, a(n) = 0, otherwise a(n) = 1 + a(A352892(n)).
For n > 1, a(n) = A286380(A156552(n)).
a(p) = 1 for all odd primes p.
For n >= 1, A352894(n) <= a(n) <= A352890(n).

A352894 Number of iterations of map x -> A352892(x) needed to reach x < n when starting from x=n, or 0 if such number is never reached. Here A352892 is the next odd term in the Collatz or 3x+1 map (A139391) conjugated by unary-binary-encoding (A156552).

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 2, 1, 4, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 1, 1, 35, 1, 40, 1, 2, 1, 5, 1, 5, 1, 1, 1, 2, 1, 6, 1, 34, 1, 1, 1, 2, 1, 1, 1, 33, 1, 6, 1, 1, 1, 17, 1, 35, 1, 1, 1, 6, 1, 1, 1, 3, 1, 35, 1, 4, 1, 1, 1, 5, 1, 10, 1, 3, 1, 10, 1, 4, 1, 1, 1, 6, 1, 24, 1, 34, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Apr 08 2022

nonn

Cf. A000040, A005940, A064989, A139391, A156552, A286380, A329603, A341515, A352890, A352891, A352892, A352893.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A348717(n) = { my(f=factor(n)); if(#f~>0, my(pi1=primepi(f[1, 1])); for(k=1, #f~, f[k, 1] = prime(primepi(f[k, 1])-pi1+1))); factorback(f); }; \\ From A348717
A352892(n) = A348717(A341515(n));
A352894(n) = if(n<=2, 0, my(k=0,x=n); while(x>=n, x = A352892(x); k++); (k));

a(2n+1) = 1 for n >= 1.
For n >= 1, a(n) <= A352891(n).
For n >= 1, a(n) <= A352893(n).

A352896 Maximum value of bigomega (A001222) computed for the terms x after the initial n, when map x -> A352892(x) is iterated starting from x=n down to the first x <= 2, or -1 if such number is never reached. Here A352892 is the next odd term in the Collatz or 3x+1 map (A139391) conjugated by unary-binary-encoding (A156552).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 3, 1, 3, 1, 3, 2, 4, 1, 2, 1, 4, 3, 4, 1, 4, 2, 3, 3, 4, 1, 1, 1, 8, 3, 8, 2, 8, 1, 8, 4, 5, 1, 3, 1, 4, 3, 6, 1, 8, 2, 4, 3, 4, 1, 3, 3, 8, 8, 5, 1, 3, 1, 8, 4, 8, 3, 3, 1, 8, 8, 8, 1, 8, 1, 8, 3, 8, 2, 4, 1, 6, 4, 7, 1, 4, 4, 7, 6, 5, 1, 3, 3, 6, 5, 8, 3, 8, 1, 3, 4, 4, 1, 3, 1, 8, 3

Offset: 1

Antti Karttunen, Apr 08 2022

Equally, maximum value of bigomega (A001222) computed for the terms x after the initial n, when map x -> A341515(x) is iterated starting from x=n.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A001222, A005940, A064989, A156552, A329603, A341515, A352892, A352895, A352897, A352899.

A352896(n) = if(n<=2,n-1, my(m=0); while(n>2, n = A352892(n); m = max(m,bigomega(n))); (m)); \\ Needs also code from A352892.

A352896(n) = if(n<=2,n-1,my(m=0); while(n>2, n = A341515(n); m = max(m,bigomega(n))); (m)); \\ Slower, but equivalent.

\\ Faster:
A139391(n) = my(x = if(n%2, 3*n+1, n/2)); x/2^valuation(x, 2); \\ From A139391
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A352895(n) = { my(mw=1); while(n>1, n = A139391(n); mw = max(hammingweight(n),mw)); (mw); };
A352896(n) = if(1==n,0,A352895(A156552(n)));

a(n) = A352897(A341515(n)) = A352897(A352892(n)).

For n > 1, a(n) = A352895(A156552(n)).

A352897 Maximum value of bigomega (A001222) computed for all the terms x (including the starting term x=n), when map x -> A352892(x) is iterated down to the first x <= 2, or -1 if such number is never reached. Here A352892 is the next odd term in the Collatz or 3x+1 map (A139391) conjugated by unary-binary-encoding (A156552).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 3, 1, 3, 2, 4, 1, 3, 1, 4, 3, 4, 1, 4, 2, 3, 3, 4, 1, 3, 1, 8, 3, 8, 2, 8, 1, 8, 4, 5, 1, 3, 1, 4, 3, 6, 1, 8, 2, 4, 3, 4, 1, 4, 3, 8, 8, 5, 1, 4, 1, 8, 4, 8, 3, 3, 1, 8, 8, 8, 1, 8, 1, 8, 3, 8, 2, 4, 1, 6, 4, 7, 1, 4, 4, 7, 6, 5, 1, 4, 3, 6, 5, 8, 3, 8, 1, 3, 4, 4, 1, 3, 1, 8, 3

Offset: 1

Antti Karttunen, Apr 08 2022

nonn

Equally, maximum value of bigomega (A001222) computed for all the terms x (including the starting term x=n), when map x -> A341515(x) is iterated starting from x=n.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for sequences computed from indices in prime factorization

Cf. A001222, A156552, A333860, A341515, A352892, A352896, A352898.

A352897(n) = { my(m=bigomega(n)); while(n>2, m = max(m,bigomega(n)); n = A352892(n)); (m); }; \\ Uses the code from A352892.

A352897(n) = { my(m=bigomega(n)); while(n>2, m = max(m,bigomega(n)); n = A341515(n)); (m); }; \\ Slightly slower.

A139391(n) = my(x = if(n%2, 3*n+1, n/2)); x/2^valuation(x, 2); \\ From A139391
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A333860(n) = { my(mw=1); while(n>1, mw = max(hammingweight(n),mw); n = A139391(n)); (mw); };
A352897(n) = if(1==n,0,A333860(A156552(n)));

a(n) = max(A001222(n), A352896(n)).

For n > 1, a(n) = A333860(A156552(n)).

cp's OEIS Frontend

A352892 Next even term in the trajectory of map x -> A341515(x), when starting from x=n; a(1) = 1. Here A341515 is the Collatz or 3x+1 map (A006370) conjugated by unary-binary-encoding (A156552).

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A347115 Möbius transform of A341515.

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A352890 Number of iterations of map x -> A341515(x) needed to reach x <= 2 when starting from x=n, or -1 if such number is never reached. Here A341515 is the Collatz or 3x+1 map (A006370) conjugated by unary-binary-encoding (A156552).

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A352891 Number of iterations of map x -> A341515(x) needed to reach x < n when starting from x=n, or 0 if such number is never reached. Here A341515 is the Collatz or 3x+1 map (A006370) conjugated by unary-binary-encoding (A156552).

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A329603 a(n) = A005940(1+(1+(3*A156552(n)))) = (1/2) * A005940(1+(3*A156552(2*n))).

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A341510 Symmetric square array A(n,k) = A005940(1+A156552(n)+A156552(k)), read by antidiagonals starting with A(1,1).

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A352893 Number of iterations of map x -> A352892(x) needed to reach x <= 2 when starting from x=n, or -1 if such number is never reached. Here A352892 is the next odd term in the Collatz or 3x+1 map (A139391) conjugated by unary-binary-encoding (A156552).

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A352894 Number of iterations of map x -> A352892(x) needed to reach x < n when starting from x=n, or 0 if such number is never reached. Here A352892 is the next odd term in the Collatz or 3x+1 map (A139391) conjugated by unary-binary-encoding (A156552).

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A329603 a(n) = A005940(1+(1+(3A156552(n)))) = (1/2) A005940(1+(3A156552(2n))).