A336774 -id:A336774 - cp's OEIS frontend

A336775 a(n) is the largest power of n such that all numbers n^k <= a(n), k=1,..,A336774(n)-1 can be exactly represented as single precision 32-bit floating point numbers according to the IEEE 754 standard.

170141183460469231731687303715884105728, 14348907, 85070591730234615865843651857942052864, 9765625, 470184984576, 5764801, 85070591730234615865843651857942052864, 4782969, 10000000000, 1771561, 15407021574586368, 4826809, 1475789056, 11390625, 21267647932558653966460912964485513216

Offset: 2

Hugo Pfoertner, Aug 04 2020

			a(3) = 14348907 = 3^15, because the next power 3^16 = 43046721 cannot be exactly represented as a binary32 floating point number, but only rounded to 43046720.

Hugo Pfoertner, Table of n, a(n) for n = 2..16384

Cf. A336774, A336778, A336779.

a(n) = n^(A336774(n)-1).

A336778 a(n) is the least number of repetitions such that the result of the repeated execution of the multiplication f <- f*n started at f=1 differs from the exact value n^a(n), when the multiplication is performed using 64-bit double precision floats according to the IEEE 754 standard.

Original entry on oeis.org

1024, 34, 512, 23, 34, 19, 342, 17, 23, 16, 34, 15, 19, 14, 256, 13, 17, 13, 23, 13, 16, 12, 34, 12, 15, 12, 19, 11, 14, 11, 205, 11, 13, 11, 17, 11, 13, 11, 23, 10, 13, 10, 16, 10, 12, 10, 34, 10, 12, 10, 15, 10, 12, 10, 19, 10, 11, 10, 14, 9, 11, 9, 171, 9, 11

Offset: 2

Hugo Pfoertner, Aug 04 2020

See A336774 for more information and links.

For n a power of two the first deviation is caused by exceeding the maximum representable number of the binary64 double precision floating point format with result +infinity in standard conforming implementations.

Hugo Pfoertner, Table of n, a(n) for n = 2..1600
Hugo Pfoertner, Comparison with exact results, range n=2..128, computed on x86-64 architecture.

Cf. A336774, A336775, A336776, A336779, A336780.

A336779 a(n) is the largest power of n such that all numbers n^k <= a(n), k=1,..,A336778(n)-1 can be exactly represented as double precision 64-bit floating point numbers according to the IEEE 754 standard. If a(n) is a power of 2, it is replaced by the corresponding negated exponent of 2.

Original entry on oeis.org

-1023, 5559060566555523, -1022, 2384185791015625, 47751966659678405306351616, 1628413597910449, -1023, 1853020188851841, 10000000000000000000000, 4177248169415651, 410186270246002225336426103593500672, 3937376385699289, 426878854210636742656, 1946195068359375, -1020

Offset: 2

Hugo Pfoertner, Aug 04 2020

The "power of 2" escape clause serves to avoid the corresponding numbers with more than 305 decimal digits in the DATA field.

			a(3) = 5559060566555523 = 3^33, because the next power 3^34 = 16677181699666569 cannot be exactly represented as a binary64 floating point number, but only rounded to 16677181699666568.

Hugo Pfoertner, Table of n, a(n) for n = 2..1600

Cf. A336774, A336775, A336776, A336778, A336780.

a(n) = n^(A336778(n)-1).

A336776 a(n) is the least number of repetitions such that the result of the repeated execution of the multiplication f <- f*n started at f=1 produces an overflow, when the multiplication is performed using 32-bit single precision floats according to the IEEE 754 standard.

Original entry on oeis.org

128, 81, 64, 56, 50, 46, 43, 41, 39, 38, 36, 35, 34, 33, 32, 32, 31, 31, 30, 30, 29, 29, 28, 28, 28, 27, 27, 27, 27, 26, 26, 26, 26, 25, 25, 25, 25, 25, 25, 24, 24, 24, 24, 24, 24, 24, 23, 23, 23, 23, 23, 23, 23, 23, 23, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 21

Offset: 2

Hugo Pfoertner, Aug 07 2020

See A336774 for more information and links.

The overflow usually raises the corresponding exception, with +infinity returned as result.

Hugo Pfoertner, Comparison with 128-bit results, range n=2..128, computed on x86-64 architecture.

Cf. A336774, A336775, A336777, A336778, A336779, A336780, A336781.

A336780 a(n) is the least number of repetitions such that the result of the repeated execution of the multiplication f <- f*n started at f=1 produces an overflow, when the multiplication is performed using 64-bit double precision floats according to the IEEE 754 standard.

Original entry on oeis.org

1024, 647, 512, 442, 397, 365, 342, 324, 309, 297, 286, 277, 269, 263, 256, 251, 246, 242, 237, 234, 230, 227, 224, 221, 218, 216, 214, 211, 209, 207, 205, 203, 202, 200, 199, 197, 196, 194, 193, 192, 190, 189, 188, 187, 186, 185, 184, 183, 182, 181, 180, 179, 178

Offset: 2

Hugo Pfoertner, Aug 07 2020

See A336774 for more information and links.

The overflow usually raises the corresponding exception, with +infinity returned as result.

Hugo Pfoertner, Comparison with 128-bit results, range n=2..128, computed on x86-64 architecture.

Cf. A336774, A336775, A336776, A336777, A336778, A336779, A336781.

A336777 a(n) is the least number of repetitions such that the result of the repeated execution of the division f <- f/n started at f=1 produces 0, when the division is performed using 32-bit single precision floats according to the IEEE 754 standard.

Original entry on oeis.org

151, 96, 76, 66, 59, 55, 51, 49, 47, 45, 43, 42, 41, 40, 39, 38, 37, 37, 36, 36, 35, 35, 34, 34, 33, 33, 33, 32, 32, 32, 31, 31, 31, 31, 31, 30, 30, 30, 30, 29, 29, 29, 29, 29, 29, 29, 28, 28, 28, 28, 28, 28, 28, 27, 27, 27, 27, 27, 27, 27, 27, 27, 26, 26, 26, 26

Offset: 2

Hugo Pfoertner, Aug 07 2020

See A336774 for more information and links.

In contrast to multiplication (A336776), a larger range of values can be used in division by using leading zeros in the significands. The underflow gap is filled by using denormal numbers, also called subnormal numbers.

Hugo Pfoertner, Comparison with 128-bit results, range n=2..128, computed on x86-64 architecture.
Wikipedia, Denormal number.

Cf. A336774, A336775, A336776, A336778, A337779, A336780, A336781.

A336781 a(n) is the least number of repetitions such that the result of the repeated execution of the division f <- f/n started at f=1 produces 0, when the division is performed using 64-bit double precision floats according to the IEEE 754 standard.

Original entry on oeis.org

1076, 680, 539, 464, 417, 384, 360, 341, 325, 312, 301, 292, 284, 277, 270, 264, 259, 255, 250, 246, 243, 239, 236, 233, 230, 228, 225, 223, 221, 218, 216, 215, 213, 211, 209, 208, 206, 205, 203, 202, 201, 200, 198, 197, 196, 195, 194, 193, 192, 191, 190, 189, 188

Offset: 2

Hugo Pfoertner, Aug 07 2020

See A336777 and A336774 for more information and links.

Hugo Pfoertner, Comparison with 128-bit results, range n=2..128, computed on x86-64 architecture.

Cf. A336774, A336775, A336776, A336777, A336778, A336779, A336780.

cp's OEIS Frontend

A336775 a(n) is the largest power of n such that all numbers n^k <= a(n), k=1,..,A336774(n)-1 can be exactly represented as single precision 32-bit floating point numbers according to the IEEE 754 standard.

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A336778 a(n) is the least number of repetitions such that the result of the repeated execution of the multiplication f <- f*n started at f=1 differs from the exact value n^a(n), when the multiplication is performed using 64-bit double precision floats according to the IEEE 754 standard.

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A336779 a(n) is the largest power of n such that all numbers n^k <= a(n), k=1,..,A336778(n)-1 can be exactly represented as double precision 64-bit floating point numbers according to the IEEE 754 standard. If a(n) is a power of 2, it is replaced by the corresponding negated exponent of 2.

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A336776 a(n) is the least number of repetitions such that the result of the repeated execution of the multiplication f <- f*n started at f=1 produces an overflow, when the multiplication is performed using 32-bit single precision floats according to the IEEE 754 standard.

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A336780 a(n) is the least number of repetitions such that the result of the repeated execution of the multiplication f <- f*n started at f=1 produces an overflow, when the multiplication is performed using 64-bit double precision floats according to the IEEE 754 standard.

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A336777 a(n) is the least number of repetitions such that the result of the repeated execution of the division f <- f/n started at f=1 produces 0, when the division is performed using 32-bit single precision floats according to the IEEE 754 standard.

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A336781 a(n) is the least number of repetitions such that the result of the repeated execution of the division f <- f/n started at f=1 produces 0, when the division is performed using 64-bit double precision floats according to the IEEE 754 standard.

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