A307772
Expansion of e.g.f. Sum_{k>=1} prime(k)*log(1 + x)^k/k!.
Original entry on oeis.org
2, 1, 0, -2, 14, -100, 792, -6984, 68112, -728924, 8498662, -107269546, 1457660932, -21221947564, 329615120330, -5440973779098, 95131744001392, -1756450890029772, 34152285999547328, -697588907138104978, 14934641645024407092, -334433142861340604942, 7818455679081107296154
Offset: 1
-
nmax = 23; Rest[CoefficientList[Series[Sum[Prime[k] Log[1 + x]^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
Table[Sum[StirlingS1[n, k] Prime[k], {k, 1, n}], {n, 1, 23}]
A307773
Expansion of e.g.f. Sum_{k>=1} prime(k)*log(1/(1 - x))^k/k!.
Original entry on oeis.org
2, 5, 18, 82, 454, 2960, 22212, 188568, 1786896, 18698116, 214149822, 2664667358, 35796803068, 516387315172, 7961576156330, 130655542040262, 2273957872532176, 41836619073742452, 811316702293124504, 16540499893106494782, 353673865103189710572, 7914396129656344633522
Offset: 1
-
a:= n-> add(ithprime(k)*abs(Stirling1(n, k)), k=1..n):
seq(a(n), n=1..30); # Alois P. Heinz, Apr 27 2019
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nmax = 22; Rest[CoefficientList[Series[Sum[Prime[k] Log[1/(1 - x)]^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
Table[Sum[Abs[StirlingS1[n, k]] Prime[k], {k, 1, n}], {n, 1, 22}]
A351681
Stirling transform of {1, primes}.
Original entry on oeis.org
1, 3, 10, 38, 163, 774, 4004, 22315, 132836, 838378, 5574797, 38861142, 282951538, 2146361911, 16931303262, 138694760316, 1178400013929, 10373294706788, 94511288422822, 890334527133081, 8663213736312460, 86975649078035438, 899960154388259079, 9586293761594853220
Offset: 1
-
nmax = 24; CoefficientList[Series[Exp[x] - 1 + Sum[Prime[k - 1] (Exp[x] - 1)^k/k!, {k, 2, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[n, k] If[k == 1, 1, Prime[k - 1]], {k, 1, n}], {n, 1, 24}]
A353406
Stirling transform of odd primes.
Original entry on oeis.org
3, 8, 25, 91, 376, 1715, 8471, 44838, 252903, 1514213, 9590874, 64056173, 449804453, 3312346950, 25521479277, 205300781275, 1720450321356, 14986361037495, 135393159641569, 1266006310597506, 12228936468908781, 121823473948915769, 1249794986354577736
Offset: 1
-
b:= proc(n, m) option remember;
`if`(n=0, ithprime(m+1), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n-1, 1):
seq(a(n), n=1..23); # Alois P. Heinz, May 13 2022
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nmax = 23; CoefficientList[Series[Sum[Prime[k + 1] (Exp[x] - 1)^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[n, k] Prime[k + 1], {k, 1, n}], {n, 1, 23}]
A347041
Stirling transform of pi (A000720).
Original entry on oeis.org
0, 0, 1, 5, 21, 88, 389, 1852, 9525, 52632, 310141, 1936489, 12749204, 88149847, 637769490, 4812457992, 37763509549, 307453610201, 2592851608305, 22626572045811, 204197274002794, 1905132039608335, 18370391387293756, 183001650861913887, 1882207129695280320
Offset: 0
-
b:= proc(n, m) option remember; `if`(n=0,
numtheory[pi](m), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..27);
Showing 1-5 of 5 results.
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