A371094 -id:A371094 - cp's OEIS frontend

A372282 Array read by upward antidiagonals: A(n, k) = A371094(A(n-1, k)) for n > 1, k >= 1; A(1, k) = 2*k-1.

1, 21, 3, 5461, 21, 5, 357913941, 5461, 341, 7, 1537228672809129301, 357913941, 1398101, 45, 9, 28356863910078205288614550619314017621, 1537228672809129301, 23456248059221, 1109, 117, 11, 9649340769776349618630915417390658987772498722136713669954798667326094136661, 28356863910078205288614550619314017621, 6602346876188694799461995861, 873813, 11605, 69, 13

Offset: 1

Antti Karttunen, Apr 28 2024

			Array begins:
n\k|    1     2        3     4      5     6        7     8      9     10
---+----------------------------------------------------------------------
1  |    1,    3,       5,    7,     9,   11,      13,   15,    17,    19,
2  |   21,   21,     341,   45,   117,   69,     341,   93,   213,   117,
3  | 5461, 5461, 1398101, 1109, 11605, 3413, 1398101, 2261, 87381, 11605,

Cf. A005408 (row 1), A372351 (row 2, bisection of A371094), A372444 (column 14).
Arrays derived from this one:
  A372283,
  A372285 the number of terms of A086893 in the interval [A(n, k), A(1+n, k)],
  A372287 the column index of A(n, k) in array A257852,
  A372288 the sum of digits of A(n, k) in "Jacobsthal greedy base",
  A372353 differences between A(n,k) and the largest term of A086893 <= A(n,k),
  A372354 floor(log_2(.)) of terms, A372356 (and their columnwise first differences),
  A372359 terms xored with binary words of the same length, either of the form 10101...0101 or 110101...0101, depending on whether the binary length is odd or even.
Cf. also arrays A371096, A371102 that give subsets of columns of this array, and array A371100 that gives the terms of the row 2 in different order.

up_to = 28;
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372282sq(n,k) = if(1==n,2*k-1,A371094(A372282sq(n-1,k)));
A372282list(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] = A372282sq((a-(col-1)),col))); (v); };
v372282 = A372282list(up_to);
A372282(n) = v372282[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)).

A371096 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(1, k) = 8*k-7, and A(n+1, k) = A371094(A(n, k)), n,k >= 1.

Original entry on oeis.org

1, 21, 9, 5461, 117, 17, 357913941, 11605, 213, 25, 1537228672809129301, 72701269, 87381, 309, 33, 28356863910078205288614550619314017621, 3752999689475413, 91625968981, 30037, 405, 41, 9649340769776349618630915417390658987772498722136713669954798667326094136661, 27043212804868893898596335048021, 100743818301219097892181, 760567125, 79189, 501, 49

Offset: 1

Antti Karttunen, Apr 21 2024

			Array begins:
n\k|         1         2            3          4           5          6
---+--------------------------------------------------------------------
1  |         1,        9,          17,        25,         33,        41,
2  |        21,      117,         213,       309,        405,       501,
3  |      5461,    11605,       87381,     30037,      79189,     48469,
4  | 357913941, 72701269, 91625968981, 760567125, 1968526677, 299193685,

Cf. A371094, A017077 (row 1).
Every fourth column (1, 5, 9, 13, 17, ...) of array A372282.
Cf. also arrays A257852, A371100 and A371102.

up_to = 28;
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A371096sq(n,k) = if(1==n,8*k-7,A371094(A371096sq(n-1,k)));
A371096list(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] = A371096sq((a-(col-1)),col))); (v); };
v371096 = A371096list(up_to);
A371096(n) = v371096[n];

A371102 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(1, k) = 4*k-1, and A(n+1, k) = A371094(A(n, k)), n,k >= 1.

Original entry on oeis.org

3, 21, 7, 5461, 45, 11, 357913941, 1109, 69, 15, 1537228672809129301, 873813, 3413, 93, 19, 28356863910078205288614550619314017621, 1466015503701, 22369621, 2261, 117, 23, 9649340769776349618630915417390658987772498722136713669954798667326094136661, 25790417485112089060398421, 6004799503160661, 873813, 11605, 141, 27

Offset: 1

Antti Karttunen, Apr 21 2024

			Array begins:
n\k|         1       2         3       4         5         6         7
---+--------------------------------------------------------------------
1  |         3,      7,       11,     15,       19,       23,       27,
2  |        21,     45,       69,     93,      117,      141,      165,
3  |      5461,   1109,     3413,   2261,    11605,     3413,     8021,
4  | 357913941, 873813, 22369621, 873813, 72701269, 22369621, 12408149,

Cf. A004767 (row 1), A102603 (row 2), A371094.

Cf. also arrays A257852, A371096, A371100, A371103.

up_to = 105;
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A371102sq(n,k) = if(1==n,4*k-1,A371094(A371102sq(n-1,k)));
A371102list(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] = A371102sq((a-(col-1)),col))); (v); };
v371102 = A371102list(up_to);
A371102(n) = v371102[n];

A372290 Numbers that occur in the odd bisection of A371094.

Original entry on oeis.org

21, 45, 69, 93, 117, 141, 165, 189, 213, 237, 261, 285, 309, 333, 341, 357, 381, 405, 429, 453, 477, 501, 525, 549, 573, 597, 621, 645, 669, 693, 717, 725, 741, 765, 789, 813, 837, 861, 885, 909, 933, 957, 981, 1005, 1029, 1053, 1077, 1101, 1109, 1125, 1149, 1173, 1197, 1221, 1245, 1269, 1293, 1317, 1341, 1365, 1389

Offset: 1

Antti Karttunen, Apr 26 2024

nonn

Numbers that occur in array A371100.

			21 is present because A371094(1) = A371094(3) = 21.
45 is present because A371094(7) = 45.
87381 is present because A371094(85) = A371094(213) = A371094(7281) = A371094(14563) = 87381.

Antti Karttunen, Table of n, a(n) for n = 1..62122

Union of A372291 and A372292.

Cf. A102603 (subsequence), A371094, A371100.

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
isA372290(n) = if(!(n%2),0,forstep(k=1,n,2,if(A371094(k)==n,return(1))); (0));

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372290list(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]>0, listput(lista,i+i-1))); Vec(lista); };

A372286 a(n) = number of terms of A086893 that are in the interval [n, A371094(n)].

Original entry on oeis.org

1, 5, 2, 4, 2, 7, 1, 2, 2, 4, 2, 3, 2, 6, 1, 3, 1, 4, 2, 3, 2, 9, 1, 2, 1, 3, 1, 2, 2, 4, 2, 2, 2, 4, 2, 3, 2, 6, 2, 3, 2, 4, 2, 3, 2, 5, 2, 3, 2, 4, 2, 3, 2, 8, 1, 2, 1, 3, 1, 3, 1, 5, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 2, 4, 2, 3, 2, 5, 2, 3, 2, 4, 2, 3, 2, 11, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3

Offset: 0

Antti Karttunen (proposed by Ali Sada), Apr 28 2024

nonn

a(n) is the number of nonzero terms of A096773 that are found in the interval [n, A371094(n)], inclusive.

			A086893 = 1, 3, 5, 13, 21, 53, 85, 213, 341, 853, 1365, 3413, 5461, ...
For n=0, A371094(0) = 1, and only the initial term of A086893 is in the interval [0, 1], therefore a(0) = 1.
For n=1, A371094(1) = 21, so the first five terms of A086893 are in the interval [1, 21], thus a(1) = 5.
For n=6, A371094(6) = 19, and in the interval [6, 19] there is only term 13 of A086893, thus a(6) = 1.
For n=35, A371094(35) = 213, so we count terms 53, 85, 213 of A086893, therefore a(35) = 3.

Antti Karttunen, Table of n, a(n) for n = 0..21845

Cf. A086893, A096773, A371094, A372285.

Cf. A002450 (seems to give the positions of records).

A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3); \\ From A086893
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372286(n) = { my(u=A371094(n), k1); for(i=1,oo,if(A086893(i)>=n,k1=i-1; break)); for(i=k1,oo,if(A086893(i)>u,return(i-k1-1))); };

A372449 a(n) = A000523(A372444(n)); One less than the length of binary expansion of the n-th iterate of 27 with A371094.

Original entry on oeis.org

4, 7, 12, 23, 44, 84, 165, 326, 650, 1297, 2590, 5177, 10349, 20695, 41386, 82766, 165527, 331048, 662093, 1324181, 2648358, 5296712, 10593418, 21186832, 42373658, 84747311, 169494616, 338989224, 677978441, 1355956875, 2711913744, 5423827481, 10847654953, 21695309901, 43390619796, 86781239588, 173562479173, 347124958346

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, A372448.

Column 14 of A372354.

A000265(n) = (n>>valuation(n,2));
A000523(n) = logint(n,2);
A372443(n) = { my(x=27); while(n, x=A000265(3*x+1); n--); (x); };
A371093(n) = valuation(1+3*n,2);
A372447(n) = A000523(A372443(n));
A372448(n) = if(!n,1,2*A372448(n-1)+A371093(A372443(n)));
A372449(n) = if(0==n,A372447(n),A372447(n)+2*A372448(n-1));

a(n) = A000523(A372444(n)).

a(0) = A372447(0) = 4, and for n > 0, a(n) = A372447(n) + 2*A372448(n-1).

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).

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)).

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));

cp's OEIS Frontend

A372282 Array read by upward antidiagonals: A(n, k) = A371094(A(n-1, k)) for n > 1, k >= 1; A(1, k) = 2*k-1.

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A372444 The n-th iterate of 27 with A371094.

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A371096 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(1, k) = 8*k-7, and A(n+1, k) = A371094(A(n, k)), n,k >= 1.

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A371102 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(1, k) = 4*k-1, and A(n+1, k) = A371094(A(n, k)), n,k >= 1.

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A372290 Numbers that occur in the odd bisection of A371094.

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A372286 a(n) = number of terms of A086893 that are in the interval [n, A371094(n)].

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A372449 a(n) = A000523(A372444(n)); One less than the length of binary expansion of the n-th iterate of 27 with A371094.

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A372351 Odd bisection of 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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