A246456 -id:A246456 - cp's OEIS frontend

A246910 Numbers n such that sigma(n+sigma(n)) = 3*sigma(n).

1, 7, 26, 30, 42, 54, 69, 78, 84, 94, 102, 103, 114, 138, 140, 174, 222, 258, 354, 364, 474, 476, 498, 520, 532, 534, 582, 618, 644, 650, 762, 764, 812, 834, 847, 894, 978, 1002, 1036, 1038, 1050, 1182, 1185, 1194, 1204, 1214, 1362, 1372, 1398, 1434, 1487

Offset: 1

Jaroslav Krizek, Sep 07 2014

nonn

A246914 gives the primes in this sequence.

			Number 26 (with sigma(26) = 42) is in sequence because sigma(26+sigma(26)) = sigma(68) = 126 = 3*42.

Martin Møller Skarbiniks Pedersen, Table of n, a(n) for n = 1..1000

Cf. A000203, A246456, A246909, A246911, A246912, A246913, A246914, A272027.

[n:n in[1..10000] | SumOfDivisors(n+SumOfDivisors(n)) eq 3*SumOfDivisors(n)]

with(numtheory): A246910:=n->`if`(sigma(n+sigma(n)) = 3*sigma(n),n,NULL): seq(A246910(n), n=1..5000); # Wesley Ivan Hurt, Sep 07 2014

for(n=1,10^4,if(sigma(n+sigma(n))==3*sigma(n),print1(n,", "))) \\ Derek Orr, Sep 07 2014

A246912 Numbers n such that sigma(n+sigma(n)) = 5*sigma(n).

Original entry on oeis.org

15456, 16920, 48576, 59520, 107160, 153360, 232596, 281916, 306720, 332280, 332640, 358560, 360360, 373104, 383400, 514080, 548772, 556920, 788256, 876960, 884520, 930384, 943344, 950040, 955296, 1234464, 1357020, 1396440, 1421280, 1534080, 1539720, 1582866

Offset: 1

Jaroslav Krizek, Sep 07 2014

nonn

			Number 15456 (with sigma(15456) = 48384) is in sequence because sigma(15456+sigma(15456)) = sigma(63840) = 241920 = 5*48384.

Cf. A000203, A246456, A246909, A246910, A246911, A246913, A246914, A274535 (5*sigma(n)).

[n:n in[1..10^7] | SumOfDivisors(n+SumOfDivisors(n))eq 5*SumOfDivisors(n)]

with(numtheory): A246912:=n->`if`(sigma(n+sigma(n)) = 5*sigma(n),n,NULL): seq(A246912(n), n=1..10^6); # Wesley Ivan Hurt, Sep 07 2014

Select[Range[16*10^5],DivisorSigma[1,#+DivisorSigma[1,#]] == 5*DivisorSigma[ 1,#]&] (* Harvey P. Dale, Mar 13 2016 *)

for(n=1,10^7,if(sigma(n+sigma(n))==5*sigma(n),print1(n,", "))) \\ Derek Orr, Sep 07 2014

A246913 Numbers n such that sigma(n+sigma(n)) = 6*sigma(n).

Original entry on oeis.org

831376, 3944688, 16956576, 17843616, 22591296, 25371360, 27870976, 51878736, 58877280, 64641984, 142990848, 164898720, 172821456, 181821024, 204330672, 276371200, 281613024, 301571424, 319848480, 326207700, 342237456, 346502520, 389165568, 389450880, 392110992

Offset: 1

Jaroslav Krizek, Sep 07 2014

nonn

a(310) > 10^11. - Hiroaki Yamanouchi, Sep 11 2015

			Number 831376 (with sigma(831376) = 1985984) is in sequence because sigma(831376+sigma(831376)) = sigma(2817360) = 11915904 = 6*1985984.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..309

Cf. A000203, A246456, A246909-A246912, A246914.

[n:n in[1..10^7] | SumOfDivisors(n+SumOfDivisors(n))eq 6*SumOfDivisors(n)]

for(n=1,10^7,if(sigma(n+sigma(n))==6*sigma(n),print1(n,", "))) \\ Derek Orr, Sep 07 2014

a(5)-a(25) from Hiroaki Yamanouchi, Sep 11 2015

A246914 Primes p such that sigma(2p+1) = 3*(p+1).

Original entry on oeis.org

7, 103, 1487, 9679, 73727, 603679

Offset: 1

Jaroslav Krizek, Sep 07 2014

Primes p such that sigma(p+sigma(p)) = 3*sigma(p). Subsequence of A246910.
The next term, if it exists, must be greater than 10^9.
Conjecture: Also primes p such that sigma(2p+1) mod p = 3. - Jaroslav Krizek, Sep 28 2014
No other terms up to 10^11. - Michel Marcus, Feb 21 2020

			Prime 7 is in sequence because sigma(2*7 + 1) = sigma(15) = 24 = 3*(7+1).

Cf. A000203, A246456, A246910.

[n:n in[1..10^7] | SumOfDivisors(n+SumOfDivisors(n))eq 3*SumOfDivisors(n) and IsPrime(n)]

with(numtheory): A246914:=n->`if`(isprime(n) and sigma(2*n+1) = 3*(n+1), n, NULL): seq(A246914(n), n=1..10^5); # Wesley Ivan Hurt, Oct 01 2014

Select[Prime[Range[1500]], DivisorSigma[1, 2# + 1] == 3# + 3 &] (* Alonso del Arte, Sep 07 2014 *)

for(n=1,10^6,p=prime(n);if(sigma(p+sigma(p))==3*sigma(p),print1(p,", "))) \\ Derek Orr, Sep 07 2014

forprime(p=2,10^7,if(sigma(2*p+1)==3*(p+1),print1(p,","))) \\ Edward Jiang, Sep 07 2014

A246911 Numbers n such that sigma(n+sigma(n)) = 4*sigma(n).

Original entry on oeis.org

28, 66, 348, 496, 840, 920, 1320, 1416, 1602, 1770, 1896, 1920, 2040, 2280, 2556, 3000, 3360, 3720, 4440, 4920, 5456, 5640, 5826, 7080, 7392, 8010, 8040, 8128, 8298, 10528, 10680, 11424, 12768, 12840, 13080, 15108, 15504, 17880, 18120, 18720, 18840, 20832

Offset: 1

Jaroslav Krizek, Sep 07 2014

nonn

			Number 28 (with sigma(28) = 56) is in sequence because sigma(26+sigma(26)) = sigma(84) = 224 = 4*56.

Cf. A000203, A239050, A246456, A246909, A246910, A246912, A246913, A246914.

[n:n in[1..10000] | SumOfDivisors(n+SumOfDivisors(n)) eq 4*SumOfDivisors(n)]

with(numtheory): A246911:=n->`if`(sigma(n+sigma(n)) = 4*sigma(n),n,NULL): seq(A246911(n), n=1..3*10^4); # Wesley Ivan Hurt, Sep 07 2014

for(n=1,10^4,if(sigma(n+sigma(n))==4*sigma(n),print1(n,", "))) \\ Derek Orr, Sep 07 2014

A246857 Numbers k such that sigma(k + sigma(k)) = 2*sigma(k).

Original entry on oeis.org

2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 329, 359, 413, 419, 431, 443, 491, 509, 593, 623, 641, 653, 659, 683, 719, 743, 761, 809, 869, 911, 953, 979, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451

Offset: 1

Jaroslav Krizek, Sep 05 2014

nonn

Union of A005384 (Sophie Germain primes) and A246858.

First composite number in sequence is 329 (see A246858).

			Composite number 329 (with sigma(329) = 384) is in sequence because sigma(329+sigma(329)) = sigma(713) = 768 = 2*384.
Prime 359 (with sigma(359) = 360) is in sequence because sigma(359+sigma(359)) = sigma(719) = 720 = 2*360.

Cf. A074400, A246456.

Cf. A005384, A246858.

[n:n in[1..10000] | SumOfDivisors(n+SumOfDivisors(n)) eq 2*SumOfDivisors(n)]

Select[Range[1500], DivisorSigma[1, # + DivisorSigma[1, #]] == 2 DivisorSigma[1, #] &] (* Michael De Vlieger, Aug 05 2021 *)

select(n -> sigma(n+sigma(n))==2*sigma(n),[1..1000]) \\ Edward Jiang, Sep 05 2014

A246858 Composite numbers k such that sigma(k + sigma(k)) = 2*sigma(k).

Original entry on oeis.org

329, 413, 623, 869, 979, 1819, 2585, 3107, 3173, 3197, 3887, 4235, 4997, 5771, 6149, 6187, 6443, 7409, 8399, 8759, 14429, 15323, 18515, 19019, 21181, 21413, 23989, 26491, 29749, 30355, 31043, 32623, 34009, 34177, 39737, 47321, 47845, 51389, 53311, 56419

Offset: 1

Jaroslav Krizek, Sep 05 2014

nonn

Complement of A005384 (Sophie Germain primes) with respect to A246857.

			Number 329 (with sigma(329) = 384) is in sequence because sigma(329 + sigma(329)) = sigma(713) = 768 = 2*384.

Cf. A074400, A246456.

Cf. A005384, A246858.

[n:n in[1..1000] | SumOfDivisors(n+SumOfDivisors(n)) eq 2*SumOfDivisors(n) and not IsPrime(n)]

Select[Range[57000], And[CompositeQ[#], DivisorSigma[1, # + DivisorSigma[1, #]] == 2 DivisorSigma[1, #]] &] (* Michael De Vlieger, Aug 05 2021 *)

lista(nn) = {forcomposite(n=2, nn, if (sigma(n+sigma(n)) == 2*sigma(n), print1(n, ", ")););} \\ Michel Marcus, Sep 05 2014

A246908 a(n) = sigma(n + sigma(n)) - sigma(n).

Original entry on oeis.org

2, 3, 4, 5, 6, 27, 16, 9, 23, 38, 12, 62, 26, 36, 32, 17, 30, 41, 36, 54, 22, 54, 24, 164, 89, 84, 28, 168, 30, 144, 72, 57, 73, 126, 36, 37, 86, 111, 64, 162, 42, 192, 76, 171, 90, 108, 72, 184, 105, 75, 96, 274, 54, 240, 56, 252, 58, 176, 84, 392, 106, 144

Offset: 1

Jaroslav Krizek, Sep 07 2014

nonn

			For n = 6; a(n) = sigma(6 + sigma(6)) - sigma(6) = sigma(18) - sigma(6) = 39 - 12 = 27.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000203, A078762, A246456.

[SumOfDivisors(n+SumOfDivisors(n))-SumOfDivisors(n):n in[1..1000]]

sig[n_]:=Module[{d6=DivisorSigma[1,n]},DivisorSigma[1,n+d6]-d6]; Array[ sig,70] (* Harvey P. Dale, Feb 20 2015 *)

vector(100,n,sigma(n+sigma(n))-sigma(n)) \\ Derek Orr, Sep 07 2014

a(n) = n + 1 for number in A078762 (numbers n such that n + sigma(n) is prime).

A246915 Numbers n such that sigma(n + sigma(n)) = sigma((n+1) + sigma(n+1)).

Original entry on oeis.org

4, 7, 16, 50, 494, 4485, 12585, 20606, 45590, 46761, 48614, 64785, 72609, 137853, 169898, 196934, 224186, 321986, 363037, 466545, 474573, 532441, 702374, 811004, 910125, 982310, 1141281, 1282436, 1288557, 1531245, 1602801, 1635854, 1695705, 1842405, 2246781, 2725802, 3018277, 3343515

Offset: 1

Jaroslav Krizek, Sep 07 2014

nonn

Numbers n such that A246456(n) = A246456(n+1).

Conjecture: sequence of numbers A246456(a(n)): 12, 24, 48, 168, 2160, 17280, 54720, 77280, 221184, 202176, 185328, 249984, 312480, 599040, 725760, 967680, 864864, 1327104, 1489488, 2048256, 1958400, 2439360, 3110400, 3902976, 4852224, 4713984, … is sequence of any multiples of 12.

			Number 16 is in sequence because A246456(4) = A246456(5) = 12.

Cf. A000203, A246456.

[n:n in[1..1000000] | SumOfDivisors(n+SumOfDivisors(n)) eq SumOfDivisors(n+1+SumOfDivisors(n+1))]

SequencePosition[Table[DivisorSigma[1,n+DivisorSigma[1,n]],{n,3344000}],{x_,x_}][[All,1]] (* The program takes a long time to run. To generate fewer terms but more quickly, reduce the "n" constant. *) (* Harvey P. Dale, Mar 07 2022 *)

for(n=1,10^7,if(sigma(n+sigma(n))==sigma(n+1+sigma(n+1)),print1(n,", "))) \\ Derek Orr, Sep 07 2014

More terms from Derek Orr, Sep 07 2014

A383392 Numbers k such that (sigma(k) + sigma(k + sigma(k))) / k is an integer where sigma(k) = A000203(k) is the sum of the divisors of k.

Original entry on oeis.org

1, 3, 14, 19, 27, 28, 48, 139, 164, 243, 496, 1428, 1440, 3360, 3480, 5932, 8128, 11004, 19683, 25296, 27144, 31756, 35616, 45436, 47520, 51480, 84000, 115506, 218520, 221088, 288288, 290520, 303309, 414528, 445788, 605880, 1019070, 1122432, 2100000, 2136288

Offset: 1

Ctibor O. Zizka, Apr 25 2025

nonn

			k = 3: (sigma(3) + sigma(3 + sigma(3)))/3 = (4 + 8)/3 = 4.

Cf. A000203, A007691, A246456.

q[k_] := Module[{s = DivisorSigma[1, k]}, Divisible[s + DivisorSigma[1, k + s], k]]; Select[Range[2200000], q] (* Amiram Eldar, Apr 25 2025 *)

isok(k) = my(s=sigma(k)); ((s+sigma(k+s)) % k) == 0; \\ Michel Marcus, Apr 25 2025

cp's OEIS Frontend

A246910 Numbers n such that sigma(n+sigma(n)) = 3*sigma(n).

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A246912 Numbers n such that sigma(n+sigma(n)) = 5*sigma(n).

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A246913 Numbers n such that sigma(n+sigma(n)) = 6*sigma(n).

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A246914 Primes p such that sigma(2p+1) = 3*(p+1).

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A246911 Numbers n such that sigma(n+sigma(n)) = 4*sigma(n).

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A246857 Numbers k such that sigma(k + sigma(k)) = 2*sigma(k).

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A246858 Composite numbers k such that sigma(k + sigma(k)) = 2*sigma(k).

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