A372449 -id:A372449 - cp's OEIS frontend

A372451 a(n) = A372449(1+n) - A372449(n).

3, 5, 11, 21, 40, 81, 161, 324, 647, 1293, 2587, 5172, 10346, 20691, 41380, 82761, 165521, 331045, 662088, 1324177, 2648354, 5296706, 10593414, 21186826, 42373653, 84747305, 169494608, 338989217, 677978434, 1355956869, 2711913737, 5423827472, 10847654948, 21695309895, 43390619792, 86781239585, 173562479173, 347124958345

Offset: 0

Antti Karttunen, May 05 2024

nonn

a(n) tells how many bits the length of the binary expansion grows when we go from A372444(n) to A372444(1+n).

Antti Karttunen, Table of n, a(n) for n = 0..1001

First differences of A372449.
Column 14 of A372356.
Cf. A000523, A372443, A372444.

A372451(n) = (A372449(1+n)-A372449(n)); \\ Needs also code from A372449.

a(n) = A000523(A372444(1+n)) - A000523(A372444(n)).

A372454 a(n) = A372444(n) - A086893(1+A372449(n)).

Original entry on oeis.org

6, -48, 2560, -1572864, -3848290697216, 6649092007880460460883968, -18999521285301737936647902825311679255527123058688, 76895533293152762966220781422103876125697362804839499718093497881599910128103059800826635129716736

Offset: 0

Antti Karttunen, May 05 2024

sign

The difference between A372444(n) and the term of A086893 with the same binary length.

			The term of A086893 that has same binary length as A372444(0) = 27 is 21 [as 21 = 10101_2 in binary, and 27 = 11011_2 in binary], therefore a(0) = 27-21 = 6.
The term of A086893 that has same binary length as A372444(1) = 165 is 213, therefore a(1) = 165-213 = -48.

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

Cf. A000523, A086893, A372444, A372448, A372449, A372453.

A000523(n) = logint(n,2);
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); };
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3);
A372454(n) = { my(x=A372444(n)); (x - A086893(1+A000523(x))); };

A372454(n) = if(!n, A372453(n), (4^A372448(n-1))*A372453(n));

a(n) = A372444(n) - A086893(1+A000523(A372444(n))).

a(0) = A372453(0) = 6; and for n > 0, a(n) = 4^A372448(n-1) * A372453(n).

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)).

A372448 a(n) is the 2-adic valuation of 1 + 3*{the n-th iterate of 27 with A371094}.

Original entry on oeis.org

1, 4, 9, 19, 39, 79, 160, 322, 645, 1292, 2585, 5171, 10344, 20689, 41379, 82759, 165520, 331043, 662087, 1324175, 2648352, 5296705, 10593412, 21186825, 42373651, 84747303, 169494607, 338989215, 677978433, 1355956867, 2711913735, 5423827471, 10847654946, 21695309894, 43390619790, 86781239584, 173562479171, 347124958343

Offset: 0

Antti Karttunen, May 04 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1001

Cf. A371093, A371094, A372443, A372444, A372447, A372449.

A000265(n) = (n>>valuation(n,2));
A372443(n) = { my(x=27); while(n, x=A000265(3*x+1); n--); (x); };
A371093(n) = valuation(1+3*n,2);
A372448(n) = if(!n,1,2*A372448(n-1)+A371093(A372443(n)));

a(n) = A371093(A372444(n)).

a(0) = 1, and for n > 0, a(n) = 2*a(n-1) + A371093(A372443(n)).

A372452 Number of terms of A086893 in the interval [A372444(n), A372444(1+n)].

Original entry on oeis.org

2, 6, 10, 21, 41, 80, 162, 324, 646, 1294, 2586, 5173, 10345, 20691, 41381, 82760, 165522, 331044, 662089, 1324177, 2648353, 5296707, 10593413, 21186827, 42373652, 84747305, 169494609, 338989216, 677978435, 1355956869, 2711913736, 5423827472, 10847654948, 21695309896, 43390619791, 86781239586, 173562479173, 347124958344

Offset: 0

Antti Karttunen, May 05 2024

nonn

The formula involving A372451 and A372453 shows that each term is at most +-1 from the corresponding term of A372451, that are the first differences of A372449.

			Between A372444(0)=27 and A372444(1)=165 there are two terms (53 and 85) of A086893, therefore a(0) = 2.
Between A372444(1)=165 and A372444(2)=8021 there are six terms (213, 341, 853, 1365, 3413, 5461) of A086893, therefore a(1) = 6.
Between A372444(2)=8021 and A372444(3)=12408149 there are 10 terms (13653, 21845, 54613, 87381, 218453, 349525, 873813, 1398101, 3495253, 5592405) of A086893, therefore a(2) = 10.

Antti Karttunen, Table of n, a(n) for n = 0..1001

Column 14 of A372285.

Cf. A086893, A372286, A372444, A372449, A372451, A372453.

A372452(n) = A372451(n)+(A372453(n)<=0)-(A372453(1+n)<0); \\ Uses also code from A372451 and A372453.

a(n) = A372286(A372444(n)).

a(n) = A372451(n) + [A372453(n)<=0] - [A372453(1+n)<0], where [ ] is the Iverson bracket, yielding 1 or 0 depending on whether the given inequivalence holds or does not hold.

cp's OEIS Frontend

A372451 a(n) = A372449(1+n) - A372449(n).

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A372454 a(n) = A372444(n) - A086893(1+A372449(n)).

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A372444 The n-th iterate of 27 with A371094.

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A372448 a(n) is the 2-adic valuation of 1 + 3*{the n-th iterate of 27 with A371094}.

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A372452 Number of terms of A086893 in the interval [A372444(n), A372444(1+n)].

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