A090481 -id:A090481 - cp's OEIS frontend

A175144 a(n) = d(p(n)-1) + d(p(n)+1), where p(n) is the n-th prime, and where d(m) is the number of divisors of m.

3, 5, 7, 8, 10, 10, 11, 12, 12, 14, 14, 13, 16, 14, 14, 14, 16, 16, 14, 20, 16, 18, 16, 20, 18, 17, 16, 16, 20, 18, 20, 20, 16, 20, 18, 20, 16, 16, 20, 14, 22, 26, 22, 18, 21, 24, 22, 20, 16, 20, 20, 28, 26, 26, 17, 20, 22, 26, 16, 24, 14, 18, 24, 24, 20, 14, 22, 26, 16, 24, 20

Offset: 1

Leroy Quet, Feb 24 2010

nonn

If a(n) is a record, then the n-th prime is in sequence A090481.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000005, A006093, A008864, A090481, A090482, A175146.

taudiff := proc(n) numtheory[tau](n-1)+numtheory[tau](n+1) ; end proc: A175144 := proc(n) taudiff(ithprime(n)) ; end proc: seq(A175144(n),n=1..80) ; # R. J. Mathar, Mar 03 2010

Table[p = Prime[n]; DivisorSigma[0, p - 1] + DivisorSigma[0, p + 1], {n, 100}]
Total[DivisorSigma[0,{#-1,#+1}]]&/@Prime[Range[80]] (* Harvey P. Dale, Feb 25 2012 *)

a(n) = numdiv(prime(n)-1) + numdiv(prime(n)+1); \\ Amiram Eldar, Apr 17 2024

lista(pmax) = forprime(p = 1, pmax, print1(numdiv(p-1) + numdiv(p+1), ", ")); \\ Amiram Eldar, Apr 17 2024

a(n) = A000005(A006093(n)) + A000005(A008864(n)). - R. J. Mathar, Mar 03 2010

More terms from R. J. Mathar, Mar 03 2010

A090482 Smallest prime p such that tau(p-1) + tau(p+1) is n, or 0 if no such number exists.

Original entry on oeis.org

0, 0, 2, 0, 3, 0, 5, 7, 0, 11, 17, 19, 37, 29, 0, 41, 101, 79, 0, 71, 197, 179, 401, 199, 2917, 181, 577, 239, 3137, 883, 4357, 419, 1297, 701, 12101, 839, 62501, 881, 30977, 1429, 21317, 2351, 16901, 1259, 287297, 1871, 1008017, 2161, 7057, 4049, 215297, 3079

Offset: 1

Amarnath Murthy, Dec 02 2003

nonn

a(9)=0. Proof: Both p-1 and p+1 are even and composite hence 9=1+8 and 9=2+7 are ruled out, the only possibilities that remain are 9 = 3+6, or 9=4+5. 3+6 is ruled out as 4 is the only even number with 3 divisors. 4+5 is ruled out as 16 is the only even number with 5 divisors.

a(15) = a(19) = 0 is also provable. - David Wasserman, Nov 17 2005

			a(10) = 11, tau(10) = 4 and tau(12) = 6, 4+6=10.
a(16) = 41, a(17) = 101.

Cf. A090481, A090483, A175144.

nn = 60; t = Table[-1, {nn}]; t[[{1,2,4,6,9,15,19}]] = 0; cnt = 7; p = 1; While[cnt < nn, p = NextPrime[p]; s = DivisorSigma[0, p-1] + DivisorSigma[0, p+1]; If[s <= nn && t[[s]] == -1, t[[s]] = p; cnt++]]; t (* T. D. Noe, Apr 28 2011 *)

Least prime p such that A175144(p) = n.

More terms from David Wasserman, Nov 17 2005

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

Original entry on oeis.org

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 *)

A347192 Integers k such that the number of divisors of k^2 - 1 (A347191) sets a new record.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 29, 41, 71, 109, 161, 169, 181, 379, 449, 649, 701, 881, 1079, 1189, 1871, 2449, 3079, 4159, 5851, 11969, 19601, 23561, 23869, 24751, 43471, 82081, 94249, 157249, 222641, 252449, 313039, 627199, 677249, 790399, 1276001, 2308879, 4058209

Offset: 1

Bernard Schott, Sep 16 2021

nonn

The first ten terms are the same as A090481 and A189828, then a(11) = 109 while A090481(11) = 179 and A189828(11) = 161.
The first eleven terms are the same as A335325, then a(12) = 161, which is nonprime, while A335325(12) = 181.
The corresponding records obtained are 2, 4, 8, 10, 16, 18, 24, 32, 40, 60, 64, 70, 80, 96, ...

			tau(71^2-1) = 60 and there is no integer k < 71 such that tau(k^2-1) >= 60, hence 71 is a term and a(10) = 71.

Diophante, A1885, Cachés derrière leurs diviseurs (in French).
Adrian Dudek, On the Number of Divisors of n^2-1, arXiv:1507.08893 [math.NT], 2015.

Cf. A000005, A005563, A014574, A129296, A129297.
Cf. A347191, A347193, A347194.
Cf. A090481, A189828, A335325 (similar, with k = p prime).

s[n_] := DivisorSigma[0, n^2 - 1]; sm = 0; seq = {}; Do[If[(sn = s[n]) > sm, sm = sn; AppendTo[seq, n]], {n, 2, 10^6}]; seq (* Amiram Eldar, Sep 16 2021 *)
DeleteDuplicates[Table[{k,DivisorSigma[0,k^2-1]},{k,2,4060000}],GreaterEqual[#1[[2]],#2[[2]]]&] [[;;,1]] (* Harvey P. Dale, Dec 04 2023 *)

A090483 Indices of zero in A090482.

Original entry on oeis.org

1, 2, 4, 6, 9, 15, 19

Offset: 1

Amarnath Murthy, Dec 02 2003

nonn

I conjecture there are no more terms; I've checked up to 200. - David Wasserman, Nov 17 2005

Cf. A090481, A090482.

Two more terms from David Wasserman, Nov 17 2005

A094342 Successive record-setters for tau(n+1)*tau(n-1)/tau(n)^2, where tau(n) is the number of divisors of n.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 29, 41, 71, 181, 239, 379, 449, 701, 881, 1429, 1871, 2729, 3079, 4159, 10529, 11969, 23561, 40699, 51679, 90271, 104651, 146719, 226799, 244529, 252449, 388961, 403649, 825551, 906751, 1276001, 2408561, 2648449, 3807649, 4058209, 4406401

Offset: 1

Isabel C. Lugo (isabel(AT)mit.edu), Jun 04 2004

Most terms are primes. These are numbers with few factors which are sandwiched between numbers with many factors. Terms <379 are same as those of A090481.

			tau(16)*tau(18)/tau(17)^2 = 5*6/2^2 = 15/2 and this is larger than for any n < 17, so 17 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..61

Cf. A090481.

f := x -> tau(x-1)*tau(x+1)/tau(x)^2:?print m := 1: A := []: for k from 2 to 10^6 do if f(k) > m then m := f(k): A := [op(A), [k, f(k)]]: fi; od;

s = {}; d1 = 1; d2 = 2; rm = 0; Do[d3 = DivisorSigma[0, n]; r = d1*d3/d2^2; If[r > rm, rm = r; AppendTo[s, n - 1]]; d1 = d2; d2 = d3, {n, 3, 10000}]; s (* Amiram Eldar, Aug 28 2019 *)

a(1) = 2 and more terms added by Amiram Eldar, Aug 28 2019

A175146 a(n) = d(A175145(n)-1)+d(A175145(n)+1), where d(m) is the number of divisors of m.

Original entry on oeis.org

3, 5, 7, 8, 10, 11, 12, 14, 16, 20, 22, 26, 28, 32, 34, 36, 38, 44, 46, 48, 56, 60, 68, 72, 76, 80, 84, 92, 96, 100, 106, 108, 128, 136, 148, 156, 160, 168, 174, 176, 196, 200, 208, 236, 244, 248, 260, 304, 328, 344, 392, 416, 436, 456, 484, 520, 528, 536, 592, 608

Offset: 1

Leroy Quet, Feb 24 2010

nonn

This sequence is strictly monotonically increasing.

Cf. A000005, A090481, A175144.

Extended by Ray Chandler, Mar 04 2010

cp's OEIS Frontend

A175144 a(n) = d(p(n)-1) + d(p(n)+1), where p(n) is the n-th prime, and where d(m) is the number of divisors of m.

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A090482 Smallest prime p such that tau(p-1) + tau(p+1) is n, or 0 if no such number exists.

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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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A347192 Integers k such that the number of divisors of k^2 - 1 (A347191) sets a new record.

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A090483 Indices of zero in A090482.

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A094342 Successive record-setters for tau(n+1)*tau(n-1)/tau(n)^2, where tau(n) is the number of divisors of n.

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A175146 a(n) = d(A175145(n)-1)+d(A175145(n)+1), where d(m) is the number of divisors of m.

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