A284105 -id:A284105 - cp's OEIS frontend

A284099 a(n) = Sum_{d|n, d == 1 (mod 7)} d.

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 16, 9, 1, 1, 1, 1, 1, 23, 1, 9, 1, 1, 1, 1, 30, 16, 1, 9, 1, 1, 1, 37, 1, 1, 1, 9, 1, 1, 44, 23, 16, 1, 1, 9, 1, 51, 1, 1, 1, 1, 1, 9, 58, 30, 1, 16, 1, 1, 1, 73, 1, 23, 1, 1, 1, 1, 72, 45, 1, 1, 16, 1, 1, 79, 1, 9, 1, 1

Offset: 1

Seiichi Manyama, Mar 20 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A109703.
Cf. Sum_{d|n, d == 1 (mod k)} d: A000593 (k=2), A078181 (k=3), A050449 (k=4), A284097 (k=5), A284098 (k=6), this sequence (k=7), A284100 (k=8).
Cf. Sum_{d|n, d == k (mod 7)} d: this sequence (k=1), A284443 (k=2), A284444 (k=3), A284445 (k=4), A284446 (k=5), A284105 (k=6).

Table[Sum[If[Mod[d, 7] == 1, d, 0], {d, Divisors[n]}], {n, 82}] (* Indranil Ghosh, Mar 21 2017 *)
Table[DivisorSum[n,#&,Mod[#,7]==1&],{n,90}] (* Harvey P. Dale, Aug 08 2021 *)

for(n=1, 82, print1(sumdiv(n, d, if(Mod(d, 7)==1, d, 0)), ", ")) \\ Indranil Ghosh, Mar 21 2017

from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%7==1]) # Indranil Ghosh, Mar 21 2017

G.f.: Sum_{k>=0} (7*k + 1)*x^(7*k+1)/(1 - x^(7*k+1)). - Ilya Gutkovskiy, Mar 21 2017
G.f.: Sum_{n >= 1} x^n*(1 + 6*x^(7*n))/(1 - x^(7*n))^2. - Peter Bala, Dec 19 2021
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/84 = 0.117495... . - Amiram Eldar, Nov 26 2023

A284103 a(n) = Sum_{d|n, d == 4 (mod 5)} d.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 4, 9, 0, 0, 4, 0, 14, 0, 4, 0, 9, 19, 4, 0, 0, 0, 28, 0, 0, 9, 18, 29, 0, 0, 4, 0, 34, 0, 13, 0, 19, 39, 4, 0, 14, 0, 48, 9, 0, 0, 28, 49, 0, 0, 4, 0, 63, 0, 18, 19, 29, 59, 4, 0, 0, 9, 68, 0, 0, 0, 38, 69, 14, 0, 37, 0, 74, 0, 23, 0, 39, 79

Offset: 1

Seiichi Manyama, Mar 20 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A013661, A109700.

Cf. Sum_{d|n, d=k-1 mod k} d: A000593 (k=2), A078182 (k=3), A050452 (k=4), this sequence (k=5), A284104 (k=6), A284105 (k=7).

Table[Sum[If[Mod[d, 5] == 4, d, 0], {d, Divisors[n]}], {n, 79}] (* Indranil Ghosh, Mar 21 2017 *)
Table[Total[Select[Divisors[n],Mod[#,5]==4&]],{n,80}] (* Harvey P. Dale, Sep 24 2024 *)

for(n=1, 79, print1(sumdiv(n, d, if(Mod(d, 5)==4, d, 0)), ", ")) \\ Indranil Ghosh, Mar 21 2017

from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%5==4]) # Indranil Ghosh, Mar 21 2017

G.f.: Sum_{k>=1} (5*k - 1)*x^(5*k-1)/(1 - x^(5*k-1)). - Ilya Gutkovskiy, Mar 21 2017

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/60 = 0.164493... (A013661 / 10). - Amiram Eldar, Nov 26 2023

A363808 Number of divisors of n of the form 7*k + 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0

Offset: 1

Seiichi Manyama, Jun 23 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A279061, A363795, A363805, A363806, A363807.
Cf. A284105.
Cf. A001620, A016630, A354632 (psi(6/7)).

a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 6 &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)

PARI
```
a(n) = sumdiv(n, d, d%7==6);
```

G.f.: Sum_{k>0} x^(6*k)/(1 - x^(7*k)).
G.f.: Sum_{k>0} x^(7*k-1)/(1 - x^(7*k-1)).
Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(6,7) - (1 - gamma)/7 = -0.218328..., gamma(6,7) = -(psi(6/7) + log(7))/7 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A284443 a(n) = Sum_{d|n, d == 2 (mod 7)} d.

Original entry on oeis.org

0, 2, 0, 2, 0, 2, 0, 2, 9, 2, 0, 2, 0, 2, 0, 18, 0, 11, 0, 2, 0, 2, 23, 2, 0, 2, 9, 2, 0, 32, 0, 18, 0, 2, 0, 11, 37, 2, 0, 2, 0, 2, 0, 46, 9, 25, 0, 18, 0, 2, 51, 2, 0, 11, 0, 2, 0, 60, 0, 32, 0, 2, 9, 18, 65, 2, 0, 2, 23, 2, 0, 83, 0, 39, 0, 2, 0, 2, 79, 18, 9

Offset: 1

Seiichi Manyama, Mar 27 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A109704.

Cf. Sum_{d|n, d == k (mod 7)} d: A284099 (k=1), this sequence (k=2), A284444 (k=3), A284445 (k=4), A284446 (k=5), A284105 (k=6).

Table[Sum[If[Mod[d, 7] == 2, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 27 2017 *)

a(n) = sumdiv(n, d, d*((d % 7) == 2)); \\ Amiram Eldar, Nov 26 2023

G.f.: Sum_{k>=0} (7*k + 2)*x^(7*k+2)/(1 - x^(7*k+2)). - Ilya Gutkovskiy, Mar 28 2017

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/84 = 0.117495... . - Amiram Eldar, Nov 26 2023

A284444 a(n) = Sum_{d|n, d == 3 (mod 7)} d.

Original entry on oeis.org

0, 0, 3, 0, 0, 3, 0, 0, 3, 10, 0, 3, 0, 0, 3, 0, 17, 3, 0, 10, 3, 0, 0, 27, 0, 0, 3, 0, 0, 13, 31, 0, 3, 17, 0, 3, 0, 38, 3, 10, 0, 3, 0, 0, 48, 0, 0, 27, 0, 10, 20, 52, 0, 3, 0, 0, 3, 0, 59, 13, 0, 31, 3, 0, 0, 69, 0, 17, 3, 10, 0, 27, 73, 0, 3, 38, 0, 3, 0, 90, 3

Offset: 1

Seiichi Manyama, Mar 27 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A109705.

Cf. Sum_{d|n, d == k (mod 7)} d: A284099 (k=1), A284443 (k=2), this sequence (k=3), A284445 (k=4), A284446 (k=5), A284105 (k=6).

Table[Sum[If[Mod[d, 7] == 3, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 27 2017 *)

a(n) = sumdiv(n, d, d*((d % 7) == 3)); \\ Amiram Eldar, Nov 26 2023

G.f.: Sum_{k>=0} (7*k + 3)*x^(7*k+3)/(1 - x^(7*k+3)). - Ilya Gutkovskiy, Mar 28 2017

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/84 = 0.117495... . - Amiram Eldar, Nov 26 2023

A284445 a(n) = Sum_{d|n, d == 4 (mod 7)} d.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 11, 4, 0, 0, 0, 4, 0, 18, 0, 4, 0, 11, 0, 4, 25, 0, 0, 4, 0, 0, 0, 36, 11, 0, 0, 22, 0, 0, 39, 4, 0, 0, 0, 15, 0, 46, 0, 4, 0, 25, 0, 4, 53, 18, 11, 4, 0, 0, 0, 64, 0, 0, 0, 36, 0, 11, 67, 4, 0, 0, 0, 22, 0, 74, 25, 4, 11, 39, 0, 4, 81

Offset: 1

Seiichi Manyama, Mar 27 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A109706.

Cf. Sum_{d|n, d == k (mod 7)} d: A284099 (k=1), A284443 (k=2), A284444 (k=3), this sequence (k=4), A284446 (k=5), A284105 (k=6).

Table[Sum[If[Mod[d, 7] == 4, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 27 2017 *)

a(n) = sumdiv(n, d, d*((d % 7) == 4)); \\ Amiram Eldar, Nov 26 2023

G.f.: Sum_{k>=0} (7*k + 4)*x^(7*k+4)/(1 - x^(7*k+4)). - Ilya Gutkovskiy, Mar 28 2017

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/84 = 0.117495... . - Amiram Eldar, Nov 26 2023

A284446 a(n) = Sum_{d|n, d == 5 (mod 7)} d.

Original entry on oeis.org

0, 0, 0, 0, 5, 0, 0, 0, 0, 5, 0, 12, 0, 0, 5, 0, 0, 0, 19, 5, 0, 0, 0, 12, 5, 26, 0, 0, 0, 5, 0, 0, 33, 0, 5, 12, 0, 19, 0, 45, 0, 0, 0, 0, 5, 0, 47, 12, 0, 5, 0, 26, 0, 54, 5, 0, 19, 0, 0, 17, 61, 0, 0, 0, 5, 33, 0, 68, 0, 5, 0, 12, 0, 0, 80, 19, 0, 26, 0, 45, 0, 82

Offset: 1

Seiichi Manyama, Mar 27 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A109707.

Cf. Sum_{d|n, d == k (mod 7)} d: A284099 (k=1), A284443 (k=2), A284444 (k=3), A284445 (k=4), this sequence (k=5), A284105 (k=6).

f:= n -> convert(select(t -> t mod 7 = 5, numtheory:-divisors(n)),`+`):
map(f, [$1..1000]); # Robert Israel, Mar 27 2017

Table[DivisorSum[n, # &, Mod[#, 7] == 5 &], {n, 82}] (* Giovanni Resta, Mar 27 2017 *)

for(n=1, 82, print1(sumdiv(n, d, if(Mod(d, 7)==5, d, 0)), ", ")) \\ Indranil Ghosh, Mar 27 2017

G.f.: Sum_{k>=0} (5+7k) x^(5+7k)/(1-x^(5+7k)). - Robert Israel, Mar 27 2017

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/84 = 0.117495... . - Amiram Eldar, Nov 26 2023

A109708 Number of partitions of n into parts each equal to 6 mod 7.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 2, 1, 0, 0, 1, 1, 2, 2, 1, 0, 1, 1, 2, 3, 3, 1, 1, 1, 2, 3, 4, 3, 2, 1, 2, 3, 5, 5, 5, 2, 2, 3, 5, 6, 8, 5, 3, 3, 5, 7, 10, 9, 7, 4, 5, 7, 11, 12, 12, 8, 6, 7, 12, 14, 17, 15, 11, 8, 12, 15, 20, 21, 19, 13, 13, 16, 22, 26, 28, 23

Offset: 0

Erich Friedman, Aug 07 2005

nonn

			a(45)=3 because we have 45=27+6+6+6=20+13+6+6=13+13+13+6.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A284105.

Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), A035462 (m=4), A109700 (m=5), A109702 (m=6), this sequence (m=7).

g:=1/product(1-x^(6+7*j),j=0..20): gser:=series(g,x=0,98): seq(coeff(gser,x,n),n=0..95); # Emeric Deutsch, Apr 14 2006

nmax=100; CoefficientList[Series[Product[1/(1-x^(7*k+6)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)

G.f.: 1/product(1-x^(6+7j), j=0..infinity). - Emeric Deutsch, Apr 14 2006
a(n) ~ Gamma(6/7) * exp(Pi*sqrt(2*n/21)) / (2^(27/14) * 3^(3/7) * 7^(1/14) * Pi^(1/7) * n^(13/14)) * (1 - (39*sqrt(3/14)/(7*Pi) + 13*Pi/(168*sqrt(42))) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284105(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017

Changed offset to 0 and added a(0)=1 by Vaclav Kotesovec, Feb 27 2015

A284104 a(n) = Sum_{d|n, d == 5 (mod 6)} d.

Original entry on oeis.org

0, 0, 0, 0, 5, 0, 0, 0, 0, 5, 11, 0, 0, 0, 5, 0, 17, 0, 0, 5, 0, 11, 23, 0, 5, 0, 0, 0, 29, 5, 0, 0, 11, 17, 40, 0, 0, 0, 0, 5, 41, 0, 0, 11, 5, 23, 47, 0, 0, 5, 17, 0, 53, 0, 16, 0, 0, 29, 59, 5, 0, 0, 0, 0, 70, 11, 0, 17, 23, 40, 71, 0, 0, 0, 5, 0, 88, 0, 0, 5

Offset: 1

Seiichi Manyama, Mar 20 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A086729, A109702.

Cf. Sum_{d|n, d=k-1 mod k} d: A000593 (k=2), A078182 (k=3), A050452 (k=4), A284103 (k=5), this sequence (k=6), A284105 (k=7).

Table[Sum[If[Mod[d, 6] == 5, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 21 2017 *)
Table[Total[Select[Divisors[n],Mod[#,6]==5&]],{n,80}] (* Harvey P. Dale, Dec 30 2017 *)

for(n=1, 80, print1(sumdiv(n, d, if(Mod(d,6)==5, d, 0)),", ")) \\ Indranil Ghosh, Mar 21 2017

from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%6==5]) # Indranil Ghosh, Mar 21 2017

G.f.: Sum_{k>=1} (6*k - 1)*x^(6*k-1)/(1 - x^(6*k-1)). - Ilya Gutkovskiy, Mar 21 2017

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/72 = 0.137077... (A086729). - Amiram Eldar, Nov 26 2023

A284504 Expansion of Product_{k>=0} (1 - x^(7*k+6)) in powers of x.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 2, -1, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, -1, 3, -1, 0, 0, 0, 0, -2, 3, -1, 0, 0, 0, 0, -3, 4, -1, 0, 0, 0, 1, -4, 4, -1, 0, 0, 0, 1, -5, 5, -1, 0, 0, 0, 2, -7, 5

Offset: 0

Seiichi Manyama, Mar 28 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. Product_{k>=0} (1 - x^(7*k+m)): A284499 (m=1), A284500 (m=2), A284501 (m=3), A284502 (m=4), A284503 (m=5), this sequence (m=6).

Cf. A281245.

CoefficientList[Series[Product[1 - x^(7k + 6), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 28 2017 *)

Vec(prod(k=0, 100, 1 - x^(7*k + 6)) + O(x^101)) \\ Indranil Ghosh, Mar 28 2017

a(n) = -(1/n) * Sum_{k=1..n} A284105(k) * a(n-k), a(0) = 1.

cp's OEIS Frontend

A284099 a(n) = Sum_{d|n, d == 1 (mod 7)} d.

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A284103 a(n) = Sum_{d|n, d == 4 (mod 5)} d.

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A363808 Number of divisors of n of the form 7*k + 6.

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A284443 a(n) = Sum_{d|n, d == 2 (mod 7)} d.

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A284444 a(n) = Sum_{d|n, d == 3 (mod 7)} d.

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A284445 a(n) = Sum_{d|n, d == 4 (mod 7)} d.

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A284446 a(n) = Sum_{d|n, d == 5 (mod 7)} d.

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A109708 Number of partitions of n into parts each equal to 6 mod 7.

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