A074757 -id:A074757 - cp's OEIS frontend

A090502 Numbers k such that tau(k) = tau(k-1) + tau(k+1).

6, 20, 28, 32, 52, 56, 68, 152, 174, 176, 184, 186, 208, 212, 248, 266, 268, 272, 292, 322, 328, 338, 464, 470, 472, 476, 518, 534, 536, 578, 582, 585, 590, 602, 606, 650, 656, 658, 666, 670, 686, 710, 754, 772, 788, 790, 804, 814, 836, 846, 848, 894, 988

Offset: 1

Amarnath Murthy, Dec 05 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..4000
Vaclav Kotesovec, Plot of a(n)/n for n = 1..1000000
Vaclav Kotesovec, Plot of a(n)/(n*log(log(n))) for n = 1..1000000

Cf. A074757.

Select[Range[1000],DivisorSigma[0,#]==DivisorSigma[0,#-1]+ DivisorSigma[ 0,#+1]&] (* Harvey P. Dale, Feb 27 2012 *)
Flatten[Position[Partition[DivisorSigma[0,Range[1000]],3,1],?(#[[2]]==#[[1]]+#[[3]]&),1,Heads->False]]+1 (* Much faster than the above Mathematica program because tau of each number only has to be calculated once. *) (* _Harvey P. Dale, Jan 03 2020 *)

is(n)=numdiv(n)==numdiv(n-1)+numdiv(n+1) \\ Charles R Greathouse IV, Apr 24 2013

More terms from Ray Chandler, Dec 09 2003

A190612 Numbers k such that (tau(k-1) + tau(k+1))/tau(k) is an integer, where tau(k)=A000005(k).

Original entry on oeis.org

6, 7, 11, 13, 19, 20, 23, 25, 28, 29, 31, 32, 34, 39, 41, 43, 46, 47, 51, 52, 53, 55, 56, 57, 59, 61, 62, 67, 68, 71, 73, 74, 79, 83, 85, 86, 87, 89, 94, 95, 97, 103, 107, 109, 113, 119, 127, 129, 131, 133, 134, 137, 139, 141, 142, 149, 151, 152, 155, 157

Offset: 1

Juri-Stepan Gerasimov, May 14 2011

nonn

			6 is a term because (tau(5) + tau(7))/tau(6) = 1;
7 is a term because (tau(6) + tau(8))/tau(7) = 4;
11 is a term because (tau(10) + tau(12))/tau(11) = 5;
13 is a term because (tau(12) + tau(14))/tau(13) = 5.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n)/n for n = 1..2000000

Cf. A000005 (number of divisors of n), A074757, A090502.

with(numtheory): A190612 := proc(n) option remember: local k: if(n=1)then return 6:fi: for k from procname(n-1)+1 do if(tau(k-1)+tau(k+1) mod tau(k) = 0)then return k: fi: od: end: seq(A190612(n),n=1..70); # Nathaniel Johnston, May 14 2011

Select[Range[200], IntegerQ[(DivisorSigma[0, #-1] + DivisorSigma[0, #+1]) / DivisorSigma[0, #]] &] (* Vaclav Kotesovec, Feb 14 2019 *)

A190266 Numbers k such that tau(k-1) = (tau(k))^2 = tau(k+1), where tau(k) = A000005(k) (number of divisors of k).

Original entry on oeis.org

7, 1241, 1673, 1751, 1769, 2471, 2839, 3161, 3305, 3497, 3711, 4135, 4265, 4279, 4471, 4711, 5191, 5433, 5561, 6017, 6041, 6103, 6313, 6809, 6953, 7031, 7241, 7463, 7671, 8023, 8057, 8345, 8791, 8889, 9079, 10167, 10793, 10841, 11111, 11209, 11391, 11751, 12297, 12729

Offset: 1

Juri-Stepan Gerasimov, May 06 2011

nonn

			a(1)=7 because tau(6) = (tau(7))^2 = tau(8) = 4;
a(2)=1241 because tau(1240) = (tau(1241))^2 = tau(1242) = 16.

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

Cf. A000005, A074757, A090502. Subsequence of A036436.

Transpose[Select[Partition[Range[15000], 3, 1], DivisorSigma[0, #[[2]]]^2 == DivisorSigma[0, First[#]] == DivisorSigma[0, Last[#]]&]][[1]] + 1 (* Amiram Eldar, Jul 17 2019 after Harvey P. Dale at A175116 *)

isA190266(n)=my(t=numdiv(n-1)); issquare(t) & t==numdiv(n+1) & t==numdiv(n)^2 \\ Charles R Greathouse IV, May 14 2011

A000005(a(n)-1) = (A000005(a(n)))^2 = A000005(a(n)+1).

Data corrected by Amiram Eldar, Jul 17 2019

cp's OEIS Frontend

A090502 Numbers k such that tau(k) = tau(k-1) + tau(k+1).

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A190612 Numbers k such that (tau(k-1) + tau(k+1))/tau(k) is an integer, where tau(k)=A000005(k).

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A190266 Numbers k such that tau(k-1) = (tau(k))^2 = tau(k+1), where tau(k) = A000005(k) (number of divisors of k).

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