A372285 -id:A372285 - cp's OEIS frontend

A372355 Array read by upward antidiagonals: A(n,k) = A372285(1+n, k)-A372285(n, k), n,k >= 1.

4, 8, 5, 16, 8, 6, 32, 16, 12, 3, 64, 32, 24, 5, 2, 128, 64, 48, 12, 7, 3, 256, 128, 96, 23, 13, 8, 7, 512, 256, 192, 44, 28, 15, 12, 1, 1024, 512, 384, 88, 55, 28, 24, 5, 6, 2048, 1024, 768, 176, 108, 56, 48, 13, 11, 3, 4096, 2048, 1536, 352, 216, 112, 96, 23, 20, 7, 8, 8192, 4096, 3072, 704, 432, 224, 192, 44, 40, 13, 16, 3

Offset: 1

Antti Karttunen, Apr 29 2024

			Array begins:
n\k|    1     2      3     4     5     6      7     8      9    10     11    12
---+----------------------------------------------------------------------------
1  |    4,    5,     6,    3,    2,    3,     7,    1,     6,    3,     8,    3,
2  |    8,    8,    12,    5,    7,    8,    12,    5,    11,    7,    16,    9,
3  |   16,   16,    24,   12,   13,   15,    24,   13,    20,   13,    32,   15,
4  |   32,   32,    48,   23,   28,   28,    48,   23,    40,   28,    64,   28,
5  |   64,   64,    96,   44,   55,   56,    96,   44,    80,   55,   128,   56,
6  |  128,  128,   192,   88,  108,  112,   192,   88,   160,  108,   256,  112,
7  |  256,  256,   384,  176,  216,  224,   384,  176,   320,  216,   512,  224,
8  |  512,  512,   768,  352,  432,  448,   768,  352,   640,  432,  1024,  448,
9  | 1024, 1024,  1536,  704,  864,  896,  1536,  704,  1280,  864,  2048,  896,
10 | 2048, 2048,  3072, 1408, 1728, 1792,  3072, 1408,  2560, 1728,  4096, 1792,
11 | 4096, 4096,  6144, 2816, 3456, 3584,  6144, 2816,  5120, 3456,  8192, 3584,
12 | 8192, 8192, 12288, 5632, 6912, 7168, 12288, 5632, 10240, 6912, 16384, 7168,

Columnwise first differences of A372285.
Cf. A086893, A372282, A372286.
Cf. also A372353.

up_to = 78;
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)); };
A372282sq(n,k) = if(1==n,2*k-1,A371094(A372282sq(n-1,k)));
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))); };
A372355sq(n,k) = (A372286(A372282sq(1+n,k))-A372286(A372282sq(n,k)));
A372355list(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] = A372355sq((a-(col-1)),col))); (v); };
v372355 = A372355list(up_to);
A372355(n) = v372355[n];

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.

Original entry on oeis.org

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];

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))); };

A372353 Array read by upward antidiagonals: A(n, k) = A372352(A372282(n, k)), n,k >= 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 24, 4, 0, 0, 0, 256, 32, 6, 0, 0, 0, 0, 6144, 16, 0, 0, 0, 0, 0, 16777216, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 896, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6144, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16777216, 0, 56, 4

Offset: 1

Antti Karttunen, Apr 29 2024

Zeros occur in the same locations where 1's occur in array A372287.

			Array begins:
n\k| 1  2  3    4         5   6  7    8  9        10 11  12                 13
---+---------------------------------------------------------------------------
1  | 0, 0, 0,   2,        4,  6, 0,   2, 4,        6, 0,  2,                 4,
2  | 0, 0, 0,  24,       32, 16, 0,   8, 0,       32, 0, 56,                96,
3  | 0, 0, 0, 256,     6144,  0, 0, 896, 0,     6144, 0,  0,              8192,
4  | 0, 0, 0,   0, 16777216,  0, 0,   0, 0, 16777216, 0,  0,         402653184,
5  | 0, 0, 0,   0,        0,  0, 0,   0, 0,        0, 0,  0, 72057594037927936,
6  | 0, 0, 0,   0,        0,  0, 0,   0, 0,        0, 0,  0,                 0,

Cf. A086893, A096773, A372352, A372282, A372287.

Cf. also A372285 and A372355 (columnwise first differences).

up_to = 91;
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3); \\ From A086893
A372352(n) = { my(k); for(i=1,oo,k=A086893(i); if(k>n, return(n-A086893(i-1)))); };
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)));
A372353sq(n,k) = A372352(A372282sq(n,k));
A372353list(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] = A372353sq((a-(col-1)),col))); (v); };
v372353 = A372353list(up_to);
A372353(n) = v372353[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

A372355 Array read by upward antidiagonals: A(n,k) = A372285(1+n, k)-A372285(n, k), n,k >= 1.

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

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A372353 Array read by upward antidiagonals: A(n, k) = A372352(A372282(n, k)), n,k >= 1.

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

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