A339685
a(n) = Sum_{d|n} 5^(d-1).
Original entry on oeis.org
1, 6, 26, 131, 626, 3156, 15626, 78256, 390651, 1953756, 9765626, 48831406, 244140626, 1220718756, 6103516276, 30517656381, 152587890626, 762939846906, 3814697265626, 19073488282006, 95367431656276, 476837167968756, 2384185791015626, 11920929003987656
Offset: 1
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A339685:= func< n | (&+[5^(d-1): d in Divisors(n)]) >;
[A339685(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024
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Table[Sum[5^(d - 1), {d, Divisors[n]}], {n, 1, 24}]
nmax = 24; CoefficientList[Series[Sum[x^k/(1 - 5 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
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a(n) = sumdiv(n, d, 5^(d-1)); \\ Michel Marcus, Dec 13 2020
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def A339685(n): return sum(5^(k-1) for k in (1..n) if (k).divides(n))
[A339685(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024
A339686
a(n) = Sum_{d|n} 6^(d-1).
Original entry on oeis.org
1, 7, 37, 223, 1297, 7819, 46657, 280159, 1679653, 10078999, 60466177, 362805091, 2176782337, 13060740679, 78364165429, 470185264735, 2821109907457, 16926661132171, 101559956668417, 609359750089711, 3656158440109669, 21936950700844039, 131621703842267137
Offset: 1
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A339686:= func< n | (&+[6^(d-1): d in Divisors(n)]) >;
[A339686(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024
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Table[Sum[6^(d - 1), {d, Divisors[n]}], {n, 1, 23}]
nmax = 23; CoefficientList[Series[Sum[x^k/(1 - 6 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
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a(n) = sumdiv(n, d, 6^(d-1)); \\ Michel Marcus, Dec 13 2020
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def A339686(n): return sum(6^(k-1) for k in (1..n) if (k).divides(n))
[A339686(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024
A339687
a(n) = Sum_{d|n} 7^(d-1).
Original entry on oeis.org
1, 8, 50, 351, 2402, 16864, 117650, 823894, 5764851, 40356016, 282475250, 1977343950, 13841287202, 96889128064, 678223075300, 4747562333837, 33232930569602, 232630519768872, 1628413597910450, 11398895225729502, 79792266297729700, 558545864365759264
Offset: 1
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A339687:= func< n | (&+[7^(d-1): d in Divisors(n)]) >;
[A339687(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024
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Table[Sum[7^(d - 1), {d, Divisors[n]}], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[x^k/(1 - 7 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
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a(n) = sumdiv(n, d, 7^(d-1)); \\ Michel Marcus, Dec 13 2020
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def A339687(n): return sum(7^(k-1) for k in (1..n) if (k).divides(n))
[A339687(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024
A339688
a(n) = Sum_{d|n} 8^(d-1).
Original entry on oeis.org
1, 9, 65, 521, 4097, 32841, 262145, 2097673, 16777281, 134221833, 1073741825, 8589967945, 68719476737, 549756076041, 4398046515265, 35184374186505, 281474976710657, 2251799830495305, 18014398509481985, 144115188210078217, 1152921504607109185
Offset: 1
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A339688:= func< n | (&+[8^(d-1): d in Divisors(n)]) >;
[A339688(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024
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Table[Sum[8^(d - 1), {d, Divisors[n]}], {n, 1, 21}]
nmax = 21; CoefficientList[Series[Sum[x^k/(1 - 8 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
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a(n) = sumdiv(n, d, 8^(d-1)); \\ Michel Marcus, Dec 13 2020
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def A339688(n): return sum(8^(k-1) for k in (1..n) if (k).divides(n))
[A339688(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024
A339689
a(n) = Sum_{d|n} 9^(d-1).
Original entry on oeis.org
1, 10, 82, 739, 6562, 59140, 531442, 4783708, 43046803, 387427060, 3486784402, 31381119478, 282429536482, 2541866359780, 22876792461604, 205891136878357, 1853020188851842, 16677181742772430, 150094635296999122, 1350851718060419878, 12157665459057460324
Offset: 1
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A339689:= func< n | (&+[9^(d-1): d in Divisors(n)]) >;
[A339689(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024
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Table[Sum[9^(d - 1), {d, Divisors[n]}], {n, 1, 21}]
nmax = 21; CoefficientList[Series[Sum[x^k/(1 - 9 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
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a(n) = sumdiv(n, d, 9^(d-1)); \\ Michel Marcus, Dec 13 2020
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def A339689(n): return sum(9^(k-1) for k in (1..n) if (k).divides(n))
[A339689(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024
A308814
a(n) = Sum_{d|n} n^(d-1).
Original entry on oeis.org
1, 3, 10, 69, 626, 7819, 117650, 2097673, 43046803, 1000010011, 25937424602, 743008621405, 23298085122482, 793714780783695, 29192926025441476, 1152921504875286545, 48661191875666868482, 2185911559749718382455, 104127350297911241532842, 5242880000000512000168021
Offset: 1
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Table[Total[n^(Divisors[n]-1)],{n,20}] (* Harvey P. Dale, Aug 08 2019 *)
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{a(n) = sumdiv(n, d, n^(d-1))}
A344821
Square array A(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{j=1..n} floor(n/j) * k^(j-1).
Original entry on oeis.org
1, 1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 9, 8, 5, 1, 6, 15, 20, 10, 6, 1, 7, 23, 46, 37, 14, 7, 1, 8, 33, 92, 128, 76, 16, 8, 1, 9, 45, 164, 349, 384, 141, 20, 9, 1, 10, 59, 268, 790, 1394, 1114, 280, 23, 10, 1, 11, 75, 410, 1565, 3946, 5491, 3332, 541, 27, 11
Offset: 1
Square array, A(n, k), begins:
1, 1, 1, 1, 1, 1, 1, ...
2, 3, 4, 5, 6, 7, 8, ...
3, 5, 9, 15, 23, 33, 45, ...
4, 8, 20, 46, 92, 164, 268, ...
5, 10, 37, 128, 349, 790, 1565, ...
6, 14, 76, 384, 1394, 3946, 9384, ...
Antidiagonal triangle, T(n, k), begins:
1;
1, 2;
1, 3, 3;
1, 4, 5, 4;
1, 5, 9, 8, 5;
1, 6, 15, 20, 10, 6;
1, 7, 23, 46, 37, 14, 7;
1, 8, 33, 92, 128, 76, 16, 8;
1, 9, 45, 164, 349, 384, 141, 20, 9;
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A:= func< n,k | k eq n select n else (&+[Floor(n/j)*k^(j-1): j in [1..n]]) >;
A344821:= func< n,k | A(k+1, n-k-1) >;
[A344821(n,k): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Jun 27 2024
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A[n_, k_] := Sum[If[k == 0 && j == 1, 1, k^(j - 1)] * Quotient[n, j], {j, 1, n}]; Table[A[k, n - k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, May 29 2021 *)
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A(n, k) = sum(j=1, n, n\j*k^(j-1));
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A(n, k) = sum(j=1, n, sumdiv(j, d, k^(d-1)));
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def A(n,k): return n if k==n else sum((n//j)*k^(j-1) for j in range(1,n+1))
def A344821(n,k): return A(k+1, n-k-1)
flatten([[A344821(n,k) for k in range(n)] for n in range(1,13)]) # G. C. Greubel, Jun 27 2024
Showing 1-7 of 7 results.