A356010
a(n) = n! * Sum_{k=1..n} floor(n/k)/k.
Original entry on oeis.org
1, 5, 23, 134, 814, 6324, 50028, 475824, 4806576, 54597600, 644119200, 8847100800, 121718332800, 1853505158400, 29894856364800, 518855607244800, 9197155541145600, 179420609436364800, 3537039053405491200, 75849875285280768000, 1670700245252548608000
Offset: 1
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S:= ListTools:-PartialSums([seq(numtheory:-sigma(k)/k, k=1..30)]):
seq(n! * S[n], n=1..30); # Robert Israel, Aug 10 2025
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a(n) = n!*sum(k=1, n, n\k/k);
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k*(1-x^k)))/(1-x)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-x^k))/(1-x)))
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a(n) = n!*sum(k=1, n, sigma(k)/k); \\ Seiichi Manyama, Aug 03 2022
A355991
a(n) = n! * Sum_{k=1..n} 1/(k! * floor(n/k)!).
Original entry on oeis.org
1, 2, 5, 12, 57, 158, 1101, 5442, 28811, 212502, 2337513, 9422306, 122489967, 1654319046, 13917499277, 111631450818, 1897734663891, 23705612782022, 450406642858401, 3091477152208002, 51404897928720023, 1130752882197523686, 26007316290543044757
Offset: 1
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a[n_] := n! * Sum[1/(k! * Floor[n/k]!), {k, 1, n}]; Array[a, 23] (* Amiram Eldar, Jul 22 2022 *)
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a(n) = n!*sum(k=1, n, 1/(k!*(n\k)!));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (1-x^k)*(exp(x^k)-1)/k!)/(1-x)))
A355987
a(n) = n! * Sum_{k=1..n} 1/floor(n/k)!.
Original entry on oeis.org
1, 3, 13, 61, 421, 2641, 23521, 203281, 2071441, 22407841, 286403041, 3453468481, 51122111041, 759194916481, 12216117513601, 203300293996801, 3811792426041601, 69634723878720001, 1444704854104512001, 29725332567567436801, 658231789483184716801
Offset: 1
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a[n_] := n! * Sum[1/Floor[n/k]!, {k, 1, n}]; Array[a, 21] (* Amiram Eldar, Jul 22 2022 *)
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a(n) = n!*sum(k=1, n, 1/(n\k)!);
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1,N, (1-x^k)*(exp(x^k)-1))/(1-x)))
A356458
Expansion of e.g.f. ( Product_{k>0} B(x^k) )^(1/(1-x)) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.
Original entry on oeis.org
1, 1, 6, 38, 319, 3117, 36359, 476121, 7025708, 114118746, 2029450055, 39078892305, 810834093733, 17998186069489, 425672049713174, 10676653292086790, 283014906314277059, 7901659174554937925, 231719030698518379003, 7118469816302381503209
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k)-1))^(1/(1-x))))
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a355886(n) = n!*sum(k=1, n, n\k/k!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a355886(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356459
a(n) = n! * Sum_{k=1..n} Sum_{d|k} d/(k/d)!.
Original entry on oeis.org
1, 7, 40, 281, 2006, 17677, 159020, 1678721, 18555850, 230978981, 2979853592, 43323807265, 644160764846, 10543905398405, 178896116995276, 3284281839169217, 61879477543508690, 1264313089711322821, 26333205612282941600, 588074615109602665601
Offset: 1
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Table[n! * Sum[Sum[d/(k/d)!, {d,Divisors[k]}], {k,1,n}], {n,1,20}] (* Vaclav Kotesovec, Aug 11 2025 *)
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a(n) = n!*sum(k=1, n, sumdiv(k, d, d/(k/d)!));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k*(exp(x^k)-1))/(1-x)))
A356004
a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/(d! * (k/d)!).
Original entry on oeis.org
1, 4, 14, 64, 322, 2054, 14380, 116722, 1060580, 10636042, 116996464, 1411275650, 18346583452, 256869465610, 3856674412952, 61743633813634, 1049641774831780, 18896533652098442, 359034139389870400, 7182372973523436802, 150833211474559084844
Offset: 1
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a[n_] := n! * Sum[DivisorSum[k, 1/(#!*(k/#)!) &], {k, 1, n}]; Array[a, 21] (* Amiram Eldar, Jul 22 2022 *)
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a(n) = n!*sum(k=1, n, sumdiv(k,d,1/(d!*(k/d)!)));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (exp(x^k)-1)/k!)/(1-x)))
Showing 1-6 of 6 results.