A210732 -id:A210732 - cp's OEIS frontend

A373047 Least k with exactly n partitions k = x + y + z satisfying sigma(k) = sigma(x) + sigma(y) + sigma(z), where sigma*(k) is the sum of the anti-divisors of k.

11, 33, 16, 20, 26, 37, 40, 19, 43, 46, 93, 91, 80, 76, 39, 78, 155, 103, 74, 135, 128, 152, 116, 117, 190, 104, 187, 138, 168, 147, 160, 223, 208, 403, 281, 173, 163, 170, 250, 243, 272, 257, 258, 232, 222, 278, 266, 245, 352, 253, 279, 256, 288, 295, 231, 291

Offset: 1

Paolo P. Lava, Aug 02 2024

			a(7) = 40 and 40 has 7 partitions of three numbers, x, y and
z, for which sigma*(65) = sigma*(x) + sigma*(y) + sigma*(z) = 55. In fact:
sigma*(1) + sigma*+(4) + sigma*(35) = 0 + 3 + 52 = 55;
sigma*(1) + sigma*(12) + sigma*(27) = 0 + 13 + 42 = 55;
sigma*(1) + sigma*(14) + sigma*(25) = 0 + 16 + 39 = 55;
sigma*(4) + sigma*(14) + sigma*(22) = 3 + 16 + 36 = 55;
sigma*(5) + sigma*(8) + sigma*(27) = 5 + 8 + 42 = 55;
sigma*(9) + sigma*(13) + sigma*(18) = 8 + 19 + 28 = 55;
sigma*(10) + sigma*(12) + sigma*(18) = 14 + 13 + 28 = 55;
Furthermore 40 is the minimum number to have this property.

Cf. A066417, A210732, A211223, A211224, A211225, A375154.

A227977 Numbers n for which n = sigma(x) = sigma(y), where n = x + y and sigma*(n) is the sum of the anti-divisors of n.

Original entry on oeis.org

154, 3136, 5536, 20066, 136036, 9550080, 78011830

Offset: 1

Paolo P. Lava, Oct 07 2013

nonn

Up to a(7) the triples (n, x, y) are (154, 77, 77), (3136, 1568, 1568)(5536, 2768, 2768), (20066, 10368, 9698), (136036, 80753, 55283), (9550080, 4775040, 4775040), (78011830, 39348342, 38663488). - Giovanni Resta, Oct 08 2013

			n = 20066 = 9698 + 10368.
Anti-divisors of 9698 are 3, 4, 5, 7, 9, 15, 17, 45, 52, 119, 163, 431, 1141, 1293, 1492, 2155, 2771, 3879, 6465 and their sum is 20066 that is equal to n.
Anti-divisors of 10368 are 5, 11, 13, 29, 55, 65, 89, 143, 145, 233, 256, 319, 377, 715, 768, 1595, 1885, 2304, 4147, 6912 and their sum is 20066 that is equal to n.

Cf. A066272, A066417, A210732.

with(numtheory); P:=proc(q) local a,b, i, j, k, n;
for n from 1 to q do for i from 1 to trunc(n/2) do
k:=0; j:=i; while j mod 2<>1 do k:=k+1; j:=j/2; od;
a:=sigma(2*i+1)+sigma(2*i-1)+sigma(i/2^k)*2^(k+1)-6*i-2;
k:=0; j:=n-i; while j mod 2<>1 do k:=k+1; j:=j/2; od;
b:=sigma(2*(n-i)+1)+sigma(2*(n-i)-1)+sigma((n-i)/2^k)*2^(k+1)-6*(n-i)-2;
if a=b and a=n then print(n); fi; od; od; end: P(10^6);

a(5)-a(7) from Giovanni Resta, Oct 08 2013

cp's OEIS Frontend

A373047 Least k with exactly n partitions k = x + y + z satisfying sigma(k) = sigma(x) + sigma(y) + sigma(z), where sigma*(k) is the sum of the anti-divisors of k.

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A227977 Numbers n for which n = sigma(x) = sigma(y), where n = x + y and sigma*(n) is the sum of the anti-divisors of n.

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cp's OEIS Frontend

A373047 Least k with exactly n partitions k = x + y + z satisfying sigma*(k) = sigma*(x) + sigma*(y) + sigma*(z), where sigma*(k) is the sum of the anti-divisors of k.

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A227977 Numbers n for which n = sigma*(x) = sigma*(y), where n = x + y and sigma*(n) is the sum of the anti-divisors of n.

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Author

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Programs

Maple

Extensions

A373047 Least k with exactly n partitions k = x + y + z satisfying sigma(k) = sigma(x) + sigma(y) + sigma(z), where sigma*(k) is the sum of the anti-divisors of k.

A227977 Numbers n for which n = sigma(x) = sigma(y), where n = x + y and sigma*(n) is the sum of the anti-divisors of n.