A090502 -id:A090502 - cp's OEIS frontend

A074757 Numbers n such that tau(n) = (tau(n+1)+tau(n-1))/2.

34, 46, 62, 74, 86, 94, 142, 188, 202, 214, 218, 231, 243, 244, 262, 278, 302, 356, 375, 394, 422, 423, 428, 436, 446, 459, 478, 494, 584, 596, 604, 621, 628, 634, 664, 698, 716, 837, 861, 885, 903, 916, 922, 944, 956, 982, 1004, 1017, 1018, 1028, 1042, 1052

Offset: 1

Benoit Cloitre, Sep 28 2002

Harvey P. Dale, Table of n, a(n) for n = 1..1000
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. A090502.

okQ[{a_,b_,c_}]:=(a+c)/2==b; Flatten[Position[Partition[DivisorSigma[0, Range[1100]],3,1],?okQ]]+1 (* _Harvey P. Dale, Aug 17 2013 *)

a(n) seems to be asymptotic to c*n with c around 23. [This conjecture is false. The limit a(n)/n, if it exists, is > 29, see graphs. - Vaclav Kotesovec, Feb 14 2019]

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

A347076 Numbers m such that tau(m) = tau(m-1) + tau(m+1) and simultaneously sigma(m) = sigma(m-1) + sigma(m+1).

Original entry on oeis.org

89484, 167784, 8587065618, 24033737496, 41249560520, 161721015522, 206958258156, 441151731162, 600656241732, 1013494535238, 4648478084262, 5099258875122, 7897343836494, 21060284613738, 26847208137084

Offset: 1

Jaroslav Krizek, Aug 15 2021

Intersection of A073500 and A090502.

a(n) is even. If a(n) is odd then two consecutive numbers are perfect squares. This only happens with (0, 1) which does not give terms. - David A. Corneth, Aug 16 2021

			tau(89484) = tau(89483) + tau(89485); 12 = 4 + 8.
sigma(89484) = sigma(89483) + sigma(89485); 208824 = 91608 + 117216.

Cf. A000005 (tau), A000203 (sigma), A073500, A090502.

[m: m in [2..10^5] | #Divisors(m) eq #Divisors(m - 1) + #Divisors(m + 1) and &+Divisors(m) eq &+Divisors(m - 1) + &+Divisors(m + 1)]

Select[Range[200000], DivisorSigma[{0, 1}, # - 1] + DivisorSigma[{0, 1}, # + 1] - DivisorSigma[{0, 1}, # ] == {0, 0} &] (* Amiram Eldar, Aug 16 2021 *)

a(14)-a(15) from Martin Ehrenstein, Sep 24 2021

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A074757 Numbers n such that tau(n) = (tau(n+1)+tau(n-1))/2.

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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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A347076 Numbers m such that tau(m) = tau(m-1) + tau(m+1) and simultaneously sigma(m) = sigma(m-1) + sigma(m+1).

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