A065900 -id:A065900 - cp's OEIS frontend

A076530 Numbers n such that sigma(n) = sigma(n+1) - sigma(n-1).

2, 3, 23, 1967, 3263, 5015, 60455, 1016507, 4420163, 12055511, 14365607, 25726727, 27896423, 66562307, 72764735, 98734967, 175186655, 224868311, 253694927, 288657203, 386668343, 421575407, 504737747, 630645455, 1493547999

Offset: 1

Joseph L. Pe, Oct 18 2002

nonn

			sigma(24) = 60. sigma(23) = 24. sigma(22) = 36 and 24 = 60 - 36; hence 23 is a term of the sequence.

Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..112 (terms < 10^13, first 50 terms from Donovan Johnson)

Cf. A000203, A065900, A104149.

Select[Range[2, 10^5], DivisorSigma[1, # ] == DivisorSigma[1, # + 1] - DivisorSigma[1, # - 1] &]
Flatten[Position[Partition[DivisorSigma[1,Range[1493548000]],3,1],?(#[[2]] == #[[3]]-#[[1]]&),1,Heads->False]]+1 (* _Harvey P. Dale, Apr 13 2020 *)

a(n) = A065900(n) - 1 = A104149(n) + 1. - Alex Ratushnyak, Jul 06 2013

a(8)-a(25) from Donovan Johnson, Feb 01 2009

A291126 Psibonacci numbers: solutions n of the equation psi(n) = psi(n-1) + psi(n-2), where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

3, 6, 210, 88200, 101970, 193290, 289680, 993990, 11264550, 59068230, 72776970, 98746230, 122460690, 126500910, 132766770, 234150930, 514442214, 531391650, 638082390, 650428020, 790769790, 1249160790, 3727074450, 4775972850, 8299675650, 9530202210

Offset: 1

Amiram Eldar, Aug 19 2017

nonn

Analogous to phibonacci numbers (A065557) and other sequences (see crossrefs).

			psi(210) = 576 = 240 + 336 = psi(209) + psi(208), therefore 210 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..70 (terms < 4*10^12)

Cf. A001615, A065557, A065900, A076136, A076251, A145469.

psi[n_]:=If[n < 1, 0, n Sum[ MoebiusMu[ d]^2 / d, {d, Divisors @ n}]];
Select[Range[10^6], psi[#]==psi[#-1]+psi[#-2] &]

a(21)-a(26) from Giovanni Resta, Aug 26 2018

A291176 Numbers k such that s(k) = s(k-1) + s(k-2), where s(k) is the sum of proper divisors of k (A001065).

Original entry on oeis.org

3, 8, 20, 146139, 584835, 44814015, 1436395095, 9988999095, 25997557299, 193861767939, 2105722150095, 3921293253003, 8234992646643

Offset: 1

Amiram Eldar, Aug 19 2017

a(14) > 10^13. - Giovanni Resta, Feb 25 2020

			s(146139) = 76581 = 75802 + 779 = s(146138) + s(146137), therefore 146139 is in the sequence.

Cf. A001065, A065557, A065900, A076136, A076251, A145469.

s[n_]:=DivisorSigma[1,n]-n; Select[Range[10^6], s[#]==s[#-1]+s[#-2] &]

a(7)-a(10) from Giovanni Resta, Aug 29 2017

a(11)-a(13) from Giovanni Resta, Feb 25 2020

A104149 Numbers k such that sigma(k+2) = sigma(k+1) + sigma(k).

Original entry on oeis.org

1, 2, 22, 1966, 3262, 5014, 60454, 1016506, 4420162, 12055510, 14365606, 25726726, 27896422, 66562306, 72764734, 98734966, 175186654, 224868310, 253694926, 288657202, 386668342, 421575406, 504737746, 630645454, 1493547998, 1653797794, 2120325010, 2221315150

Offset: 1

Neven Juric (neven.juric(AT)apis-it.hr), Aug 16 2010

nonn

Apparently all terms > 1 are even. - Zak Seidov, Mar 23 2015

For n <= 95, no a(n) is divisible by 3; a(2), a(25) and a(57) == 2 (mod 3), the rest == 1 (mod 3). - Robert Israel, Mar 23 2015

			sigma(22) = 1+2+11+22 = 36.
sigma(23) = 1+23 = 24.
sigma(24) = 1+2+3+4+6+8+12+24 = 60.
sigma(24) = sigma(23) + sigma(22).

Giovanni Resta, Table of n, a(n) for n = 1..112 (terms < 10^13; first 50 terms from Donovan Johnson)

[n: n in [1..2*10^6] | SumOfDivisors(n+2) eq (SumOfDivisors(n+1)+SumOfDivisors(n))]; // Vincenzo Librandi, Mar 24 2015

Select[Range@ 100000, DivisorSigma[1, # + 2] == DivisorSigma[1, # + 1] + DivisorSigma[1, #] &] (* Michael De Vlieger, Mar 23 2015 *)
Position[Partition[DivisorSigma[1,Range[3*10^7]],3,1],?(#[[1]]+#[[2]]==#[[3]]&),1,Heads->False]//Flatten (* The program generates the first 13 terms *) (* _Harvey P. Dale, May 08 2018 *)

s1=1; s2=3; for(n=1, 10^8, s3=sigma(n+2); if(s3==s1+s2, print1(n ", ")); s1=s2; s2=s3) /* Donovan Johnson, Apr 08 2013 */

a(n) = A065900(n) - 2. - R. J. Mathar, Aug 19 2010

a(n) = A076530(n) - 1. - M. F. Hasler, Aug 19 2010

More terms from Zak Seidov and R. J. Mathar, Aug 19 2010

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 *)

A332973 Solutions k of the equation usigma(k) = usigma(k-1) + usigma(k-2) where usigma(k) is the sum of unitary divisors of k (A034448).

Original entry on oeis.org

3, 42, 188970, 998670, 51670374, 91397154, 236280786, 259172826, 792554574, 1106710914, 1468869930, 1957827498, 2467823442, 2496238590, 3324585210, 4055970282, 4183629690, 4384566870, 13479861630, 20681058270, 29343074178, 43449285210, 68705958690, 71418085926

Offset: 1

Amiram Eldar, Mar 04 2020

nonn

			42 is a term since s(42) = 96 and s(40) + s(41) = 54 + 42 = 96.

Giovanni Resta, Table of n, a(n) for n = 1..64 (terms < 10^13)

The unitary version of A065900.

Cf. A034448, A065557, A075565, A076136, A145469, A291126, A291176, A292033, A294995, A332974.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[3, 10^8], usigma[#] == usigma[# - 1] + usigma[# - 2] &]

usigma(k) = sumdivmult(k, d, if(gcd(d, k/d)==1, d)); \\ A034448
isok(k) = usigma(k) == usigma(k-1) + usigma(k-2); \\ Jinyuan Wang, Mar 08 2020

Terms a(22) and beyond from Giovanni Resta, Mar 10 2020

A332975 Solutions k of the equation isigma(k) = isigma(k-1) + isigma(k-2) where isigma(k) is the sum of the infinitary divisors of k (A049417).

Original entry on oeis.org

3, 24, 360, 5016, 28440, 42066, 50568, 60456, 187176, 998670, 1454706, 12055512, 14365608, 25726728, 27896424, 51670374, 91702962, 141084774, 236280786, 249854952, 386668344, 439362504, 792554574, 1115866152, 1931976696, 2467823442, 2496238590, 2655297558, 2715505440

Offset: 1

Amiram Eldar, Mar 04 2020

nonn

			24 is a term since isigma(24) = 60 and isigma(22) + isigma(23) = 36 + 24 = 60.

Amiram Eldar, Table of n, a(n) for n = 1..38

The infinitary version of A065900.

Cf. A049417, A065557, A075565, A076136, A076251, A145469, A291126, A291176, A292033, A294995, A332976.

fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[3, 10^5], isigma[#] == isigma[# - 1] + isigma[# - 2] &]

A348335 a(n) = smallest k such that the sum of the divisors of the n numbers from k to k+n-1 equals sigma(k+n), or -1 if no such k exists.

Original entry on oeis.org

14, 1, 591357

Offset: 1

Metin Sariyar, Oct 13 2021

a(4) > 10^9, if it exists. - Amiram Eldar, Oct 13 2021

			a(1) = 14 because sigma(14) = sigma(15) = 24; a(1) = A002961(1).
a(2) = 1 because sigma(1) + sigma(2) = 1 + 3 = 4, the same as sigma(3) = 4; a(2) = A104149(1).
a(3) = 591357 because sigma(591357) + sigma(591358) + sigma(591359) = 866880 + 890352 + 599760 = 2356992, the same as sigma(591360) = 2356992.

Cf. A000203, A002961, A065900, A092403, A104149, A252922.

a[n_] := Module[{sig = DivisorSigma[1, Range[n]], k = n + 1}, While[(s = DivisorSigma[1, k]) != Plus @@ sig, sig = Join[Drop[sig, 1], {s}]; k++]; k - n]; Array[a, 3] (* Amiram Eldar, Oct 29 2021 *)

isok(m, nb) = sum(i=1, nb, sigma(m+i-1)) == sigma(m+nb);
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Oct 28 2021

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

cp's OEIS Frontend

A076530 Numbers n such that sigma(n) = sigma(n+1) - sigma(n-1).

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A291126 Psibonacci numbers: solutions n of the equation psi(n) = psi(n-1) + psi(n-2), where psi is the Dedekind psi function (A001615).

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A291176 Numbers k such that s(k) = s(k-1) + s(k-2), where s(k) is the sum of proper divisors of k (A001065).

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A104149 Numbers k such that sigma(k+2) = sigma(k+1) + sigma(k).

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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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A332973 Solutions k of the equation usigma(k) = usigma(k-1) + usigma(k-2) where usigma(k) is the sum of unitary divisors of k (A034448).

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A332975 Solutions k of the equation isigma(k) = isigma(k-1) + isigma(k-2) where isigma(k) is the sum of the infinitary divisors of k (A049417).

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