A002961 -id:A002961 - cp's OEIS frontend

A275337 Least k such that sigma(kn) = sigma(kn+1), or 0 if no such k exists.

14, 7, 319, 341, 537

Offset: 1

Altug Alkan, Jul 24 2016

Terms 6 through 30, with 0 signifying no such k <= 10^6: {0, 2, 220708, 1649, 23039, 87, 0, 6141, 1, 179, 110354, 873, 0, 86, 0, 5813, 62, 58, 0, 37497, 17149, 699, 0, 33, 0}. - Michael De Vlieger, Jul 25 2016
a(28) = 16806458, from A002961 b-file. - Michel Marcus, Jul 26 2016
If a(n) = 0, then a(m*n) = 0 for all m > 0. If c is a divisor of a(n), then a(c*n) = a(n)/c. - Chai Wah Wu, Jul 26 2016

			a(3) = 319 because sigma(319*3) = sigma(319*3+1) = 1440.

Michel Marcus, Some terms

Cf. A002961.

Table[SelectFirst[Range[10^6], DivisorSigma[1, # n] == DivisorSigma[1, # n + 1] &], {n, 30}] /. k_ /; MissingQ@ k -> 0 (* Michael De Vlieger, Jul 25 2016, Version 10.2 *)

a(n) = {my(k = 1); while (sigma(k*n) != sigma(k*n+1), k++); k;} \\ Michel Marcus, Jul 26 2016

A333949 Numbers k such that s(k) = s(k+1), where s(k) is the sum of recursive divisors of k (A333926).

Original entry on oeis.org

14, 206, 957, 1334, 1364, 1485, 1634, 2685, 2974, 4136, 4364, 14841, 20145, 24957, 33998, 36566, 42818, 64672, 74918, 79826, 79833, 84134, 86343, 92685, 109864, 111506, 122073, 138237, 147454, 159711, 162602, 166934, 187863, 190773, 193893, 201597, 274533, 288765

Offset: 1

Amiram Eldar, Apr 11 2020

nonn

			14 is a term since A333926(14) = A333926(15) = 24.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A333926.

Analogous sequences: A002961, A064115 (nonunitary), A064125 (unitary), A293183 (bi-unitary), A306985 (infinitary).

recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], recDivSum[#] == recDivSum[# + 1] &]

A340713 Numbers k such that sigma(k+1) = 4 * sigma(k).

Original entry on oeis.org

37033919, 141162839, 264995639, 596672999, 606523679, 630777839, 791656319, 920424119, 1060332839, 1379454719, 1954690919, 3799661039, 4024838999, 4633959959, 5393988599, 5935994063, 8831231639, 9866482079, 11237657759, 11273710139, 12266364599, 14440498379, 14952625379

Offset: 1

Seiichi Manyama, Jan 17 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..40 (using A058072 data)

Cf. A000203, A002961, A058072, A067081, A077087, A340715.

isok(k) = sigma(k+1) == 4*sigma(k); \\ Michel Marcus, Jan 18 2021

a(10)-a(23) from Seiichi Manyama using A058072 data, Jan 17 2021

A347603 Numbers k such that tau(k) = 2*tau(k-1) and sigma(k) = sigma(k-1), where tau(k) and sigma(k) are respectively the number and sum functions of the divisors of k.

Original entry on oeis.org

4365, 74919, 79827, 111507, 347739, 445875, 739557, 2168907, 4481986, 7263945, 7845387, 9309465, 10838247, 12290055, 12673095, 18151479, 22083215, 25645707, 39175955, 62634519, 69076995, 72794967, 80889207, 81166839, 87215967, 94682133, 107522943, 110768835, 119192283

Offset: 1

Claude H. R. Dequatre, Sep 08 2021

Conjecture: the asymptotic density of terms is equal to 0 and this sequence is infinite.

			a(1) = 4365 because the divisors of 4365 are: 1, 3, 5, 9, 15, 45, 97, 291, 485, 873, 1455, 4365; so, tau(4365) = 12 and sigma(4365) = 7644. The divisors of 4364 are: 1, 2, 4, 1091, 2182, 4364; so, tau(4364) = 6 and sigma(4364) = 7644. Thus tau(4365) = 2*tau(4364), sigma(4365) = sigma(4364) and so 4365 is a term.

Kevin P. Thompson, Table of n, a(n) for n = 1..288

Cf. A027750, A000005, A000203.

Similar sequences: A002961, A005237, A005238, A006601, A049051, A347076.

Select[Range[2, 10^6], DivisorSigma[0, #] == 2*DivisorSigma[0, # - 1] && DivisorSigma[1, #] == DivisorSigma[1, # - 1] &] (* Amiram Eldar, Sep 08 2021 *)

for(k=2,100000000,if(numdiv(k)==2*numdiv(k-1) && sigma(k)==sigma(k-1),print1(k", ")))

from sympy import divisor_count as tau, divisor_sigma as sigma
print([k for k in range(2, 10**6) if tau(k) == 2*tau(k-1) and sigma(k) == sigma(k-1)]) # Karl-Heinz Hofmann, Jan 15 2022

A172333 Numbers m such that m and m+22 have the same sum of divisors.

Original entry on oeis.org

57, 85, 213, 224, 354, 476, 568, 594, 812, 1218, 1235, 1316, 1484, 2103, 2470, 2492, 2643, 2840, 2996, 3836, 3978, 4026, 4544, 4810, 4844, 5012, 6125, 6356, 6524, 7364, 7532, 7648, 8876, 9272, 9328, 10098, 11107, 11797, 12572, 12594, 13412, 13640

Offset: 1

Michel Lagneau, Feb 01 2010

nonn

If 3*k-1 and 14*k-1 are both prime with k>1, then n = 28*(3*k-1) belongs to this sequence. The number of such integers n <= x would be asymptotically cx/(log x)^2 for some constant c > 0 from the Hardy-Littlewood conjecture D in Partitio Numerorum. - Tomohiro Yamada, Oct 03 2018

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 62, p. 22, Ellipses, Paris 2008.
W. Sierpinski, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 110.
Tomohiro Yamada, On equations sigma(n) = sigma(n+k) and phi(n) = phi(n+k), J. Comb. Number Theory 9 (2017), 15-21.

Tomohiro Yamada, Table of n, a(n) for n = 1..46702 (All terms < 2^28, first 2000 terms from Muniru A Asiru)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1-70.
Tomohiro Yamada, On equations sigma(n) = sigma(n+k) and phi(n) = phi(n+k)<, arXiv:1001.2511 [math.NT], 2010.

Cf. A000203, A015861, A002961, A015865, A015867, A015858, A015859, A015860.

Filtered([1..13700],k->Sigma(k)=Sigma(k+22)); # Muniru A Asiru, Oct 20 2018

with(numtheory):for n from 1 to 20000 do;if sigma(n) = sigma(n+22) then print(n); else fi ; od;

isok(k) = sigma(k)==sigma(k+22); \\ Altug Alkan, Oct 03 2018

A175874 a(n) = least number such that sigma(a(n)+n)=n*sigma(a(n)).

Original entry on oeis.org

14, 118, 3, 2

Offset: 1

Zak Seidov, Oct 06 2010

a(1)= A002961(1).
a(2)= A175876(1), a(3)= A175875(1). - Zak Seidov, Jul 07 2013
Is n = 2 the only solution for sigma(n+4)=4*sigma(n)? - Zak Seidov, Jul 07 2013
A further solution to sigma(n+4)=4*sigma(n), and the terms a(5) and a(6), if they exist, they are all larger than 3*10^12. - Giovanni Resta, Jun 06 2016

			sigma(14+1)/sigma(14)=24/24=1, sigma(118+2)/sigma(118)=360/180=2,
sigma(3+3)/sigma(3)=12/4=3, sigma(4+4)/sigma(2)=12/3=4.

Cf. A000203 sigma(n), A002961 sigma(n) = sigma(n+1).

Cf. A175875, A175876. - Zak Seidov, Jul 07 2013

A192283 Sum of prime anti-divisors of n = sum of prime anti-divisors of n+1 with n > 1.

Original entry on oeis.org

237, 4019, 7401, 14178, 14339, 18435, 19146, 21405, 54562, 56348, 60125, 82967, 98447, 99347, 109157, 113391, 125333, 132096, 132386, 145063, 173399, 195213, 260288, 278271, 343848, 384169, 396813, 434375, 460758, 474105, 477707, 528845, 550400, 587211

Offset: 1

Paolo P. Lava, Jul 27 2011

nonn

Like A006145 but using anti-divisors.

			Anti-divisors of 7401 are 2, 6, 19, 41, 113, 131, 361, 779, 4934. The primes are 2, 19, 41, 113 and 131 whose sum is 306.
Anti-divisors of 7402 are 3, 4, 5, 7, 9, 15, 21, 35, 45, 47, 63, 105, 113, 131, 141, 235, 315, 329, 423, 705, 987, 1645, 2115, 2961, 4935. The primes are 3, 5, 7, 47, 113 and 131 whose sum is 306.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A002961, A006145, A066272, A192282

with(numtheory);
P:=proc(n)
local a,b,i,k;
b:=2;
for i from 4 to n do
  a:=0;
  for k from 2 to i-1 do
    if abs((i mod k)- k/2) < 1 then if isprime(k) then a:=a+k; fi; fi;
  od;
  if a=b then print(i-1); fi;
  b:=a;
od;
end:
P(200000);

A290303 Values of usigma(n) = usigma(n+1).

Original entry on oeis.org

24, 60, 72, 180, 1440, 2160, 1872, 2640, 2400, 3000, 2880, 3024, 4320, 4320, 4320, 5280, 5280, 7400, 8640, 10080, 10200, 11520, 11880, 11520, 11088, 12960, 12096, 14400, 25920, 21600, 26640, 34560, 25200, 40320, 34560, 36000, 51840, 60480, 63360, 60480, 65280

Offset: 1

Amiram Eldar, Jul 26 2017

nonn

The sum of unitary divisors of numbers n such that n and n+1 have the same sum.

The unitary version of A053215.

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

Cf. A002961, A034448, A053215, A064125.

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]];a={}; u1=0; For[k=0, k<10^5, k++; u2=usigma[k]; If[u1==u2, a = AppendTo[a, u1]]; u1=u2]; a

a(n) = A034448(A064125(n)).

A335071 Numbers m such that the delta(m) = abs(sigma(m+1)/(m+1) - sigma(m)/(m)) is smaller than delta(k) for all k < m.

Original entry on oeis.org

1, 2, 14, 21, 62, 81, 117, 206, 897, 957, 1334, 1634, 2685, 2974, 4364, 14282, 14841, 18873, 19358, 24957, 33998, 36566, 42818, 56564, 64665, 74918, 79826, 79833, 92685, 109214, 111506, 116937, 122073, 138237, 145215, 15511898, 16207345, 17714486, 17983593, 18077605

Offset: 1

Amiram Eldar, May 22 2020

nonn

Can two consecutive numbers have the same abundancy: sigma(m)/m = sigma(m+1)/(m+1)? If yes, then this sequence is finite.

There is no disproof of existence, but this would require both of the consecutive numbers to be k-perfect with the same k >= 2, and it is conjectured that such numbers are multiples of k!. It is very unlikely that an odd k-perfect number will ever be found, and even much less probable that it will be just next to an even k-perfect number. - M. F. Hasler, Jun 06 2020

			The values of delta(k) for the first terms are 0.5, 0.166..., 0.114..., 0.112..., 0.102..., ...

Amiram Eldar, Table of n, a(n) for n = 1..260
SeqFan thread, A335071 question, SeqFan Mailing List, May 2020.

Cf. A000203, A002961, A017665/A017666.

ab[n_] := DivisorSigma[1, n]/n; dm = 2; ab1 = ab[1]; s = {}; Do[ab2 = ab[n]; d = Abs[ab2 - ab1]; If[d < dm, dm = d; AppendTo[s, n]]; ab1 = ab2, {n, 2, 10^5}]; s

lista(nn) = {my(d=oo, newd, lastm=1, ab=1); for (m=2, nn, nab = sigma(m)/m; if ((newd=abs(nab-ab)) < d, print1(m-1, ", "); d = newd;); ab = nab;);} \\ Michel Marcus, May 24 2020

A340715 Least positive number k such that sigma(k+1) = n * sigma(k).

Original entry on oeis.org

14, 5, 1, 37033919, 14182439039

Offset: 1

Seiichi Manyama, Jan 17 2021

			  n | sigma(a(n)) | sigma(a(n)+1)
----+-------------+--------------
  1 |          24 |            24
  2 |           6 |            12
  3 |           1 |             3
  4 |    39940992 |     159763968
  5 | 14182439040 |   70912195200

Cf. A000203, A002961, A058072, A067081, A077087, A080371, A340713, A340720.

k = 1;n = 1;Print[While[DivisorSigma[1, k + 1] != n*DivisorSigma[1, k], k;k k+]; k] (* Robert P. P. McKone, Jan 17 2021 *)

{a(n) = my(k=1); while(sigma(k+1)!=n*sigma(k), k++); k}

cp's OEIS Frontend

A275337 Least k such that sigma(k*n) = sigma(k*n+1), or 0 if no such k exists.

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A333949 Numbers k such that s(k) = s(k+1), where s(k) is the sum of recursive divisors of k (A333926).

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A340713 Numbers k such that sigma(k+1) = 4 * sigma(k).

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Extensions

A347603 Numbers k such that tau(k) = 2*tau(k-1) and sigma(k) = sigma(k-1), where tau(k) and sigma(k) are respectively the number and sum functions of the divisors of k.

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Python

A172333 Numbers m such that m and m+22 have the same sum of divisors.

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A175874 a(n) = least number such that sigma(a(n)+n)=n*sigma(a(n)).

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A192283 Sum of prime anti-divisors of n = sum of prime anti-divisors of n+1 with n > 1.

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Maple

A290303 Values of usigma(n) = usigma(n+1).

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Mathematica

A275337 Least k such that sigma(kn) = sigma(kn+1), or 0 if no such k exists.