A256440 -id:A256440 - cp's OEIS frontend

A256464 a(n) is the index, in the A256440 triangle, of the first row where n is the n-th term of the row.

1, 2, 5, 4, 13, 6, 23, 9, 19, 13, 43, 12, 59, 22, 27, 19, 79, 19, 97, 22, 41, 35, 113, 24, 89, 43, 59, 33, 157, 31, 173, 41, 77, 61, 101, 36, 211, 71, 95, 45, 239, 46, 257, 59, 71, 86, 281, 48, 209, 73, 127, 73, 331, 62, 167, 67, 143, 111, 373, 60, 389, 121

Offset: 1

Michel Marcus, Mar 30 2015

nonn

The integers for which a(n)=n are 1, 2, 4, 6, 12, 24, 36, 48, 60, ... : A004394.

The integers for which a(n)=n-1 are 3, 8, 18, 30, 72, ...: A247022 (except 3).

			Triangle A256440 starts:
1;
1, 2;
1, 3, 2;
1, 3, 2, 4;
1, 5, 3, 2, 4;
1, 5, 3, 2, 4, 6;
1, 7, 5, 3, 2, 4, 6;
...
a(1)=1 because row 1 is the first row where 1 is the 1st term of the row.
a(2)=2 because row 2 is the first row where 2 is the 2nd term of the row.
a(3)=5 because row 5 is the first row where 3 is the 3rd term of the row.

Cf. A004394, A247022, A256440.

posf(n) = {i = n; stop = 0; while (! stop, v = vector(i, k, sigma(k)/k); w = vecsort(v, , 1); pos = posv(n, w); if (pos == n, stop=1, i++);); i;}

A256566 a(n) is the central term of the (2n-1)-st row of triangle A256440.

Original entry on oeis.org

1, 3, 3, 3, 9, 9, 9, 2, 2, 2, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 38, 46, 46, 46, 46, 46, 46, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 62, 62, 74, 74, 74, 74, 82, 82, 74, 74, 74, 62, 74, 74, 74, 74, 74, 62, 74, 74, 82, 82, 82, 82, 82, 82, 82

Offset: 1

Michel Marcus, Apr 02 2015

nonn

All terms are deficient and after the 4th term sigma(n)/n appears to stay between 1.5 and 1.6.

Note that sequence is not monotonic and can return later to previous terms values.

Michel Marcus, Table of n, a(n) for n = 1..1000

Cf. A005100 (deficient numbers), A256440.

lista(nn) = {forstep(n=1, nn, 2, v = vector(n, k, sigma(k)/k); w = vecsort(v,,1); print1(w[(n+1)/2], ", "););}

A256562 Number of deficient numbers <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 6, 7, 8, 9, 10, 10, 11, 12, 13, 14, 15, 15, 16, 16, 17, 18, 19, 19, 20, 21, 22, 22, 23, 23, 24, 25, 26, 27, 28, 28, 29, 30, 31, 31, 32, 32, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 42, 42, 43, 43, 44, 45, 46, 46, 47, 48, 49, 50, 51, 51, 52, 53

Offset: 1

Michel Marcus, Apr 02 2015

nonn

			For k=1 to 5, all numbers are deficients so a(k) = k in this range.
a(6) = 5 since 6 is the first number that is not deficient.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Marc Deléglise, Bounds for the density of abundant integers, Experiment. Math. Volume 7, Issue 2 (1998), 137-143.
Charles R. Wall, Phillip L. Crews and Donald B. Johnson, Density bounds for the sum of divisors function, Math. Comp. 26 (1972), 773-777.
Eric Weisstein's World of Mathematics, Abundant Number

Partial sums of A294934.
Cf. A000396 (perfect), A005100 (deficient), A005101 (abundant).
Cf. A091194 (number of abundant numbers <= n).
Cf. A256440, A318172.

[#[k:k in [1..n]| DivisorSigma(1,k) lt 2*k]:n in [1..70]]; // Marius A. Burtea, Nov 06 2019

a[n_]:=Length[Select[Range[n],DivisorSigma[1,#]/#<2&]];a/@Range[68] (* Ivan N. Ianakiev, Apr 03 2015 *)

PARI
```
a(n) = sum(k=1, n, sigma(k)/k < 2);
```

a(n) ~ c*n, where c = 0.752380... is the asymptotic density of the deficient numbers (A318172). - Amiram Eldar, Mar 21 2021

cp's OEIS Frontend

A256464 a(n) is the index, in the A256440 triangle, of the first row where n is the n-th term of the row.

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A256566 a(n) is the central term of the (2n-1)-st row of triangle A256440.

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A256562 Number of deficient numbers <= n.

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