A002961 -id:A002961 - cp's OEIS frontend

A242962 a(1) = a(2) = 0; for n >= 3: a(n) = (n*(n+1)/2) mod antisigma(n) = A000217(n) mod A024816(n).

0, 0, 0, 1, 6, 3, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84

Offset: 1

Jaroslav Krizek, May 29 2014

nonn

A000217(n) = triangular numbers, A024816(n) = sum of numbers less than n which do not divide n.

a(n) = sigma(n) = A000203(n) for n = 5 and n>= 7 (see A242963).

			a(6) = 3 because A000217(6) mod A024816(6) = 21 mod 9 = 3.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000217, A002961, A024816, A242963, A243117, A243118.

[((n*(n+1)div 2) mod (n*(n+1)div 2-SumOfDivisors(n))): n in [3..1000]]

Array[If[# < 3, 0, Mod[PolygonalNumber@ #, Total@ Complement[Range@ #, Divisors@ #]]] &, 65] (* Michael De Vlieger, Jan 28 2020 *)

A242963 Numbers n such that A242962(n) = sigma(n) = A000203(n).

Original entry on oeis.org

5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Jaroslav Krizek, May 29 2014

nonn

A242962(n) = (n*(n+1)/2) mod antisigma(n) = A000217(n) mod A024816(n).
Union of number 5 and numbers >= 7.
Conjecture: this sequence lists all the positive integers n such that, for some integer k, (sin(k*Pi/n))^2 is irrational. - Lorenzo Sauras Altuzarra, Jan 27 2020

Michael De Vlieger, Table of n, a(n) for n = 1..14995

Cf. A000203, A000217, A002961, A024816, A242962, A243117, A243118, A217290, A009005.

[n: n in [3..100000] | ((n*(n+1)div 2) mod (n*(n+1)div 2-SumOfDivisors(n))) eq (SumOfDivisors(n))]

Select[Range[3, 71], DivisorSigma[1, #] == Mod[PolygonalNumber@ #, Total@ Complement[Range@ #, Divisors@ #]] &] (* Michael De Vlieger, Jan 28 2020 *)

A358817 Numbers k such that A046660(k) = A046660(k+1).

Original entry on oeis.org

1, 2, 5, 6, 10, 13, 14, 21, 22, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 49, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 80, 82, 85, 86, 93, 94, 98, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 122, 129, 130, 133, 135, 137, 138, 141, 142, 145, 147, 154, 157

Offset: 1

Amiram Eldar, Dec 02 2022

nonn

First differs from its subsequence A007674 at n=18.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 38, 369, 3655, 36477, 364482, 3644923, 36449447, 364494215, 3644931537, ... . Apparently, the asymptotic density of this sequence exists and equals 0.36449... .

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

Cf. A046660.
Subsequences: A007674, A052213, A085651, A358818.
Similar sequences: A002961, A005237, A006049, A045920.

seq[kmax_] := Module[{s = {}, e1 = 0, e2}, Do[e2 = PrimeOmega[k] - PrimeNu[k]; If[e1 == e2, AppendTo[s, k - 1]]; e1 = e2, {k, 2, kmax}]; s]; seq[160]

e(n) = {my(f = factor(n)); bigomega(f) - omega(f)};
lista(nmax) = {my(e1 = e(1), e2); for(n=2, nmax, e2=e(n); if(e1 == e2, print1(n-1,", ")); e1 = e2);}

A054005 Sum of divisors of k such that k and k+1 have the same number and sum of divisors.

Original entry on oeis.org

24, 2160, 2640, 4320, 51840, 65280, 115200, 138240, 194400, 186048, 276480, 483840, 622080, 700416, 950400, 984960, 1118880, 1128960, 1612800, 2661120, 3937248, 3617280, 5019840, 6128640, 5806080, 7375680, 8467200, 11583936

Offset: 1

Asher Auel, Jan 12 2000

nonn

			See example in A054004.

Amiram Eldar, Table of n, a(n) for n = 1..1831

Cf. A000005, A000203, A002961, A005237, A053249, A054004, A054006, A054007.

Select[Partition[Table[{n,DivisorSigma[0,n],DivisorSigma[1,n]},{n,116*10^5}],2,1],#[[1,2]]== #[[2,2]] && #[[1,3]]==#[[2,3]]&][[All,1,3]] (* Harvey P. Dale, May 16 2023 *)

a(n) = sigma(A054004(n)).

More terms from Jud McCranie, Oct 15 2000

Definition clarified by Harvey P. Dale, May 16 2023

A054006 Number of divisors of k and k+1 which have the same number and sum of divisors.

Original entry on oeis.org

4, 8, 8, 8, 8, 8, 16, 16, 16, 8, 16, 16, 16, 16, 16, 16, 16, 16, 32, 16, 16, 16, 16, 32, 16, 16, 16, 24, 16, 16, 16, 32, 32, 32, 16, 16, 32, 16, 32, 16, 16, 32, 32, 16, 32, 16, 16, 16, 16, 16, 32, 32, 16, 32, 16, 16, 64, 32, 16, 32, 16, 32, 16, 64, 32, 32, 16, 32, 32, 32, 32

Offset: 1

Asher Auel, Jan 12 2000

nonn

			See example in A054004.

Amiram Eldar, Table of n, a(n) for n = 1..1831 (calculated using the b-file at A054004)

Cf. A000005, A000203, A002961, A005237, A053249, A054004, A054005, A054007.

Select[Partition[Array[DivisorSigma[{0, 1}, #] &, 10^6], 2, 1], SameQ @@ # &][[All, 1, 1]] (* Michael De Vlieger, Nov 21 2019 *)

a(n) = tau(A054004(n)).

More terms from Jud McCranie, Oct 15 2000

A054007 Numbers k such that k and k+1 have the same sum but an unequal number of divisors.

Original entry on oeis.org

206, 957, 1364, 2974, 4364, 14841, 18873, 19358, 20145, 24957, 36566, 56564, 74918, 79826, 79833, 92685, 111506, 116937, 138237, 147454, 161001, 162602, 174717, 190773, 193893, 201597, 230390, 274533, 347738, 416577, 422073, 430137

Offset: 1

Asher Auel, Jan 12 2000

nonn

			The divisors of 206 are 1, 2, 103, 206, so tau(206) = 4 and sigma(206) = 312; the divisors of 207 are 1, 3, 9, 23, 69, 207, so tau(207) = 6 and sigma(207) = 312. Hence, the integer 206 belongs to this sequence. - _Bernard Schott_, Oct 18 2019

Amiram Eldar, Table of n, a(n) for n = 1..8304 (terms below 10^13, calculated from the b-file at A002961)

Cf. A000005, A000203, A002961, A005237, A053249, A054004, A054005, A054006.

Select[Range[100000], DivisorSigma[0, #] != DivisorSigma[0, # + 1] && DivisorSigma[1, #] == DivisorSigma[1, # + 1] &] (* Jayanta Basu, Mar 20 2013 *)

Members of A002961 which are not members of A054004

More terms from Jud McCranie, Oct 15 2000

A175876 Numbers k such that sigma(k+2) = 2*sigma(k).

Original entry on oeis.org

118, 1558, 2938, 17758, 19918, 32218, 33838, 55963, 71038, 186778, 308038, 511498, 523774, 553498, 699358, 838213, 1048903, 1159378, 1328938, 1333246, 1700038, 2462686, 2703886, 2956078, 3115318, 3561094, 3764206, 3972694, 7625878, 7852918, 8048962

Offset: 1

Zak Seidov, Oct 06 2010

nonn

a(1) = A175874(2).

Vincenzo Librandi and Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 171 terms from Vincenzo Librandi)

Cf. A000203, A002961, A175874, A175875.

Do[If[DivisorSigma[1, n+2]==2 DivisorSigma[1, n], Print[n]], {n, 2, 10^7}] (* or *) Select[Range[8000000], DivisorSigma[1, # + 2] == 2 DivisorSigma[1, #]&](* Vincenzo Librandi, Apr 04 2013 *)

A333261 Numbers k such that A071324(k) = A071324(k+1).

Original entry on oeis.org

1, 5, 51, 68, 87, 116, 171, 176, 591, 2108, 2403, 7143, 8787, 9308, 18548, 19371, 27387, 32127, 34887, 37928, 40131, 140667, 180548, 192428, 200996, 433311, 521727, 934449, 2476671, 2617563, 3960896, 8198156, 9670748, 11892512, 16585748, 19113651, 25367396, 25643012

Offset: 1

Amiram Eldar, Mar 13 2020

From Shreyansh Jaiswal, Jun 14 2025: (Start)
If the density of the set containing all even terms exists, then it is less than 0.15. (Proposition 3 in Jaiswal.)
Let k denote any even term. Then, the least prime factor of k+1 is either 3 or 5. (Theorem 11 in Jaiswal.)
Each even term satisfies at least one of three specific congruences. (Theorem 2 in Jaiswal.)
10519952096 and 16159802432 are also terms of this sequence.
Conjecture: There are infinitely many terms of this sequence. (Conjecture 15 in Jaiswal.) (End)

			1 is a term since A071324(1) = A071324(2) = 1.
5 is a term since A071324(5) = A071324(6) = 4.

Amiram Eldar, Table of n, a(n) for n = 1..89 (terms below 10^10)
Shreyansh Jaiswal, On the Even Solutions of A071324(n) = A071324(n+1), Jun 14, 2025.

Cf. A002961, A071324, A206368.

f[n_] := Plus @@ (-(d = Divisors[n])*(-1)^(Range[Length[d],1,-1])); seq = {}; f1 = f[1]; Do[f2 = f[n]; If[f1 == f2, AppendTo[seq, n-1]]; f1 = f2, {n, 2, 50000}]; seq
SequencePosition[Table[Total[Times@@@Partition[Riffle[Reverse[Divisors[n]],{1,-1},{2,-1,2}],2]],{n,2565*10^4}],{x_,x_}][[All,1]] (* Harvey P. Dale, Nov 06 2022 *)

from sympy import divisors;  from functools import lru_cache
cached_divisors = lru_cache()(divisors)
def c(n):  return sum(d if i%2==0 else -d for i, d in enumerate(reversed(cached_divisors(n))))
for n in range(1,2201):
    if c(n) == c(n+1):
        print(n, end=", ") # Shreyansh Jaiswal, Apr 14 2025

A348346 Numbers k such that k and k+1 have the same positive sum of noninfinitary divisors (A348271).

Original entry on oeis.org

20150, 52767, 99296, 835515, 1241504, 2199392, 6294015, 11158496, 12770450, 17016416, 19127907, 20128544, 23686748, 24790688, 26580554, 33366015, 34385247, 39687651, 42106976, 44157087, 45466676, 59825349, 60832449, 73780244, 75268775, 81654650, 84696849, 111457213

Offset: 1

Amiram Eldar, Oct 13 2021

nonn

Numbers k such that A348271(k) = A348271(k+1) > 0.

The terms are restricted to have a positive sum of noninfinitary divisors, since there are many consecutive numbers without noninfinitary divisors (these are the terms of A036537).

			20150 is a term since A348271(20150) = A348271(20151) = 6720.

Amiram Eldar, Table of n, a(n) for n = 1..154

Cf. A036537, A348271, A348345.
Subsequence of A162643.
Similar sequences: A002961, A064115, A064125, A293183, A306985.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; s[n_] := DivisorSigma[1,n] - isigma[n]; Select[Range[10^5], (s1 = s[#]) > 0 && s1 == s[# + 1] &]

A054002 Number of divisors of n such that n and n-1 have the same sum of divisors.

Original entry on oeis.org

4, 6, 4, 8, 16, 8, 8, 12, 12, 8, 4, 10, 8, 4, 8, 12, 8, 16, 16, 16, 16, 8, 16, 12, 16, 16, 8, 8, 4, 16, 8, 24, 16, 4, 16, 8, 4, 32, 8, 16, 16, 16, 8, 8, 8, 8, 16, 32, 8, 8, 16, 16, 16, 12, 16, 16, 12, 8, 48, 32, 8, 24, 24, 16, 16, 16, 8, 16, 24, 64, 8, 16, 16, 16, 16, 64, 24, 32, 60

Offset: 1

Asher Auel, Jan 12 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10135 (from the b-file at A002961)

Cf. A000005, A000203, A002961, A053249, A054003.

[#Divisors(n):n in [2..3500000]| SumOfDivisors(n) eq SumOfDivisors(n-1)]; // Marius A. Burtea, Sep 07 2019

a(n) = tau(A002961(n) + 1).

More terms from Naohiro Nomoto, Jun 23 2001

cp's OEIS Frontend

A242962 a(1) = a(2) = 0; for n >= 3: a(n) = (n*(n+1)/2) mod antisigma(n) = A000217(n) mod A024816(n).

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A242963 Numbers n such that A242962(n) = sigma(n) = A000203(n).

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Magma

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A358817 Numbers k such that A046660(k) = A046660(k+1).

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A054005 Sum of divisors of k such that k and k+1 have the same number and sum of divisors.

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A054006 Number of divisors of k and k+1 which have the same number and sum of divisors.

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A054007 Numbers k such that k and k+1 have the same sum but an unequal number of divisors.

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A175876 Numbers k such that sigma(k+2) = 2*sigma(k).

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A333261 Numbers k such that A071324(k) = A071324(k+1).

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