A140118
Extrapolation for (n + 1)-st odd prime made by fitting least-degree polynomial to first n odd primes.
Original entry on oeis.org
3, 7, 9, 19, 3, 49, -39, 151, -189, 381, -371, 219, 991, -4059, 11473, -26193, 53791, -100639, 175107, -281581, 410979, -506757, 391647, 401587, -2962157, 9621235, -24977199, 57408111, -120867183, 236098467, -428880285, 719991383, -1096219131, 1442605443, -1401210665, 99178397, 4340546667
Offset: 1
Jonathan Wellons (wellons(AT)gmail.com), May 08 2008, May 19 2008
The lowest-order polynomial having points (1,3), (2,5), (3,7) and (4,11) is f(x) = 1/3 (x^3 - 6x^2 + 17x - 3). When evaluated at x = 5, f(5) = 19.
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a(n) = sum(i=1, n, prime(i+1)*(-1)^(n-i)*binomial(n, i-1)); \\ Michel Marcus, Jul 07 2025
A226805
P_n(n+1) where P_n(x) is the polynomial of degree n-1 which satisfies P_n(i) = i^i for i = 1,...,n.
Original entry on oeis.org
1, 7, 70, 877, 13316, 237799, 4885980, 113566121, 2946476764, 84417530491, 2647176188372, 90183424037293, 3316840864313484, 130985236211745959, 5528094465439087876, 248308899812296990033, 11827417687501017074876, 595470029978391175571923
Offset: 1
P_3(x) = 18 - 27*x + 10*x^2; a(3) = P_3(3+1) = 70.
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P[n_][x_] = Sum[a[i]*x^i, {i, 0, n - 1}];ecu[n_] := Table[P[n][i] == i^i, {i, 1, n}];PP[n_][x_] := P[n][x] /. Solve[ecu[n]][[1]];Table[PP[i][i + 1], {i, 1, 22}]
a[n_] := InterpolatingPolynomial[Table[{i, i^i}, {i, n}], n+1]; Array[a, 20] (* Giovanni Resta, Jun 18 2013 *)
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a(n)=subst(polinterpolate(vector(n,i,i^i)),'x,n+1) \\ Charles R Greathouse IV, Nov 19 2013
A379542
Second term of the n-th differences of the prime numbers.
Original entry on oeis.org
3, 2, 0, 2, -6, 14, -30, 62, -122, 220, -344, 412, -176, -944, 4112, -11414, 26254, -53724, 100710, -175034, 281660, -410896, 506846, -391550, -401486, 2962260, -9621128, 24977308, -57407998, 120867310, -236098336, 428880422, -719991244, 1096219280
Offset: 0
For all primes (not just odd) we have
A007442.
Including 1 in the primes gives
A030016.
Cf.
A064113,
A065890,
A084758,
A140119,
A173390,
A258025,
A258026,
A293467,
A333214,
A333254,
A377041.
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nn=40;Table[Differences[Prime[Range[nn+2]],n][[2]],{n,0,nn}]
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a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * prime(k+2)); \\ Michel Marcus, Jan 12 2025
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