A376644
Numbers k such that !k * k! + 1 = A003422(k) * A000142(k) + 1 = A061640(k) + 1 is prime.
Original entry on oeis.org
1, 2, 4, 6, 13, 14, 21, 41, 64, 110, 268, 1196, 3083, 5201, 7496, 9698
Offset: 1
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Select[Range[300], PrimeQ[#! * Sum[k!, {k, 0, #-1}] + 1] &]
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is(k) = isprime(k! * sum(i = 0, k-1, i!) + 1);
A143217
a(n) = n! * (!(n+1)) = n! * Sum_{k=0..n} k!.
Original entry on oeis.org
1, 2, 8, 60, 816, 18480, 629280, 29806560, 1864154880, 148459288320, 14652782323200, 1754531527795200, 250496911136102400, 42032247888401971200, 8188505926989625036800, 1832839841629043799552000, 467088574163459753336832000, 134454052266325985991942144000
Offset: 0
a(4) = 816 = 4! * 34, where 34 = A003422(4) and A000142 = (1, 1, 2, 6, 24, 120, ...).
a(4) = 816 = sum of row 4 terms of triangle A143216: (24 + 24 + 48 + 144 + 576).
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[Factorial(n)*(&+[Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jul 12 2022
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Table[n!*Sum[i!, {i, 0, n}], {n, 0, 16}]
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f=factorial; [f(n)*sum(f(k) for k in (0..n)) for n in (0..40)] # G. C. Greubel, Jul 12 2022
A217238
a(n) = n! * Sum_{k=1..n} k!.
Original entry on oeis.org
0, 1, 6, 54, 792, 18360, 628560, 29801520, 1864114560, 148458925440, 14652778694400, 1754531487878400, 250496910657100800, 42032247882174950400, 8188505926902446745600, 1832839841627736125184000, 467088574163438830546944000
Offset: 0
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[0] cat [Factorial(n)*&+[Factorial(k) : k in [1..n]] : n in [1..20]]; // Wesley Ivan Hurt, Jul 03 2020
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Table[n!*Sum[i!, {i, n}], {n, 0, 16}]
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f=factorial; [f(n)*sum(f(k) for k in (1..n)) for n in (0..40)] # G. C. Greubel, Jul 12 2022
A217239
a(n) = n!*(!n - 1) = n! * Sum_{k=1..n-1} k!.
Original entry on oeis.org
0, 0, 2, 18, 216, 3960, 110160, 4399920, 238412160, 16777031040, 1484589254400, 161180565638400, 21054377854540800, 3256459838542310400, 588451470350449305600, 122827588903536701184000, 29325437466043778002944000
Offset: 0
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[0,0] cat [Factorial(n)*(&+[Factorial(k): k in [1..n-1]]): n in [2..30]]; // G. C. Greubel, Jul 12 2022
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Table[n!*Sum[i!, {i, n-1}], {n, 0, 16}]
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f=factorial; [f(n)*sum(f(k) for k in (1..n-1)) for n in (0..40)] # G. C. Greubel, Jul 12 2022
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