A294926 -id:A294926 - cp's OEIS frontend

A080226 Number of deficient divisors of n.

1, 2, 2, 3, 2, 3, 2, 4, 3, 4, 2, 4, 2, 4, 4, 5, 2, 4, 2, 5, 4, 4, 2, 5, 3, 4, 4, 5, 2, 6, 2, 6, 4, 4, 4, 5, 2, 4, 4, 6, 2, 6, 2, 6, 6, 4, 2, 6, 3, 6, 4, 6, 2, 5, 4, 6, 4, 4, 2, 7, 2, 4, 6, 7, 4, 6, 2, 6, 4, 7, 2, 6, 2, 4, 6, 6, 4, 6, 2, 7, 5, 4, 2, 7, 4, 4, 4, 7, 2, 8, 4, 6, 4, 4, 4, 7, 2, 6, 6, 7, 2, 6, 2, 7, 8

Offset: 1

Reinhard Zumkeller, Feb 07 2003

nonn

Number of divisors d of n with sigma(d)<2*d (sigma = A000203).

			All 4 divisors of n=21 are deficient: 1=A005100(1), 3=A005100(3), 7=A005100(6) and 21=A005100(17), therefore a(21)=4.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Deficient Number.

Cf. A000203, A005100, A187793, A294926, A294934.

a[n_] := Sum[If[DivisorSigma[1, d] < 2d, 1, 0], {d, Divisors[n]}];
Array[a, 105] (* Jean-François Alcover, Dec 02 2021 *)

A080226(n) = sumdiv(n, d, (sigma(d)<(2*d))); \\ Antti Karttunen, Nov 14 2017

A080224(n) + A080225(n) + a(n) = A000005(n).

a(n) = Sum_{d|n} A294934(d) = A294926(n) + A294934(n). - Antti Karttunen, Nov 14 2017

A294927 Number of proper divisors of n that are nondeficient (A023196).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 1, 0, 0, 0

Offset: 1

Antti Karttunen, Nov 14 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A023196, A032741, A071395, A294887, A294926, A294928, A294929, A294936.

a[n_] := DivisorSum[n, 1 &, # < n && DivisorSigma[1, #] >= 2*# &]; Array[a, 100] (* Amiram Eldar, Mar 14 2024 *)

PARI
```
A294927(n) = sumdiv(n, d, (d=(2*d)));
```

a(n) = Sum_{d|n, dA294936(d).

a(n) + A294926(n) = A032741(n).

A294929 Number of proper divisors of n that are abundant (A005101).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 6

Offset: 1

Antti Karttunen, Nov 14 2017

nonn

			The proper divisors of 24 are 1, 2, 3, 4, 6, 8, 12. Only one of these, 12, is abundant (in A005101), thus a(24) = 1.
The proper divisors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60. Six of these are abundant: 12, 20, 24, 30, 40, 60, thus a(120) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A005101, A032741, A080224, A294889, A294926, A294927, A294928, A294930, A294937.

Cf. also A294902.

a[n_] := Count[Most[Divisors[n]], ?(DivisorSigma[1, #] > 2*# &)]; Array[a, 100] (* _Amiram Eldar, Mar 14 2024 *)

PARI
```
A294929(n) = sumdiv(n, d, (d(2*d)));
```

a(n) = Sum_{d|n, dA294937(d).
a(n) = A080224(n) - A294937(n).
a(n) + A294928(n) = A032741(n).

A294886 Sum of deficient proper divisors of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 10, 1, 10, 9, 15, 1, 15, 1, 22, 11, 14, 1, 18, 6, 16, 13, 28, 1, 36, 1, 31, 15, 20, 13, 19, 1, 22, 17, 30, 1, 48, 1, 40, 33, 26, 1, 34, 8, 43, 21, 46, 1, 42, 17, 36, 23, 32, 1, 40, 1, 34, 41, 63, 19, 72, 1, 58, 27, 74, 1, 27, 1, 40, 49, 64, 19, 84, 1, 46, 40, 44, 1, 52, 23, 46, 33, 92, 1, 90, 21

Offset: 1

Antti Karttunen, Nov 14 2017

nonn

Sum of divisors of n smaller than n that are deficient numbers (in A005100).

			Proper divisors of 90 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45. Of these 1, 2, 3, 5, 9, 10, 15 and 45 are in A005100, thus a(90) = 1+2+3+5+9+10+15+45 = 90.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sums of divisors

Cf. A001065, A005100, A001065, A187793, A294926, A294934, A294887, A294888, A294889.

Cf. A125310 (fixed points).

a[n_] := DivisorSum[n, # &, # < n && DivisorSigma[1, #] < 2*# &]; Array[a, 100] (* Amiram Eldar, Mar 14 2024 *)

PARI
```
A294886(n) = sumdiv(n, d, (d
				
```

a(n) = Sum_{d|n, dA294934(d)*d.
a(n) = A187793(n) - (A294934(n)*n).
a(n) + A294887(n) = A001065(n).

A294928 Number of proper divisors of n that are nonabundant (A263837).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 6, 2, 3, 3, 5, 1, 7, 1, 5, 3, 3, 3, 6, 1, 3, 3, 6, 1, 7, 1, 5, 5, 3, 1, 7, 2, 5, 3, 5, 1, 6, 3, 7, 3, 3, 1, 8, 1, 3, 5, 6, 3, 7, 1, 5, 3, 7, 1, 7, 1, 3, 5, 5, 3, 7, 1, 7, 4, 3, 1, 9, 3, 3, 3, 7, 1, 9, 3, 5, 3, 3, 3, 8, 1, 5, 5, 7, 1, 7, 1, 7, 7

Offset: 1

Antti Karttunen, Nov 14 2017

nonn

			The seven proper divisors of 24 are 1, 2, 3, 4, 6, 8, 12. All except 12 are nonabundant (in A263837), thus a(24) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A263837, A294888, A294926, A294927, A294929, A294935.
Differs from A032741 for the first time at n=24.
Cf. also A294901.

a[n_] := DivisorSum[n, 1 &, # < n && DivisorSigma[1, #] <= 2*# &]; Array[a, 100] (* Amiram Eldar, Mar 14 2024 *)

PARI
```
A294928(n) = sumdiv(n, d, (d
				
```

a(n) = Sum_{d|n, dA294935(d).

a(n) + A294929(n) = A032741(n).

cp's OEIS Frontend

A080226 Number of deficient divisors of n.

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A294927 Number of proper divisors of n that are nondeficient (A023196).

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A294929 Number of proper divisors of n that are abundant (A005101).

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A294886 Sum of deficient proper divisors of n.

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A294928 Number of proper divisors of n that are nonabundant (A263837).

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