A351305
a(n) = n^10 * Product_{p|n, p prime} (1 + 1/p^10).
Original entry on oeis.org
1, 1025, 59050, 1049600, 9765626, 60526250, 282475250, 1074790400, 3486843450, 10009766650, 25937424602, 61978880000, 137858491850, 289537131250, 576660215300, 1100585369600, 2015993900450, 3574014536250, 6131066257802, 10250001049600, 16680163512500, 26585860217050
Offset: 1
Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10:
A034444 (k=0),
A001615 (k=1),
A065958 (k=2),
A065959 (k=3),
A065960 (k=4),
A351300 (k=5),
A351301 (k=6),
A351302 (k=7),
A351303 (k=8),
A351304 (k=9), this sequence (k=10).
-
f[p_, e_] := p^(10*e) + p^(10*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Feb 08 2022 *)
-
a(n)=sumdiv(n, d, moebius(n/d)^2*d^10);
-
for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^10*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022
A320974
a(n) = n^n * Product_{p|n, p prime} (1 + 1/p^n).
Original entry on oeis.org
1, 5, 28, 272, 3126, 47450, 823544, 16842752, 387440172, 10009766650, 285311670612, 8918294011904, 302875106592254, 11112685048647250, 437893920912786408, 18447025548686262272, 827240261886336764178, 39346558271492178663450, 1978419655660313589123980
Offset: 1
-
Table[n^n Product[1 + Boole[PrimeQ[d]]/d^n, {d, Divisors[n]}], {n, 19}]
Table[SeriesCoefficient[Sum[MoebiusMu[k]^2 PolyLog[-n, x^k], {k, 1, n}], {x, 0, n}], {n, 19}]
Table[Sum[MoebiusMu[n/d]^2 d^n, {d, Divisors[n]}], {n, 19}]
A320973
Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = n^k * Product_{p|n, p prime} (1 + 1/p^k).
Original entry on oeis.org
1, 1, 2, 1, 3, 2, 1, 5, 4, 2, 1, 9, 10, 6, 2, 1, 17, 28, 20, 6, 4, 1, 33, 82, 72, 26, 12, 2, 1, 65, 244, 272, 126, 50, 8, 2, 1, 129, 730, 1056, 626, 252, 50, 12, 2, 1, 257, 2188, 4160, 3126, 1394, 344, 80, 12, 4, 1, 513, 6562, 16512, 15626, 8052, 2402, 576, 90, 18, 2
Offset: 1
Square array begins:
1, 1, 1, 1, 1, 1, ...
2, 3, 5, 9, 17, 33, ...
2, 4, 10, 28, 82, 244, ...
2, 6, 20, 72, 272, 1056, ...
2, 6, 26, 126, 626, 3126, ...
4, 12, 50, 252, 1394, 8052, ...
-
Table[Function[k, n^k Product[1 + Boole[PrimeQ[d]]/d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[MoebiusMu[j]^2 PolyLog[-k, x^j], {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, Sum[MoebiusMu[n/d]^2 d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
A194533
Jordan function ratio J_8(n)/J_2(n).
Original entry on oeis.org
1, 85, 820, 5440, 16276, 69700, 120100, 348160, 597780, 1383460, 1786324, 4460800, 4855540, 10208500, 13346320, 22282240, 24221380, 50811300, 47176564, 88541440, 98482000, 151837540, 148316260, 285491200, 254312500, 412720900, 435781620, 653344000, 595531444
Offset: 1
-
f[p_, e_] := p^(6*(e - 1))*(p^2 + 1)*(p^4 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 01 2022 *)
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