This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(1)=1=A091666(1), a(4)=6=A091666(5), a(5)=12=A091666(9), a(6)=10=A091666(16).
From _Zak Seidov_, Feb 20 2012: (Start) n=4 and prime(4)^2=49, preceded by prime(15)=47, so a(4)=49-47=2; n=97 and prime(97)^2=509^2=259081, preceded by prime(22765)=259033, so a(97)=259081-259033=48. (End)
f[n_]:=Module[{n2=n^2},n2-NextPrime[n2,-1]]; f/@Prime[Range[90]] (* Harvey P. Dale, Oct 19 2011 *)
a(n) = my(p=prime(n)); p^2 - precprime(p^2); \\ Michel Marcus, Feb 27 2023
p(4)=7, 7^3 = 343; for odd k and n > 1, p(n)^r - k is even and thus not prime, so we only need consider even k. for k = 2: 343 - 2 = 341, which is 11 * 31 and not prime. for k = 4: 343 - 4 = 339, which is 3 * 113, also not prime. for k = 6: 343 - 6 = 337, which is prime, so 6 is the smallest number that can be subtracted from 343 to make another prime. Hence a(4) = 6.
sk[n_]:=With[{c=Prime[n]^3},c-NextPrime[c,-1]]; Array[sk,80] (* Harvey P. Dale, May 07 2019 *)
a(n) = {k = 0; while (! isprime(prime(n)^3 - k), k++); return (k);} \\Michel Marcus, Aug 02 2013
p(1)=2, 2^3 = 8. for even k, 2^r + k is even and thus not prime, so we only need consider odd k. for k = 1: 8 + 1 = 9, which is 3^2 and not prime. for k = 3: 8 + 3 = 11, which is prime, so 3 is the smallest number that can be added to 8 to make a new prime. Hence a(1) = 3.
[NextPrime(p^3)-p^3: p in PrimesUpTo(500)]; // Bruno Berselli, Sep 03 2013
Table[NextPrime[Prime[n]^3] - Prime[n]^3, {n, 100}] (* Bruno Berselli, Sep 03 2013 *)
a(n) = {k = 0; p3 = prime(n)^3; while (! isprime(p3+k), k++); k;} \\ Michel Marcus, Sep 03 2013
a(n) = {p3 = prime(n)^3; nextprime(p3) - p3;} \\ Michel Marcus, Sep 03 2013
p(3)=5, 5^4 = 625; for odd k and n > 1, p(n)^r - k is even and thus not prime, so we only need consider even k. for k = 2: 625 - 2 = 623, which is 7 * 89 and not prime. for k = 4: 625 - 4 = 621, which is 3^3 * 23, also not prime. for k = 6: 625 - 6 = 619, which is prime, so 6 is the smallest number that can be subtracted from 625 to make another prime. Hence a(3) = 6.
sk[p_]:=Module[{k=1,c=p^4},While[CompositeQ[c-k],k++];k]; sk/@Prime[Range[100]] (* Harvey P. Dale, Nov 19 2023 *)
Table[With[{c=p^4},c-NextPrime[c,-1]],{p,Prime[Range[100]]}] (* Harvey P. Dale, Nov 20 2023 *)
p(2)=3, 3^4 = 81; for odd k and n > 1, p(n)^r + k is even and thus not prime, so we only need consider even k. for k = 2: 81 + 2 = 83, which is prime, so 2 is the smallest number that can be added to 81 to make a new prime. Hence a(2) = 2.
NextPrime[#]-#&/@(Prime[Range[80]]^4) (* Harvey P. Dale, May 17 2015 *)
p(10)=29, 29^5 = 20511149; for odd k and n > 1, p(n)^r - k is even and thus not prime, so we only need consider even k. for k = 2: 20511149 - 2 = 20511147, which is 3 * 23 * 297263 and not prime. for k = 4: 20511149 - 4 = 20511145, which is 5 * 4102229, also not prime. for k = 6: 20511149 - 6 = 20511141, which is prime, so 6 is the smallest number that can be subtracted from 20511149 to make another prime. Hence a(10) = 6.
#-NextPrime[#,-1]&/@(Prime[Range[80]]^5) (* Harvey P. Dale, Sep 27 2020 *)
p(2)=3, 3^5 = 243; for odd k and n > 1, p(n)^r - k is even and thus not prime, so we only need consider even k. for k = 2: 243 + 2 = 245, which is 5 * 7^2 and not prime. for k = 4: 243 + 4 = 247, which is 13 * 19, also not prime. for k = 6: 243 + 6 = 249, which is 3 * 83, also not prime. for k = 8: 243 + 8 = 251, which is prime, so 8 is the smallest number that can be added to 243 to make another prime. Hence a(2) = 8.
t = Table[p2 = Prime[k]^2; NextPrime[p2] - p2, {k, 100000}]; Take[Union[t], 60]
NextPrime[#]-#&/@(Prime[Range[100000]]^2)//Union (* Harvey P. Dale, Apr 19 2020 *)
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