A152771 -id:A152771 - cp's OEIS frontend

A152864 Deficiency of n, plus the number of proper divisors of n: a(n) = 2n - sigma(n) + d(n) - 1.

1, 2, 3, 3, 5, 3, 7, 4, 7, 5, 11, 1, 13, 7, 9, 5, 17, 2, 19, 3, 13, 11, 23, -5, 21, 13, 17, 5, 29, -5, 31, 6, 21, 17, 25, -11, 37, 19, 25, -3, 41, -5, 43, 9, 17, 23, 47, -19, 43, 12, 33, 11, 53, -5, 41, -1, 37, 29, 59, -37, 61, 31

Offset: 1

Omar E. Pol, Dec 14 2008

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A005843, A032741, A033879, A152770, A152771, A152772.

a[n_] := 2*n - Differences[DivisorSigma[{0, 1}, n]][[1]] - 1; Array[a, 100] (* Amiram Eldar, Apr 07 2024 *)

a(n) = 2*n - sigma(n) + numdiv(n) - 1; \\ Amiram Eldar, Apr 07 2024

a(n) = A005843(n)-A000203(n)+A000005(n)-1 = A033879(n)+A032741(n).

A366969 a(n) = Sum_{k=3..n} (k-2) * floor(n/k).

Original entry on oeis.org

0, 0, 1, 3, 6, 11, 16, 24, 32, 43, 52, 69, 80, 97, 114, 136, 151, 179, 196, 227, 252, 281, 302, 347, 373, 408, 441, 486, 513, 570, 599, 651, 692, 739, 780, 854, 889, 942, 991, 1066, 1105, 1186, 1227, 1300, 1367, 1432, 1477, 1582, 1634, 1716, 1781, 1868, 1919, 2024

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Partial sums of A152771.

Cf. A006218, A024916, A366968, A366970, A366971.

PARI
```
a(n) = sum(k=3, n, (k-2)*(n\k));
```

from math import isqrt
def A366969(n): return n+(-(s:=isqrt(n))*(s*(s-2)-7)+sum(((q:=n//w)+1)*(q+(w<<1)-8) for w in range(1,s+1))>>1) # Chai Wah Wu, Oct 30 2023

G.f.: 1/(1-x) * Sum_{k>=1} x^(3*k)/(1-x^k)^2 = 1/(1-x) * Sum_{k>=3} (k-2) * x^k/(1-x^k).

a(n) = n + A024916(n) - 2*A006218(n). - Chai Wah Wu, Oct 30 2023

A152772 a(n) = sigma(n) - 3*d(n) + 3.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 5, 6, 7, 9, 9, 13, 11, 15, 15, 19, 15, 24, 17, 27, 23, 27, 21, 39, 25, 33, 31, 41, 27, 51, 29, 48, 39, 45, 39, 67, 35, 51, 47, 69, 39, 75, 41, 69, 63, 63, 45, 97, 51, 78, 63, 83, 51, 99, 63, 99, 71, 81, 57, 135

Offset: 1

Omar E. Pol, Dec 14 2008

Cf. A000005 (d), A000203 (sigma), A152770, A152771.

a[n_] := DivisorSigma[1, n] - 3 * DivisorSigma[0, n] + 3; Array[a, 100] (* Amiram Eldar, Apr 07 2024 *)

a(n) = sigma(n) - 3 * numdiv(n) + 3; \\ Amiram Eldar, Apr 07 2024

cp's OEIS Frontend

A152864 Deficiency of n, plus the number of proper divisors of n: a(n) = 2n - sigma(n) + d(n) - 1.

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A366969 a(n) = Sum_{k=3..n} (k-2) * floor(n/k).

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Formula

A152772 a(n) = sigma(n) - 3*d(n) + 3.

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Mathematica

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