A189827 -id:A189827 - cp's OEIS frontend

A189828 Numbers k for which d(k-1) + d(k+1) is a record, where d(k) is the number of divisors of k.

2, 3, 5, 7, 11, 17, 19, 29, 41, 71, 161, 169, 181, 239, 419, 701, 721, 881, 1079, 1681, 2159, 2519, 2521, 4031, 4159, 5039, 7561, 8399, 10081, 13441, 13859, 18721, 20161, 22679, 25199, 27719, 27721, 35281, 45361, 50399, 50401, 55439, 65519, 110879, 138599

Offset: 1

T. D. Noe, Apr 28 2011

nonn

Many of these terms are in A090481, which restricts k to be prime. The record values are in A189829.

Donovan Johnson, Table of n, a(n) for n = 1..135

Cf. A090481, A189827, A189829.

DeleteDuplicates[Table[{n,DivisorSigma[0,n-1]+DivisorSigma[0,n+1]},{n,2,140000}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Jul 14 2025 *)

A189825 Least number k such that d(k-1) + d(k+1) = n, where d(k) is the number of divisors of k.

Original entry on oeis.org

2, 4, 3, 14, 5, 7, 15, 11, 17, 19, 35, 29, 65, 41, 101, 79, 143, 71, 197, 161, 323, 169, 2917, 181, 577, 239, 575, 629, 899, 419, 1297, 701, 901, 721, 25599, 881, 5183, 1121, 9215, 2351, 4901, 1079, 107585, 1681, 36863, 2159, 3601, 2881, 11663, 2519

Offset: 3

T. D. Noe, Apr 28 2011

nonn

The function d(k-1) + d(k+1) is a measure of the compositeness of the numbers next to k. There is no k for n=1 and n=2. Some terms can be quite large; for example, a(99) = 6533135.

Donovan Johnson, Table of n, a(n) for n = 3..880

Cf. A175144.

nn = 100; t = Table[-1, {nn}]; t[[1]] = t[[2]] = 0; cnt = 2; n = 1; While[cnt < nn, n++; s = DivisorSigma[0,n-1] + DivisorSigma[0,n+1]; If[s <= nn && t[[s]] == -1, t[[s]] = n; cnt++]]; Drop[t,2]

Least k such that A189827(k) = n.

A189829 Record values of d(n-1) + d(n+1), where d(k) is the number of divisors of k.

Original entry on oeis.org

3, 5, 7, 8, 10, 11, 12, 14, 16, 20, 22, 24, 26, 28, 32, 34, 36, 38, 44, 46, 48, 52, 56, 58, 60, 68, 72, 76, 78, 80, 84, 88, 92, 96, 98, 100, 104, 106, 108, 116, 124, 128, 136, 148, 156, 160, 168, 172, 174, 176, 184, 196, 200, 208, 224, 236, 248, 260, 268

Offset: 1

T. D. Noe, Apr 28 2011

nonn

The n are in A189828.

Donovan Johnson, Table of n, a(n) for n = 1..135

Cf. A189827, A189828.

A317749 a(n+1) = d(n) + d(a(n)) with a(1)=1, where d(n) is the number of the divisors of n.

Original entry on oeis.org

1, 2, 4, 5, 5, 4, 7, 4, 7, 5, 6, 6, 10, 6, 8, 8, 9, 5, 8, 6, 10, 8, 8, 6, 12, 9, 7, 6, 10, 6, 12, 8, 10, 8, 8, 8, 13, 4, 7, 6, 12, 8, 12, 8, 10, 10, 8, 6, 14, 7, 8, 8, 10, 6, 12, 10, 12, 10, 8, 6, 16, 7, 6, 10, 11, 6, 12, 8, 10, 8, 12, 8, 16, 7, 6, 10, 10, 8, 12, 8, 14, 9, 7, 4, 15, 8, 8, 8, 12, 8, 16, 9, 9, 7, 6, 8, 16, 7, 8, 10

Offset: 1

Jinyuan Wang, Aug 06 2018

nonn

If a(n+1)=4, then n and a(n) are prime numbers.

a(n+1) < 2*sqrt(a(n)) + 2*sqrt(n).

			d(1) = 1, d(2) = 2, d(3) = 2; a(1) = 1, a(2) = 2, a(3) = 4.
a(38)=4, so 37 and a(37)=13 are prime numbers.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A000005, A128863, A049820, A189827.

a[n_] := DivisorSigma[0, n - 1] + DivisorSigma[0, a[n - 1]]; a[1] = 1; Array[a, 80] (* Robert G. Wilson v, Aug 06 2018 *)

a(n) = if (n==1, 1, numdiv(n-1) + numdiv(a(n-1))); \\ Michel Marcus, Aug 25 2018

a(n+1) = d(n) + d(a(n)) where d(n) is the number of divisors of n (A000005).

Name edited by and more terms from Robert G. Wilson v, Aug 06 2018

cp's OEIS Frontend

A189828 Numbers k for which d(k-1) + d(k+1) is a record, where d(k) is the number of divisors of k.

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A189825 Least number k such that d(k-1) + d(k+1) = n, where d(k) is the number of divisors of k.

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A189829 Record values of d(n-1) + d(n+1), where d(k) is the number of divisors of k.

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A317749 a(n+1) = d(n) + d(a(n)) with a(1)=1, where d(n) is the number of the divisors of n.

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