A307771
Expansion of e.g.f. Sum_{k>=1} prime(k)*(exp(x) - 1)^k/k!.
Original entry on oeis.org
2, 5, 16, 60, 253, 1178, 5976, 32623, 189702, 1166720, 7554877, 51351254, 365560784, 2720255911, 21121563036, 170839106566, 1437200307921, 12556366592382, 113755900474652, 1067028469382353, 10346222830388738, 103538470949470066, 1067747451140472577, 11330777204488565252
Offset: 1
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a:= n-> add(ithprime(k)*Stirling2(n, k), k=1..n):
seq(a(n), n=1..30); # Alois P. Heinz, Apr 27 2019
# second Maple program:
b:= proc(n, m) option remember;
`if`(n=0, ithprime(m), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n-1, 1):
seq(a(n), n=1..24); # Alois P. Heinz, Aug 03 2021
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nmax = 24; Rest[CoefficientList[Series[Sum[Prime[k] (Exp[x] - 1)^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 24; Rest[CoefficientList[Series[Sum[Prime[k] x^k/Product[(1 - j x), {j, 1, k}], {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[StirlingS2[n, k] Prime[k], {k, 1, n}], {n, 1, 24}]
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upto(n) = my(v1, v2); v1 = vector(n, i, prime(i)); v2 = vector(n, i, 0); v2[1] = v1[1]; for(i=1, n-1, v1 = vector(#v1-1, j, j*v1[j] + v1[j+1]); v2[i+1] = v1[1]); v2 \\ Mikhail Kurkov, Mar 30 2025, after Alois P. Heinz
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
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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}]
A354002
Inverse Stirling transform of odd primes.
Original entry on oeis.org
3, 2, -2, 6, -30, 192, -1440, 12240, -115916, 1209422, -13784264, 170426380, -2272355448, 32507854434, -496746974148, 8076163535824, -139211242006108, 2536169979011432, -48695473146705746, 982863502262307532, -20805668315828056010, 460926536131613987430
Offset: 1
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nmax = 22; CoefficientList[Series[Sum[Prime[k + 1] Log[1 + x]^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] Prime[k + 1], {k, 1, n}], {n, 1, 22}]
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a(n) = sum(k=1, n, stirling(n,k,1) * prime(k+1)); \\ Michel Marcus, May 13 2022
A354003
Inverse Stirling transform of A008578 (1 together with the primes).
Original entry on oeis.org
1, 1, -1, 3, -14, 84, -604, 5020, -47144, 492408, -5653004, 70681706, -955450018, 13878511166, -215521103888, 3562431678650, -62439880637498, 1156609714838858, -22575425757129216, 463085375385002432, -9959296414838153618, 224079866356625633070, -5264190202707104532482
Offset: 1
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nmax = 23; CoefficientList[Series[Log[1 + x] + Sum[Prime[k - 1] Log[1 + x]^k/k!, {k, 2, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] If[k == 1, 1, Prime[k - 1]], {k, 1, n}], {n, 1, 23}]
Showing 1-4 of 4 results.
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