A372560 -id:A372560 - cp's OEIS frontend

A372561 Array read by upward antidiagonals: A(n, k) = A265745(A372560(n, k)) for n > 1, k >= 1.

3, 5, 5, 5, 5, 3, 5, 5, 5, 3, 5, 5, 5, 5, 5, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 10347, 6251, 2155, 1131, 619, 363, 235, 107, 43, 27, 11, 7, 5

Offset: 1

Antti Karttunen, May 08 2024

In general, it seems that for n>2, k>1, A(n, k) = A(n-1, k+1) = A(k, n), except on those two anomalous antidiagonals, first on the thirteenth antidiagonal, where for n=1..13, A(n,14-n) obtains values 5, 7, 11, 27, 43, 107, 235, 363, 619, 1131, 2155, 6251, 10347, and then on the 30th antidiagonal, where for n=1.., A(n,31-n) obtains values 5, 11, 15, 23, 39, 71, 135, 391, 647, 1671, 2695, 4743, 17031, 33415, 49799, 82567, 148103, 410247, etc. The corresponding antidiagonals in A372560 begin as:
  233, 933, 14933, 978670933, 64138178286933, 1183140560213014108063589658350933, ..., and:
  911, 58325, 933205, 238900565, 15656587449685, 67244531063362552157525, etc. I conjecture that for the former sequence of numbers x, from 933 onward, A372555(x) = 7, and for the latter sequence of numbers y, from 58325 onward, A372555(y) = 9, and that the array A372555(A372560(n, k)) is symmetric apart from its borders, i.e, that for n, k > 1, A372555(A372560(n, k)) = A372555(A372560(k, n)).

			Array begins:
n\k| 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21
---+----------------------------------------------------------------
1  | 3, 5, 3, 3, 5, 3, 5, 5, 3, 5, 5, 5, 5, 3, 5, 3, 7, 5, 7, 5, 5,
2  | 5, 5, 5, 5, 3, 5, 5, 3, 5, 5, 5, 7, 5, 5, 5, 7, 5, 7, 7, 5, 5,
3  | 5, 5, 5, 3, 5, 5, 3, 5, 5, 5, 11, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7,
4  | 5, 5, 3, 5, 5, 3, 5, 5, 5, 27, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9,
5  | 5, 3, 5, 5, 3, 5, 5, 5, 43, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7,
6  | 3, 5, 5, 3, 5, 5, 5, 107, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7,
7  | 5, 5, 3, 5, 5, 5, 235, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7,
8  | 5, 3, 5, 5, 5, 363, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9,
9  | 3, 5, 5, 5, 619, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7,
10 | 5, 5, 5, 1131, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 1671,
11 | 5, 5, 2155, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 2695, 3,
12 | 5, 6251, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 4743, 3, 5,
13 | 10347, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 17031, 3, 5, 3,
14 | 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 33415, 3, 5, 3, 5,
15 | 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 49799, 3, 5, 3, 5, 5,
etc.
From column 19 to column 41, the first 11 rows:
n\k|19 20 ........................................................... 40 41
---+-------------------------------------------------------------------------
1  | 7, 5, 5, 5, 7, 7, 5, 5, 5, 7, 7, 5,    3, 3, 3, 5, 5, 5, 5, 3, 3, 3, 1,
2  | 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 11,   3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1,
3  | 5, 5, 7, 9, 7, 7, 7, 9, 7, 15,   3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1,
4  | 5, 7, 9, 7, 7, 7, 9, 7, 23,   3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1,
5  | 7, 9, 7, 7, 7, 9, 7, 39,   3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1,
6  | 9, 7, 7, 7, 9, 7, 71,   3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1,
7  | 7, 7, 7, 9, 7, 135,  3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1, 1,
8  | 7, 7, 9, 7, 391,  3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1,
9  | 7, 9, 7, 647,  3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1,
10 | 9, 7, 1671, 3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
11 | 7, 2695, 3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

Cf. A265745, A371094, A372282, A372288, A372555, A372560.

\\ Needs also program from A372560.
up_to = 91;
A001045(n) = (2^n - (-1)^n) / 3;
A130249(n) = (#binary(3*n+1)-1);
A265745(n) = { my(s=0); while(n,s++; n -= A001045(A130249(n))); (s); };
A372561sq(n,k) = A265745(A372560sq(n,k));
A372561list(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] = A372561sq((a-(col-1)),col))); (v); };
v372561 = A372561list(up_to);
A372561(n) = v372561[n];

A372443 The n-th iterate of 27 with Reduced Collatz-function R, which gives the odd part of 3n+1.

Original entry on oeis.org

27, 41, 31, 47, 71, 107, 161, 121, 91, 137, 103, 155, 233, 175, 263, 395, 593, 445, 167, 251, 377, 283, 425, 319, 479, 719, 1079, 1619, 2429, 911, 1367, 2051, 3077, 577, 433, 325, 61, 23, 35, 53, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Antti Karttunen, May 01 2024

nonn

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A000265, A075677, A371093.
Column 14 of A372283, Row 13 of A256598 (but only up to the first 1).
Row 1 of A372560.
From term 47 to the first 1 same as A088593.
Sequences derived from this one or related to:
  A372444,
  A372445 column index of a(n) in array A257852,
  A372362 the 2-adic valuation of 1 + 3*a(n), equal to row index of a(n) in array A257852,
  A372447 binary lengths minus 1,
  A372446 a(n) xored with the term of A086893 having the same binary length,
  A372453 a(n) minus the term of A086893 having the same binary length.

R(n) = { n = 1+3*n; n>>valuation(n, 2); };
A372443(n) = { my(x=27); while(n, x=R(x); n--); (x); };

a(0) = 27; for n > 0, a(n) = R(a(n-1)), where R(n) = (3*n+1)/2^A371093(n) = A000265(3*n+1).

For n > 0, a(n) = R(A372444(n-1)) = A000265(1+3*A372444(n-1)).

A372444 The n-th iterate of 27 with A371094.

Original entry on oeis.org

27, 165, 8021, 12408149, 19607957362005, 32439509492992549521282389, 58947232705679751034215288252890081792789279233365, 259166427025070423330595967015238989905128148712607202753574381749095993394717720069452733214971221

Offset: 0

Antti Karttunen, May 01 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10

Cf. A371094.
Column 7 of A371102, column 14 of A372282.
Column 1 of A372560.
Sequences derived from this one:
  A372443 obtained when Reduced Collatz-function R is applied to a(n-1), for n > 0,
  A372445 column index of a(n) in array A257852,
  A372448 the 2-adic valuation of 1 + 3*a(n), equal to row index of a(n) in array A257852,
  A372449 binary lengths minus 1; their first differences: A372451,
  A372452 number of terms of A086893 in the interval [a(n), a(1+n)],
  A372454 the difference between a(n) and the term of A086893 with the same binary length.

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372444(n) = { my(x=27); while(n, x=A371094(x); n--); (x); };

a(0) = 27; for n > 0, a(n) = A371094(a(n-1)).

cp's OEIS Frontend

A372561 Array read by upward antidiagonals: A(n, k) = A265745(A372560(n, k)) for n > 1, k >= 1.

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A372443 The n-th iterate of 27 with Reduced Collatz-function R, which gives the odd part of 3n+1.

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A372444 The n-th iterate of 27 with A371094.

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