A305618
Expansion of e.g.f. log(1 + Sum_{k>=1} x^prime(k)/prime(k)!).
Original entry on oeis.org
0, 1, 1, -3, -9, 20, 190, -126, -6280, -10326, 293041, 1519320, -16985045, -194560444, 1013712777, 27317463952, -19210030599, -4305097718760, -17733269020226, 743855089334604, 7868686621862292, -132351392654695270, -2854492900112993039, 20150897206881256464
Offset: 1
E.g.f.: A(x) = x^2/2! + x^3/3! - 3*x^4/4! - 9*x^5/5! + 20*x^6/6! + ...
exp(A(x)) = 1 + x^2/2! + x^3/3! + x^5/5! + x^7/7! + ... + x^A000040(k)/A039716(k) + ...
exp(exp(A(x))-1) = 1 + x^2/2! + x^3/3! + 3*x^4/4! + 11*x^5/5! + ... + A190476(k)*x^k/k! + ...
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a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n)-add(a(j)*t(n-j)*
j*binomial(n, j), j=1..n-1)/n))(i-> `if`(isprime(i), 1, 0))
end:
seq(a(n), n=1..25); # Alois P. Heinz, Dec 04 2018
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nmax = 24; Rest[CoefficientList[Series[Log[1 + Sum[x^Prime[k]/Prime[k]!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]
a[n_] := a[n] = Boole[PrimeQ[n]] - Sum[k Binomial[n, k] Boole[PrimeQ[n - k]] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 24}]
A305619
Expansion of e.g.f. log(1 + Sum_{k>=1} x^prime(k)/prime(k)).
Original entry on oeis.org
0, 1, 2, -3, 4, -10, 636, -1078, -18416, -131976, 5035920, 5333592, 187347744, -4079616528, -14669908512, -140154110640, 28743506893056, -92449999037568, 2738959517576448, -52969092379214976, 34211286306178560, -16812071564735736576, 1407763084021569335808
Offset: 1
E.g.f.: A(x) = x^2/2! + 2*x^3/3! - 3*x^4/4! + 4*x^5/5! - 10*x^6/6! + ...
exp(A(x)) = 1 + x^2/2 + x^3/3 + x^5/5 + x^7/7 + ... + x^A000040(k)/A000040(k) + ...
exp(exp(A(x))-1) = 1 + x^2/2! + 2*x^3/3! + 3*x^4/4! + 44*x^5/5! + ... + A218002(k)*x^k/k! + ...
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a:=series(log(1+add(x^ithprime(k)/ithprime(k),k=1..100)),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019
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nmax = 23; Rest[CoefficientList[Series[Log[1 + Sum[x^Prime[k]/Prime[k], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]
a[n_] := a[n] = Boole[PrimeQ[n]] (n - 1)! - Sum[k Binomial[n, k] Boole[PrimeQ[n - k]] (n - k - 1)! a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 23}]
A317255
a(n) = 149836681069944461 + (n-1)*1723457117682300.
Original entry on oeis.org
149836681069944461, 151560138187626761, 153283595305309061, 155007052422991361, 156730509540673661, 158453966658355961, 160177423776038261, 161900880893720561, 163624338011402861, 165347795129085161, 167071252246767461, 168794709364449761, 170518166482132061
Offset: 1
a(26) = 149836681069944461 + 25*7725290*223092870 = 192923109012001961 is prime.
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List([1..25], n->149836681069944461+(n-1)*1723457117682300);
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seq(149836681069944461+(n-1)*1723457117682300,n=1..25);
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Table[149836681069944461 + (n - 1) 1723457117682300, {n, 1, 25}]
A317259
a(n) = 136926916457315893 + (n - 1)*9843204333812850.
Original entry on oeis.org
136926916457315893, 146770120791128743, 156613325124941593, 166456529458754443, 176299733792567293, 186142938126380143, 195986142460192993, 205829346794005843, 215672551127818693, 225515755461631543, 235358959795444393, 245202164129257243, 255045368463070093
Offset: 1
a(26) = 136926916457315893 + 25*44121555*223092870 = 383007024802637143 is prime.
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List([1..25], n->136926916457315893+(n-1)*9843204333812850);
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seq(136926916457315893+(n-1)*9843204333812850,n=1..25);
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Table[136926916457315893 + (n - 1) 9843204333812850, {n, 1, 25}]
Showing 1-4 of 4 results.
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