A318506 -id:A318506 - cp's OEIS frontend

A318505 Sum of divisors of n, up to, but not including the second largest of them A032742(n); a(1) = 0 by convention.

0, 0, 0, 1, 0, 3, 0, 3, 1, 3, 0, 10, 0, 3, 4, 7, 0, 12, 0, 12, 4, 3, 0, 24, 1, 3, 4, 14, 0, 27, 0, 15, 4, 3, 6, 37, 0, 3, 4, 30, 0, 33, 0, 18, 18, 3, 0, 52, 1, 18, 4, 20, 0, 39, 6, 36, 4, 3, 0, 78, 0, 3, 20, 31, 6, 45, 0, 24, 4, 39, 0, 87, 0, 3, 24, 26, 8, 51, 0, 66, 13, 3, 0, 98, 6, 3, 4, 48, 0, 99, 8, 30, 4, 3, 6, 108, 0, 24, 24, 67, 0, 63, 0, 54, 52

Offset: 1

Antti Karttunen, Aug 27 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sums of divisors.

Cf. A001065, A013661, A032742, A318504, A318506, A318507, A318508.

A001065(n) = (sigma(n)-n);
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318505(n) = if(1==n,0,A001065(n)-A032742(n));
A318505(n) = sumdiv(n,d,(d<A032742(n))*d); \\ Alternatively.

a(1) = 0; for n > 1, a(n) = A001065(n) - A032742(n).

Sum_{k=1..n} a(k) ~ (zeta(2)/2 - 1/2 - c) * n^2, where c is defined in the corresponding formula in A032742. - Amiram Eldar, Dec 21 2024

A318508 a(n) = A032742(n) AND A001065(n)-A032742(n), where AND is bitwise-and (A004198) and A001065 = sum of proper divisors and A032742 = the largest proper divisor of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 0, 1, 1, 0, 2, 0, 3, 4, 0, 0, 8, 0, 8, 4, 3, 0, 8, 1, 1, 0, 14, 0, 11, 0, 0, 0, 1, 6, 0, 0, 3, 4, 20, 0, 1, 0, 18, 2, 3, 0, 16, 1, 16, 0, 16, 0, 3, 2, 4, 0, 1, 0, 14, 0, 3, 20, 0, 4, 33, 0, 0, 4, 35, 0, 4, 0, 1, 24, 2, 8, 35, 0, 0, 9, 1, 0, 34, 0, 3, 4, 32, 0, 33, 8, 14, 4, 3, 2, 32, 0, 16, 0, 2, 0, 51, 0, 52, 32

Offset: 1

Antti Karttunen, Aug 28 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A001065, A004198, A032742, A318505, A318506, A318507.

Cf. also A318457, A318517.

A001065(n) = (sigma(n)-n);
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318505(n) = if(1==n,0,A001065(n)-A032742(n));
A318508(n) = bitand(A318505(n),A032742(n));

a(n) = A004198(A032742(n), A318505(n)).

For n > 1, a(n) = A001065(n) - A318506(n) = (A001065(n) - A318507(n))/2.

A318507 a(n) = A032742(n) XOR A001065(n)-A032742(n), where XOR is bitwise-or (A003987) and A001065 = sum of proper divisors and A032742 = the largest proper divisor of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 0, 1, 7, 2, 6, 1, 12, 1, 4, 1, 15, 1, 5, 1, 6, 3, 8, 1, 20, 4, 14, 13, 0, 1, 20, 1, 31, 15, 18, 1, 55, 1, 16, 9, 10, 1, 52, 1, 4, 29, 20, 1, 44, 6, 11, 21, 14, 1, 60, 13, 56, 23, 30, 1, 80, 1, 28, 1, 63, 11, 12, 1, 58, 19, 4, 1, 115, 1, 38, 1, 60, 3, 20, 1, 106, 22, 42, 1, 72, 23, 40, 25, 28, 1, 78, 5, 48, 27, 44, 21, 92, 1

Offset: 1

Antti Karttunen, Aug 28 2018

Note that here zeros occur only on even perfect numbers (even terms of A000396), in contrast to A318457, which would be zero also for any hypothetical odd perfect number. - Antti Karttunen, Aug 29 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A000396, A001065, A003987, A032742, A318505, A318506, A318508.

Cf. also A106409, A318457, A318462, A318517.

A001065(n) = (sigma(n)-n);
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318505(n) = if(1==n,0,A001065(n)-A032742(n));
A318507(n) = bitxor(A318505(n),A032742(n));

a(n) = A003987(A032742(n), A318505(n)).

For n > 1, a(n) = A001065(n) - 2*A318508(n).

A318516 a(n) = A032742(n) OR n-A032742(n), where OR is bitwise-or (A003986) and A032742 = the largest proper divisor of n.

Original entry on oeis.org

1, 1, 3, 2, 5, 3, 7, 4, 7, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10, 15, 11, 23, 12, 21, 13, 27, 14, 29, 15, 31, 16, 31, 17, 31, 18, 37, 19, 31, 20, 41, 21, 43, 22, 31, 23, 47, 24, 47, 25, 51, 26, 53, 27, 47, 28, 55, 29, 59, 30, 61, 31, 63, 32, 61, 33, 67, 34, 63, 35, 71, 36, 73, 37, 59, 38, 75, 39, 79, 40, 63, 41, 83, 42

Offset: 1

Antti Karttunen, Aug 28 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A003986, A032742, A060681, A318506, A318514, A318517, A318518.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318516(n) = bitor(A032742(n),n-A032742(n));

a(n) = A003986(A032742(n), A060681(n)).

a(n) = n - A318518(n).

A318514 a(n) = n OR (greatest proper divisor of n).

Original entry on oeis.org

1, 3, 3, 6, 5, 7, 7, 12, 11, 15, 11, 14, 13, 15, 15, 24, 17, 27, 19, 30, 23, 31, 23, 28, 29, 31, 27, 30, 29, 31, 31, 48, 43, 51, 39, 54, 37, 55, 47, 60, 41, 63, 43, 62, 47, 63, 47, 56, 55, 59, 51, 62, 53, 63, 63, 60, 59, 63, 59, 62, 61, 63, 63, 96, 77, 99, 67, 102, 87, 103, 71, 108, 73, 111, 91, 110, 79, 111, 79, 120, 91, 123, 83, 126

Offset: 1

Antti Karttunen, Aug 28 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A003986, A032742, A106409, A318506, A318515, A318516.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318514(n) = bitor(n,A032742(n));

a(n) = A003986(n, A032742(n)).

a(n) = n + A032742(n) - A318515(n).

cp's OEIS Frontend

A318505 Sum of divisors of n, up to, but not including the second largest of them A032742(n); a(1) = 0 by convention.

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A318508 a(n) = A032742(n) AND A001065(n)-A032742(n), where AND is bitwise-and (A004198) and A001065 = sum of proper divisors and A032742 = the largest proper divisor of n.

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PARI

Formula

A318507 a(n) = A032742(n) XOR A001065(n)-A032742(n), where XOR is bitwise-or (A003987) and A001065 = sum of proper divisors and A032742 = the largest proper divisor of n.

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A318516 a(n) = A032742(n) OR n-A032742(n), where OR is bitwise-or (A003986) and A032742 = the largest proper divisor of n.

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A318514 a(n) = n OR (greatest proper divisor of n).

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