A239036 -id:A239036 - cp's OEIS frontend

A238795 The unique set of eleven distinct positive odd numbers up to 105 the sum of whose reciprocals is 1.

3, 5, 7, 9, 11, 33, 35, 45, 55, 77, 105

Offset: 1

Arkadiusz Wesolowski, Mar 05 2014

For any set of distinct numbers whose reciprocals sum to 1, the largest must be neither a prime nor a prime power.

For sets of 11 distinct positive odd numbers, relaxing the a(11) <= 105 constraint allows a second solution at 115 (3, 5, 7, 9, 15, 21, 23, 45, 69, 105, 115), a third at 117 (3, 5, 7, 9, 13, 21, 35, 39, 63, 105, 117), 3 more at 135, 2 more at 143, and 2 more at 165. - Jon E. Schoenfield, Mar 07 2014

			1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/33 + 1/35 + 1/45 + 1/55 + 1/77 + 1/105 = 1.

R. Arce-Nazario, F. Castro, and R. Figueroa, On the number of solutions of Sum_{i = 1..11} 1/x_i = 1 in distinct odd natural numbers, J. Number Theory, 133 (2013), pp. 2036-2046.
Index entries for sequences related to Egyptian fractions

Cf. A201644, A239036.

A239233 Smaller member in a set of two distinct odd numbers the sum of whose reciprocals is 2/(2*n - 1), or 0 if no such integer exists.

Original entry on oeis.org

0, 0, 3, 0, 5, 0, 7, 9, 9, 0, 11, 0, 13, 15, 15, 0, 17, 21, 19, 21, 21, 0, 23, 0, 25, 27, 27, 33, 29, 0, 31, 33, 33, 0, 35, 0, 37, 39, 39, 0, 41, 0, 43, 45, 45, 49, 47, 57, 49, 51, 51, 0, 53, 0, 55, 57, 57, 69, 59, 63, 61, 63, 63, 0, 65, 0, 67, 69, 69, 0, 71

Offset: 1

Arkadiusz Wesolowski, Mar 13 2014

nonn

a(n) = 0 if and only if n = 1 or 2*n - 1 is in A002145.

			a(5) = 5 because 2/9 = 1/5 + 1/45.

Cf. A239036.

print1(0, ", "); forstep(n=3, 141, 2, f=2/n; y=0; if(!(isprime(n)&&Mod(n, 4)==3), y=2*floor((n+1)/4)+1; until(y==n, if(numerator(f-1/y)==1, break); y=y+2)); print1(y, ", "));

A240132 Number of solutions of 1/x_1 + 1/x_2 + ... + 1/x_{2n-1} = 1 in distinct odd numbers > 1.

Original entry on oeis.org

0, 0, 0, 0, 5, 379118

Offset: 1

Jonathan Sondow, Apr 21 2014

The next term is a(7) > 467871307.

			If 1 < x_1 < x_2 < x_3 are odd, then 1/x_1 + 1/x_2 + 1/x_3 < 3/3 = 1, so a(2) = 0.

R. Arce-Nazario, F. Castro, and R. Figueroa, On the number of solutions of Sum_{i = 1..11} 1/x_i = 1 in distinct odd natural numbers, J. Number Theory, 133 (2013), 2036-2046.
C. Elsholtz, Review of ``On the number of solutions of sum_{i = 1..11} 1/x_i = 1 in distinct odd natural numbers'' by R. Arce-Nazario, F. Castro, and R. Figueroa, zbMATH 1270.11033.

The five sets of 9 odd numbers are: A201644, A201646, A201647, A201648, A201649.

See A238795 and A239036 for specific sets of 11 odd numbers.

cp's OEIS Frontend

A238795 The unique set of eleven distinct positive odd numbers up to 105 the sum of whose reciprocals is 1.

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A239233 Smaller member in a set of two distinct odd numbers the sum of whose reciprocals is 2/(2*n - 1), or 0 if no such integer exists.

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A240132 Number of solutions of 1/x_1 + 1/x_2 + ... + 1/x_{2n-1} = 1 in distinct odd numbers > 1.

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