A099632 -id:A099632 - cp's OEIS frontend

A053223 Second differences of sigma(n).

-1, 2, -4, 7, -10, 11, -9, 7, -11, 22, -30, 24, -10, 7, -20, 34, -40, 41, -32, 14, -16, 48, -65, 40, -13, 18, -42, 68, -82, 71, -46, 21, -12, 49, -96, 75, -26, 38, -82, 102, -106, 92, -46, 0, -18, 100, -143, 103, -57, 47, -70, 110, -114, 96, -88, 50, -40, 138, -214, 140, -26, 15, -66

Offset: 1

Asher Auel, Jan 06 2000

sign

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

Cf. A000203, A053222, A053228, A053229, A053247, A053248.

Indices of 0: A099632 - 1.

[DivisorSigma(1, n+2) - 2*DivisorSigma(1,n+1) + DivisorSigma(1, n): n in [1..100]]; // G. C. Greubel, Sep 03 2018

Table[DivisorSigma[1,n+2] - 2*DivisorSigma[1,n+1] + DivisorSigma[1,n], {n,1,100}] (* G. C. Greubel, Sep 03 2018 *)

a(n)=sigma(n+2)-2*sigma(n+1)+sigma(n) \\ Charles R Greathouse IV, Mar 09 2014

G.f.: -1 - 1/x + (1 - x)^2*Sum_{k>=1} k*x^(k-2)/(1 - x^k). - Ilya Gutkovskiy, Jan 29 2017

A226361 Numbers n such that sigma(n) = sigma(n+1) + sigma(n+2).

Original entry on oeis.org

378624, 661152, 5479092, 5526024, 7179624, 18744216, 122321970, 168201288, 215676636, 778701984, 1482154170, 1788138780, 1974360132, 2288979096, 3361923780, 4214315484, 4757106144, 4971510492, 6264306144, 6884356716, 10730488296, 11375549304, 16851779736

Offset: 1

Alex Ratushnyak, Jun 05 2013

nonn

Giovanni Resta, Table of n, a(n) for n = 1..88 (terms < 10^13)

Cf. A000203, A055574, A065900, A073500, A099632.

nn = 10^7; t = {}; sig0 = 1; sig1 = 3; Do[sig2 = DivisorSigma[1, n + 2]; If[sig0 == sig1 + sig2, AppendTo[t, n]]; sig0 = sig1; sig1 = sig2, {n, nn}]; t (* T. D. Noe, Jun 05 2013 *)

a(17)-a(23) from Donovan Johnson, Jun 05 2013

A226475 Numbers n such that sigma(n) + sigma(n+1) = sigma(n+2) + sigma(n+3).

Original entry on oeis.org

75, 113, 295, 533, 686, 2130, 14805, 26966, 30235, 35095, 135653, 355675, 432996, 590138, 1214588, 2692853, 2952064, 3375195, 3486795, 5973014, 6880351, 7334956, 22266602, 25841659, 30483834, 37416582, 38390010, 40952513, 41109593, 57242145

Offset: 1

Alex Ratushnyak, Jun 11 2013

nonn

Sigma(n) is the sum of the divisors of n: A000203.

			sigma(75) + sigma(76) = 124 + 140 = 264, and sigma(77) + sigma(78) = 96 + 168 = 264, so 75 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..138 (terms < 10^13)

Cf. A000203, A055574, A065900, A073500, A099632, A226361.

t = {}; s = DivisorSigma[1, Range[0, 3]]; n = 3; While[Length[t] < 10, n++; s = RotateLeft[s]; s[[4]] = DivisorSigma[1, n]; If[s[[1]] + s[[2]] == s[[3]] + s[[4]], AppendTo[t, n - 3]]]; t (* T. D. Noe, Jun 12 2013 *)
Module[{ds=DivisorSigma[1,Range[6*10^7]]},Flatten[Position[Partition[ds,4,1],?(Total[Take[#,2]]==Total[Take[#,-2]]&),{1},Heads->False]]] (* _Harvey P. Dale, Sep 13 2014 *)

A076666 Numbers n such that sigma(n) + sigma(n+3) = sigma(n+1) + sigma(n+2).

Original entry on oeis.org

2012, 2096, 15892, 17888, 39916, 102784, 141008, 146227, 482144, 487865, 1321312, 1887008, 2749057, 3513881, 7141158, 16767172, 17503912, 28122834, 30534728, 37453779, 42140437, 60994100, 67777337, 78251933, 113091820, 113768920, 129868059, 199240914, 240859196, 302897372

Offset: 1

Joseph L. Pe, Oct 25 2002

nonn

Each term of the sequence marks the start of four consecutive sigma-values for which the sum of the means equals the sum of the extremes.

			sigma(2012) + sigma(2015) = 3528 + 2688 = 6216; sigma(2013) + sigma(2014) = 2976 + 3240 = 6216, so 2012 is a term of the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..90 (terms < 4*10^12)

Cf. A000203, A055574, A065900, A073500, A099632, A226361, A226475.

Select[Range[10^5], DivisorSigma[1, # ] + DivisorSigma[1, # + 3] == DivisorSigma[1, # + 1] + DivisorSigma[1, # + 2] &]

a(6)-a(26) from Donovan Johnson, Feb 01 2009

a(27)-a(30) from Alex Ratushnyak, Jun 29 2013

A099633 Numbers k such that (phi(k-2) + phi(k+2))/2 = phi(k); 2-phi/balanced numbers.

Original entry on oeis.org

5, 10, 14, 20, 25, 50, 80, 154, 200, 434, 520, 610, 815, 1205, 1345, 2680, 3200, 3560, 4669, 7330, 9290, 10130, 11705, 17630, 24410, 25090, 33385, 34205, 39754, 39785, 40870, 48470, 55979, 59090, 73165, 75470, 78035, 95990, 107680, 115685, 120230

Offset: 1

Labos Elemer, Oct 27 2004

nonn

			k=154, phi(152)=72, phi(156)=48, phi(154)=60.

Donovan Johnson, Table of n, a(n) for n = 1..499

Cf. A066167, A000010, A099631, A099632.

Flatten[Position[Partition[EulerPhi[Range[121000]],5,1],?((#[[1]]+ #[[5]]) /2 == #[[3]]&),{1},Heads->False]]+2 (* _Harvey P. Dale, Jan 20 2015 *)

isok(k) = (k>2) && (eulerphi(k-2)+eulerphi(k+2) == 2*eulerphi(k)); \\ Michel Marcus, Nov 18 2022

Incorrect term 2 removed by Michel Marcus, Nov 19 2022

A226753 Numbers n such that sigma(n) + sigma(n+1) + sigma(n+2) = sigma(n+3) + sigma(n+4) + sigma(n+5).

Original entry on oeis.org

52, 56, 2199, 17312, 92444, 31768040, 41159079, 52236776, 79563436, 683321810, 689969123, 757808252, 881421986, 910016369, 22184779055

Offset: 1

Alex Ratushnyak, Jun 16 2013

The sequence of corresponding sums begins: 272, 290, 10820, 86514, 473568, 162349956, 203060920, 258797664, 398693936, 3418531584, 3544794576, 3812186016, 4066082208, 4374966816.

			Sigma(52) + sigma(53) + sigma(54) = 98 + 54 + 120 = 272.
Sigma(55) + sigma(56) + sigma(57) = 72 + 120 + 80 = 272.
So 52 is in the sequence.

Cf. A000203, A055574, A065900, A073500, A099632, A226361, A226475.

a(15) from Giovanni Resta, Jun 17 2013

cp's OEIS Frontend

A053223 Second differences of sigma(n).

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A226361 Numbers n such that sigma(n) = sigma(n+1) + sigma(n+2).

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A226475 Numbers n such that sigma(n) + sigma(n+1) = sigma(n+2) + sigma(n+3).

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A076666 Numbers n such that sigma(n) + sigma(n+3) = sigma(n+1) + sigma(n+2).

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A099633 Numbers k such that (phi(k-2) + phi(k+2))/2 = phi(k); 2-phi/balanced numbers.

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A226753 Numbers n such that sigma(n) + sigma(n+1) + sigma(n+2) = sigma(n+3) + sigma(n+4) + sigma(n+5).

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