A372290 -id:A372290 - cp's OEIS frontend

A371094 a(n) = m*(2^e) + ((4^e)-1)/3, where m = 3n+1, and e is the 2-adic valuation of m.

1, 21, 7, 21, 13, 341, 19, 45, 25, 117, 31, 69, 37, 341, 43, 93, 49, 213, 55, 117, 61, 5461, 67, 141, 73, 309, 79, 165, 85, 725, 91, 189, 97, 405, 103, 213, 109, 1877, 115, 237, 121, 501, 127, 261, 133, 1109, 139, 285, 145, 597, 151, 309, 157, 5461, 163, 333, 169, 693, 175, 357, 181, 1493, 187, 381, 193, 789, 199

Offset: 0

Antti Karttunen (proposed by Ali Sada), Apr 19 2024

Construction: take the binary expansion of 3n+1 (A016777(n)), and substitute "01" for all trailing 0-bits that follow after its odd part (= A067745(1+n)), of which there are A371093(n) in total. See the examples.

			For n=1, 3*n+1 = 4, "100" in binary, when we substitute 01's for the two trailing 0's, we obtain 21, "10101" in binary, therefore a(1) = 21.
For n=6, 3*6+1 = 19, "10011" in binary, and there are no trailing 0's, and no changes, therefore a(6) = 19.
For n=7, 3*7+1 = 22, "10110" in binary, with one trailing 0, which when replaced with 01 gives us 45, "101101" in binary, therefore a(7) = 45.
For n=229, there are e=4 trailing bit expansions 0 -> 01,
  3n+1 = binary  101011  0 0 0 0
  a(n) = binary  101011 01010101

Antti Karttunen, Table of n, a(n) for n = 0..16384
Michael De Vlieger, Plot binary expansion of a(n) arranged with smallest bit at bottom and largest at top, n increasing from left to right for n = 0..3071. Black indicates 1, white indicates 0.
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for sequences related to binary expansion of n

Cf. A016777, A067745, A075677, A086893, A096773, A372289, A371093.
Cf. A016921, A372351 (even and odd bisection), A372290 (numbers occurring in the latter).
Cf. arrays A372282, A372283 and A371096, A371100, A371102.
Cf. also A302338.

Array[#2*(2^#3) + ((4^#3) - 1)/3 & @@ {#1, #2, IntegerExponent[#2, 2]} & @@ {#, 3 #1 + 1} &, 67, 0] (* Michael De Vlieger, Apr 19 2024 *)

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

def A371094(n): return ((m:=3*n+1)<<(e:=(~m & m-1).bit_length()))+((1<<(e<<1))-1)//3 # Chai Wah Wu, Apr 28 2024

a(n) = A372289(A016777(n)).

a(2n) = A016777(2n) = A016921(n).

A371100 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n, k) = 4^n(6k - 3 - 2*(-1)^n) + (4^n - 1)/3, n,k >= 1.

Original entry on oeis.org

21, 21, 45, 341, 117, 69, 341, 725, 213, 93, 5461, 1877, 1109, 309, 117, 5461, 11605, 3413, 1493, 405, 141, 87381, 30037, 17749, 4949, 1877, 501, 165, 87381, 185685, 54613, 23893, 6485, 2261, 597, 189, 1398101, 480597, 283989, 79189, 30037, 8021, 2645, 693, 213, 1398101, 2970965, 873813, 382293, 103765, 36181, 9557, 3029, 789, 237

Offset: 1

Antti Karttunen and Ali Sada, Apr 18 2024

			The top left corner of the array:
n\k|      1       2       3        4        5        6        7        8
---+--------------------------------------------------------------------------
1  |     21,     45,     69,      93,     117,     141,     165,     189, ...
2  |     21,    117,    213,     309,     405,     501,     597,     693, ...
3  |    341,    725,   1109,    1493,    1877,    2261,    2645,    3029, ...
4  |    341,   1877,   3413,    4949,    6485,    8021,    9557,   11093, ...
5  |   5461,  11605,  17749,   23893,   30037,   36181,   42325,   48469, ...
6  |   5461,  30037,  54613,   79189,  103765,  128341,  152917,  177493, ...
7  |  87381, 185685, 283989,  382293,  480597,  578901,  677205,  775509, ...
8  |  87381, 480597, 873813, 1267029, 1660245, 2053461, 2446677, 2839893, ...
...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals, flattened).

Cf. A007283, A079319, A257852, A371094, A371101.
Cf. A372351 (same terms, in different order), A372290 (sorted into ascending order, without duplicates), A372293 (odd numbers that do not occur here).
Leftmost column is A144864 duplicated, without its initial 1.
Row 1: A102603.

A371100[n_, k_] := 4^n*(6*k - 3 - 2*(-1)^n) + (4^n - 1)/3;
Table[A371100[n - k + 1, k], {n, 10}, {k, n}] (* Paolo Xausa, Apr 21 2024 *)

up_to = 55;
A371100sq(n,k) = 4^n*(6*k - 3 - 2*(-1)^n) + (4^n - 1)/3;
A371100list(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] = A371100sq((a-(col-1)),col))); (v); };
v371100 = A371100list(up_to);
A371100(n) = v371100[n];

A(n, k) = A007283(n)*A257852(n,k) + A079319(n).
A(n, k) = A371094(A257852(n,k)).
A(n+2, k) = 5 + 16*A(n,k).

A372351 Odd bisection of A371094.

Original entry on oeis.org

21, 21, 341, 45, 117, 69, 341, 93, 213, 117, 5461, 141, 309, 165, 725, 189, 405, 213, 1877, 237, 501, 261, 1109, 285, 597, 309, 5461, 333, 693, 357, 1493, 381, 789, 405, 3413, 429, 885, 453, 1877, 477, 981, 501, 87381, 525, 1077, 549, 2261, 573, 1173, 597, 4949, 621, 1269, 645, 2645, 669, 1365, 693, 11605, 717

Offset: 1

Antti Karttunen, Apr 28 2024

nonn

Row 2 of A372282.
Cf. A371094, and array A371100 (gives the same terms, in different order).
Cf. A372290 (the range of this sequence), A372291 (numbers that occur only once), A372292 (more than once), A372293 (odd numbers not occurring here).

Table[With[{e = IntegerExponent[6*n - 2, 2]}, (6*n - 2)*2^e + (4^e - 1)/3], {n, 100}] (* Paolo Xausa, Apr 29 2024 *)

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372351(n) = A371094(n+n-1);

def A372351(n): return ((m:=6*n-2)<<(e:=(~m & m-1).bit_length()))+((1<<(e<<1))-1)//3 # Chai Wah Wu, Apr 28 2024

a(n) = A371094(2*n-1).

A372291 Numbers that occur exactly once in the odd bisection of A371094.

Original entry on oeis.org

45, 69, 93, 141, 165, 189, 237, 261, 285, 333, 357, 381, 429, 453, 477, 525, 549, 573, 621, 645, 669, 717, 725, 741, 765, 813, 837, 861, 909, 933, 957, 1005, 1029, 1053, 1101, 1109, 1125, 1149, 1197, 1221, 1245, 1293, 1317, 1341, 1389, 1413, 1437, 1485, 1493, 1509, 1533, 1581, 1605, 1629, 1677, 1701, 1725, 1773, 1797

Offset: 1

Antti Karttunen, Apr 26 2024

nonn

Numbers that occur exactly once in array A371100.

			45 is present because A371094(k) = 45 for no other odd number than k=7.

Antti Karttunen, Table of n, a(n) for n = 1..25365

Cf. A371094, A371100.

Setwise difference A372290 \ A372292.

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
isA372291(n) = if(!(n%2),0,my(c=0); forstep(k=1,n,2,if(A371094(k)==n,c++;if(c>1,return(0)))); (c));

A372292 Numbers that occur more than once in the odd bisection of A371094.

Original entry on oeis.org

21, 117, 213, 309, 341, 405, 501, 597, 693, 789, 885, 981, 1077, 1173, 1269, 1365, 1461, 1557, 1653, 1749, 1845, 1877, 1941, 2037, 2133, 2229, 2325, 2421, 2517, 2613, 2709, 2805, 2901, 2997, 3093, 3189, 3285, 3381, 3413, 3477, 3573, 3669, 3765, 3861, 3957, 4053, 4149, 4245, 4341, 4437, 4533, 4629, 4725, 4821, 4917

Offset: 1

Antti Karttunen, Apr 26 2024

nonn

Numbers that occur more than once in array A371100.

			21 is present because A371094(1) = A371094(3) = 21.
87381 is present because A371094(85) = A371094(213) = A371094(7281) = A371094(14563) = 87381.
185685 is present because A371094(469) = A371094(15473) = A371094(30947) = 185685.

Antti Karttunen, Table of n, a(n) for n = 1..15532

Setwise difference A372290 \ A372291.

Cf. A144864 (subsequence after its initial 1), A371094, A371100.

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
isA372292(n) = if(!(n%2),0,my(c=0); forstep(k=1,n,2,if(A371094(k)==n,c++)); (c>1));

search_up_to = 1398101;
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372292list(up_to_n) = { my(v=vector((1+up_to_n)/2), x, lista=List([])); forstep(k=1,up_to_n,2,x=A371094(k); if(x <= up_to_n, v[(x+1)/2]++)); for(i=1,(1+up_to_n)/2,if(v[i]>1, listput(lista,i+i-1))); Vec(lista); };
v372292 = A372292list(search_up_to);
A372292(n) = v372292[n];

A372293 Odd numbers that do not occur in the odd bisection of A371094.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 143, 145, 147, 149, 151, 153, 155, 157

Offset: 1

Antti Karttunen, Apr 26 2024

nonn

Odd numbers that do not occur in array A371100.

Antti Karttunen, Table of n, a(n) for n = 1..14930

Setwise difference A005408 \ A372290.
Cf. A371094, A371100.
Subsequences: A004767, A017077.

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
isA372293(n) = if(!(n%2),0,forstep(k=1,n,2,if(A371094(k)==n,return(0))); (1));

cp's OEIS Frontend

A371094 a(n) = m*(2^e) + ((4^e)-1)/3, where m = 3n+1, and e is the 2-adic valuation of m.

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A371100 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n, k) = 4^n*(6*k - 3 - 2*(-1)^n) + (4^n - 1)/3, n,k >= 1.

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Programs

Mathematica

PARI

Formula

A372351 Odd bisection of A371094.

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A372291 Numbers that occur exactly once in the odd bisection of A371094.

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A372292 Numbers that occur more than once in the odd bisection of A371094.

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Examples

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Crossrefs

Programs

PARI

PARI

A372293 Odd numbers that do not occur in the odd bisection of A371094.

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Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A371100 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n, k) = 4^n(6k - 3 - 2*(-1)^n) + (4^n - 1)/3, n,k >= 1.