A246456 -id:A246456

A246910

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

A246914 gives the primes in this sequence.

LINKS

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

EXAMPLE

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

MAPLE

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

PROG

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

(PARI)

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, Sep 07 2014

STATUS

approved

A246912

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..32.

EXAMPLE

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

MAPLE

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

MATHEMATICA

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

PROG

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

(PARI)

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, Sep 07 2014

STATUS

approved

A246913

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

EXAMPLE

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

PROG

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

(PARI)

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, Sep 07 2014

EXTENSIONS

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

STATUS

approved

A246914

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

+10
6

7, 103, 1487, 9679, 73727, 603679

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..6.

EXAMPLE

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

MAPLE

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

MATHEMATICA

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

PROG

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

(PARI)

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

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

CROSSREFS

Cf. A000203, A246456, A246910.

KEYWORD

nonn,more

AUTHOR

Jaroslav Krizek, Sep 07 2014

STATUS

approved

A246911

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..42.

EXAMPLE

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

MAPLE

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

PROG

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

(PARI)

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, Sep 07 2014

STATUS

approved

A246857

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Union of A005384 (Sophie Germain primes) and A246858.

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

LINKS

Table of n, a(n) for n=1..53.

EXAMPLE

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.

MATHEMATICA

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

PROG

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

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

CROSSREFS

Cf. A074400, A246456.

Cf. A005384, A246858.

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, Sep 05 2014

STATUS

approved

A246858

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..40.

EXAMPLE

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

Cf. A074400, A246456.

Cf. A005384, A246858.

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, Sep 05 2014

STATUS

approved

A246908

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

FORMULA

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

Cf. A000203, A078762, A246456.

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, Sep 07 2014

STATUS

approved

A246915

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

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

EXAMPLE

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

MATHEMATICA

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 *)

PROG

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

(PARI)

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

CROSSREFS

Cf. A000203, A246456.

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, Sep 07 2014

EXTENSIONS

More terms from Derek Orr, Sep 07 2014

STATUS

approved