A168415 -id:A168415 - cp's OEIS frontend

A125247 Numbers n whose abundance sigma(n) - 2n = -8. Numbers n whose deficiency is 8.

22, 130, 184, 1012, 2272, 18904, 33664, 70564, 85936, 100804, 391612, 527872, 1090912, 17619844, 2147713024, 6800695312, 34360655872, 549759483904, 1661355408388, 28502765343364, 82994670582016, 99249696661504, 120646991405056, 431202442356004, 952413274955776

Offset: 1

Jason G. Wurtzel, Nov 25 2006

a(19) > 10^12. - Donovan Johnson, Dec 08 2011
a(20) > 10^13. - Giovanni Resta, Mar 29 2013
a(30) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018
a(20) <= 36028797958488064 ~ 3.6*10^16. Indeed, if k is in A057195 then 2^(k-1)*A168415(k) is in this sequence, and k=28 yields this upper bound for a(20) which is in any case a term of this sequence. - M. F. Hasler, Apr 27 2015
If n is in this sequence and p a prime not dividing n, then np is abundant if and only if p < sigma(n)/8 = n/4-1. For all n=a(k) except {22, 70564, 100804, 17619844}, there is such a p near this limit, such that n*p is a primitive weird number (A002975; in A258882 for the terms mentioned in the preceding comment). - M. F. Hasler, Jul 20 2016
Any term x of this sequence can be combined with any term y of A088833 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. - Timothy L. Tiffin, Sep 13 2016
Is there any odd number in this sequence? Is it possible to prove the contrary? - M. F. Hasler, Feb 22 2017

			The abundance of 22 = (1+2+11+22)-44 = -8

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..29

Cf. A033880, A088833 (abundance 8).

[n: n in [1..2*10^7] | (DivisorSigma(1,n)-2*n) eq - 8]; // Vincenzo Librandi, Jul 22 2016

Select[Range[10^6], DivisorSigma[1, #] - 2 # == -8 &] (* Michael De Vlieger, Jul 21 2016 *)

for(n=1,1000000,if(((sigma(n)-2*n)==-8),print1(n,",")))

a(13)-a(15) from Klaus Brockhaus, Nov 29 2006
a(16)-a(17) from Donovan Johnson, Dec 23 2008
a(18) from Donovan Johnson, Dec 08 2011
a(19) from Giovanni Resta, Mar 29 2013
a(20)-a(25) from Hiroaki Yamanouchi, Aug 21 2018

A188165 a(n) = 2^n + 9.

Original entry on oeis.org

10, 11, 13, 17, 25, 41, 73, 137, 265, 521, 1033, 2057, 4105, 8201, 16393, 32777, 65545, 131081, 262153, 524297, 1048585, 2097161, 4194313, 8388617, 16777225, 33554441, 67108873, 134217737, 268435465, 536870921, 1073741833, 2147483657, 4294967305

Offset: 0

Brad Clardy, Mar 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000079, A104070 (primes of this form), A168415.

[2^n+9: n in [0..40]]; // Vincenzo Librandi, May 13 2014

Table[2^n + 9, {n, 0, 40}] (* or *) CoefficientList[Series[(10 - 19 x)/((1 - x) (1 - 2 x)), {x, 0, 30}], x] (* Vincenzo Librandi, May 13 2014 *)

PARI
```
a(n) = 1<Michel Marcus, Jul 19 2013
```

From Bruno Berselli, Sep 26 2011:  (Start)
G.f.: (10-19*x)/(1-3*x+2*x^2).
a(n) = 2*a(n-1)-9 = 3*a(n-1)-2*a(n-2).
a(n) = A168415(n)+2.  (End)
E.g.f.: exp(2*x) + 9*exp(x). - Elmo R. Oliveira, Nov 08 2023

A240941 Numbers k that divide 2^k + 7.

Original entry on oeis.org

1, 3, 15, 75, 6308237, 871506915, 2465425275, 2937864075, 2948967789, 83313712623, 195392257275, 11126651718075, 45237726869109, 2920008144904215

Offset: 1

Derek Orr, Aug 04 2014

Some larger terms: 213736983815110866141, 23423890178454972202084722709155. - Max Alekseyev, Sep 23 2016

			2^3 + 7 = 15 is divisible by 3. Thus 3 is a term of this sequence.

OEIS Wiki, 2^n mod n

Cf. A033981, A168415.

k = 1; lst = {1,3}; While[k < 2500000001, If[ PowerMod[2, k, k] + 7 == k, AppendTo[ lst, k]; Print[ k]]; k += 2]; lst (* Robert G. Wilson v, Aug 05 2014 *)

for(n=1,10^9,if(Mod(2,n)^n==Mod(-7,n),print1(n,", ")))

a(7)-a(9) from Robert G. Wilson v, Aug 05 2014

a(10)-a(14) from Max Alekseyev, Sep 23 2016

A287640 Number T(n,k) of set partitions of [n], where k is minimal such that for all j in [n]: j is member of block b implies b = 1 or at least one of j-1, ..., j-k is member of a block >= b-1; triangle T(n,k), n >= 0, 0 <= k <= max(floor(n/2), n-2), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 13, 1, 1, 41, 9, 1, 1, 131, 59, 11, 1, 1, 428, 344, 88, 15, 1, 1, 1429, 1906, 634, 146, 23, 1, 1, 4861, 10345, 4389, 1231, 280, 39, 1, 1, 16795, 55901, 30006, 9835, 2763, 602, 71, 1, 1, 58785, 303661, 205420, 77178, 25014, 6967, 1408, 135, 1

Offset: 0

Alois P. Heinz, May 28 2017

			T(4,0) = 1: 1234.
T(4,1) = 13: 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
T(4,2) = 1: 13|2|4.
T(5,2) = 9: 124|3|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 14|2|35, 14|2|3|5, 1|24|3|5.
T(6,3) = 11: 1245|3|6, 1346|2|5, 134|26|5, 134|2|56, 134|2|5|6, 145|23|6, 145|2|36, 145|2|3|6, 14|25|3|6, 15|24|3|6, 1|245|3|6.
T(6,4) = 1: 1345|2|6.
T(7,4) = 15: 12456|3|7, 13457|2|6, 1345|27|6, 1345|2|67, 1345|2|6|7, 1456|23|7, 1456|2|37, 1456|2|3|7, 145|26|3|7, 146|25|3|7, 14|256|3|7, 156|24|3|7, 15|246|3|7, 16|245|3|7, 1|2456|3|7.
Triangle T(n,k) begins:
  1;
  1;
  1,    1;
  1,    4;
  1,   13,     1;
  1,   41,     9,    1;
  1,  131,    59,   11,    1;
  1,  428,   344,   88,   15,   1;
  1, 1429,  1906,  634,  146,  23,  1;
  1, 4861, 10345, 4389, 1231, 280, 39, 1;
  ...

Alois P. Heinz, Rows n = 0..20, flattened
Wikipedia, Partition of a set

Columns k=0-1 give: A000012, A001453.
Row sums give A000110.
Cf. A168415, A287213, A287215, A287416, A287641.

b:= proc(n, l) option remember; `if`(n=0 or l=[], 1, add(b(n-1,
      [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
    end:
T:= (n, k)-> `if`(k=0, 1, b(n, [0$k])-b(n, [0$k-1])):
seq(seq(T(n, k), k=0..max(n/2, n-2)), n=0..12);

b[n_, l_] := b[n, l] = If[n == 0 || l == {}, 1, Sum[b[n-1, Append[Table[ Max[l[[i]], j], {i, 2, Length[l]}], j]], {j, 1, l[[1]] + 1}]];
T[n_, k_] := If[k == 0, 1, b[n, Table[0, k]] - b[n, Table[0, k - 1]]];
Table[T[n, k], {n, 0, 12}, { k, 0, Max[n/2, n - 2]}] // Flatten (* Jean-François Alcover, May 22 2018, translated from Maple *)

T(n,k) = A287641(n,k) - A287641(n,k-1) for k>0, T(n,0) = 1.

T(n+4,n+1) = A168415(n) for n>0.

A257272 a(n) = 2^(n-1)*(2^n+7).

Original entry on oeis.org

4, 9, 22, 60, 184, 624, 2272, 8640, 33664, 132864, 527872, 2104320, 8402944, 33583104, 134275072, 536985600, 2147713024, 8590393344, 34360655872, 137440788480, 549759483904, 2199030595584, 8796107702272, 35184401448960, 140737547075584, 562950070861824, 2251800048566272, 9007199724503040

Offset: 0

M. F. Hasler, Apr 27 2015

For n in A057195, a(n) is of deficiency 8, i.e., in A125247.

Also, the third column (k=2) of the table given in A181444.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A000079, A007582, A028403, A256873, A257273, A256871.

Cf. A168415, A057195, A104066, A125247.

[2^(n-1)*(2^n+7): n in [0..25]]; // Vincenzo Librandi, Apr 27 2015

Table[2^(n - 1) (2^n + 7), {n, 0, 30}] (* Bruno Berselli, Apr 27 2015 *)
CoefficientList[Series[(4 - 15 x)/((1 - 4 x) (1 - 2 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 27 2015 *)

PARI
```
a(n)=2^(n-1)*(2^n+7)
```

Vec((4-15*x)/((1-4*x)*(1-2*x)) + O(x^100)) \\ Colin Barker, Apr 27 2015

a(n) = 2^(n-1)*A168415(n).
n in A057195 <=> A168415(n) in A104066 <=> a(n) in A125247.
G.f.: (4-15*x)/((1-4*x)*(1-2*x)). - Vincenzo Librandi, Apr 27 2015

A242475 a(n) = 2^n + 8.

Original entry on oeis.org

9, 10, 12, 16, 24, 40, 72, 136, 264, 520, 1032, 2056, 4104, 8200, 16392, 32776, 65544, 131080, 262152, 524296, 1048584, 2097160, 4194312, 8388616, 16777224, 33554440, 67108872, 134217736, 268435464, 536870920, 1073741832

Offset: 0

Vincenzo Librandi, May 20 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000051, A052548, A062709, A140504, A168614, A153972, A168415, A188165.

Magma
```
[2^n+8: n in [0..40]];
```

Table[2^n + 8, {n, 0, 40}] (* or *) CoefficientList[Series[(9 - 17 x)/((1 - x) (1 - 2 x)),{x, 0, 30}], x]
LinearRecurrence[{3,-2},{9,10},40] (* Harvey P. Dale, May 21 2025 *)

G.f.: (9 - 17*x)/((1 - x)*(1 - 2*x)).
a(n) = 2*a(n-1) - 8 = 3*a(n-1) - 2*a(n-2).
a(n) = A052548(n)+6 = A140504(n)+4 = A153972(n)+2.
E.g.f.: exp(2*x) + 8*exp(x). - Elmo R. Oliveira, Nov 11 2023

A246139 a(n) = 2^n + 10.

Original entry on oeis.org

11, 12, 14, 18, 26, 42, 74, 138, 266, 522, 1034, 2058, 4106, 8202, 16394, 32778, 65546, 131082, 262154, 524298, 1048586, 2097162, 4194314, 8388618, 16777226, 33554442, 67108874, 134217738, 268435466, 536870922, 1073741834, 2147483658, 4294967306

Offset: 0

Vincenzo Librandi, Aug 18 2014

First trisection of A085688. [Bruno Berselli, Aug 19 2014]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. Sequences of the form 2^n + k: A000079 (k=0), A000051 (k=1), A052548 (k=2), A062709 (k=3), A140504 (k=4), A168614 (k=5), A153972 (k=6), A168415 (k=7), A242475 (k=8), A188165 (k=9), this sequence (k=10).

Cf. A085688.

Magma
```
[2^n+10: n in [0..40]];
```
Mathematica
```
Table[2^n + 10, {n, 0, 40}]
```

vector(50, n, 2^(n-1)+10) \\ Derek Orr, Aug 18 2014

G.f.: (11 - 21*x)/(1 - 3*x + 2*x^2).
a(n) = A000079(n) + 10.
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1.
E.g.f.: exp(2*x) + 10*exp(x). - Elmo R. Oliveira, Nov 11 2023

A193579 a(n) = 2*4^n + 7.

Original entry on oeis.org

9, 15, 39, 135, 519, 2055, 8199, 32775, 131079, 524295, 2097159, 8388615, 33554439, 134217735, 536870919, 2147483655, 8589934599, 34359738375, 137438953479, 549755813895, 2199023255559, 8796093022215, 35184372088839, 140737488355335, 562949953421319, 2251799813685255

Offset: 0

Brad Clardy, Sep 20 2011

Bisection of A168415 (odd part).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A168415, A020988.

[2*4^n + 7: n in [0..30]]; // Vincenzo Librandi, Sep 30 2011

2*4^Range[0,30]+7 (* or *) LinearRecurrence[{5,-4},{9,15},30] (* Harvey P. Dale, Jun 13 2020 *)

a(n) = 2*4^n+7 \\ Felix Fröhlich, Nov 07 2018

Vec(3*(3 - 10*x)/((1 - x)*(1 - 4*x)) + O(x^20)) \\ Felix Fröhlich, Nov 07 2018

a(n) = 2^(2n + 1) + 7 = 3*(A020988(n) + 3).
From Bruno Berselli, Sep 20 2011: (Start)
G.f.: 3*(3 - 10*x)/((1 - x)*(1 - 4*x)).
a(n) = A085688(A016969(n)). (End)
E.g.f.: 7*exp(x) + 2*exp(4*x). - Franck Maminirina Ramaharo, Nov 07 2018

A195463 a(n) = 4^(n+1) + 7.

Original entry on oeis.org

11, 23, 71, 263, 1031, 4103, 16391, 65543, 262151, 1048583, 4194311, 16777223, 67108871, 268435463, 1073741831, 4294967303, 17179869191, 68719476743, 274877906951, 1099511627783, 4398046511111, 17592186044423, 70368744177671, 281474976710663, 1125899906842631

Offset: 0

Brad Clardy, Sep 19 2011

These are the even terms of A168415. Since the odd terms of A168415 are divisible by three the primes of this sequence are the same as A104066.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A104066, A168415, A188165.

[4^(n+1) + 7: n in [0..30]]; // Vincenzo Librandi, Sep 30 2011

4^Range[25] + 7 (* Paolo Xausa, Feb 21 2025 *)

a(n)=4^(n+1)+7 \\ Charles R Greathouse IV, Sep 19 2011

a(n) = 4^(n+1) + 7.
From Alexander R. Povolotsky, Sep 19 2011: (Start)
G.f.: (11 - 32*x)/(1 - 5*x + 4*x^2).
a(n+1) = 4*a(n) - 21. (End)
a(n) = A188165(2*n+2) - 2. - Bruno Berselli, Sep 26 2011
E.g.f.: exp(x)*(4*exp(3*x) + 7). - Elmo R. Oliveira, Feb 20 2025

A267615 a(n) = 2^n + 11.

Original entry on oeis.org

12, 13, 15, 19, 27, 43, 75, 139, 267, 523, 1035, 2059, 4107, 8203, 16395, 32779, 65547, 131083, 262155, 524299, 1048587, 2097163, 4194315, 8388619, 16777227, 33554443, 67108875, 134217739, 268435467, 536870923, 1073741835, 2147483659, 4294967307, 8589934603, 17179869195, 34359738379

Offset: 0

Ilya Gutkovskiy, Jan 18 2016

Recurrence relation b(n) = 3*b(n - 1) - 2*b(n - 2) for n>1, b(0) = k, b(1) = k + 1, gives the closed form b(n) = 2^n + k - 1.

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. sequences with closed form 2^n + k - 1: A168616 (k=-4), A028399 (k=-3), A036563 (k=-2), A000918 (k=-1), A000225 (k=0), A000079 (k=1), A000051 (k=2), A052548 (k=3), A062709 (k=4), A140504 (k=5), A168614 (k=6), A153972 (k=7), A168415 (k=8), A242475 (k=9), A188165 (k=10), A246139 (k=11), this sequence (k=12).

Cf. A156940.

[2^n+11: n in [0..30]]; // Vincenzo Librandi, Jan 19 2016

Table[2^n + 11, {n, 0, 35}]
LinearRecurrence[{3, -2}, {12, 13}, 40] (* Vincenzo Librandi, Jan 19 2016 *)

a(n) = 2^n + 11; \\ Altug Alkan, Jan 18 2016

G.f.: (12 - 23*x)/(1 - 3*x + 2*x^2).
a(n) = 3*a(n - 1) - 2*a(n - 2) for n>1, a(0)=12, a(1)=13.
a(n) = A000079(n) + A010850(n).
Sum_{n>=0} 1/a(n) = 0.367971714327125...
Lim_{n->oo} a(n + 1)/a(n) = 2.
E.g.f.: exp(2*x) + 11*exp(x). - Elmo R. Oliveira, Nov 08 2023

cp's OEIS Frontend

A125247 Numbers n whose abundance sigma(n) - 2n = -8. Numbers n whose deficiency is 8.

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A188165 a(n) = 2^n + 9.

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A240941 Numbers k that divide 2^k + 7.

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A287640 Number T(n,k) of set partitions of [n], where k is minimal such that for all j in [n]: j is member of block b implies b = 1 or at least one of j-1, ..., j-k is member of a block >= b-1; triangle T(n,k), n >= 0, 0 <= k <= max(floor(n/2), n-2), read by rows.

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A257272 a(n) = 2^(n-1)*(2^n+7).

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A242475 a(n) = 2^n + 8.

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A246139 a(n) = 2^n + 10.

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