A146757
Number of primes p < 10^n such that s - p is prime, where s is the next square greater than p.
Original entry on oeis.org
2, 15, 68, 363, 2084, 13567, 95164, 705036, 5444255, 43211106, 351904307, 2921904565
Offset: 1
A(2) = 15 because at 10^2 there are 15 primes that, subtracted from the next higher value square, produce prime differences: {2, 7, 11, 13, 23, 29, 31, 47, 53, 59, 61, 79, 83, 89, 97}.
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Table[Length[Select[Prime[Range[PrimePi[10^n]]], PrimeQ[Ceiling[Sqrt[#]]^2 - #] &]], {n, 6}] (* T. D. Noe, Mar 31 2013 *)
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10 'sq less pr are prime 20 N=1:O=1:C=1 30 A=3:S=sqrt(N):if N>10^3 then print N,C-1:stop 40 B=N\A 50 if B*A=N then 100 60 A=A+2 70 if A<=S then 40 80 R=O^2:Q=R-N 90 if N1 print R;N;Q;C:N=N+2:C=C+1:goto 30 100 N=N+2:if N
A146759
Number of primes p < 10^n such that c - p is prime, where c is the next cube greater than p.
Original entry on oeis.org
2, 7, 43, 224, 1355, 9306, 66200, 500249, 3883527, 31081813, 254358928, 2120975833
Offset: 1
a(2) = 7 because at 10^2 there are 7 primes that, subtracted from the next higher value cube, produce prime differences: {3, 5, 41, 47, 53, 59, 61}.
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Table[Length[Select[Prime[Range[PrimePi[10^n]]], PrimeQ[Ceiling[#^(1/3)]^3 - #] &]], {n, 6}] (* T. D. Noe, Mar 31 2013 *)
cpQ[n_]:=PrimeQ[Ceiling[Surd[n,3]]^3-n]; nn=9; Module[{c=Table[If[ cpQ[n],1,0], {n, Prime[ Range[ PrimePi[ 10^nn]]]}]}, Table[ Total[ Take[c,PrimePi[10^p]]],{p,nn}]] (* Harvey P. Dale, Aug 13 2014 *)
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a(n) = {my(nb = 0); forprime(p=2, 10^n, if (isprime((sqrtnint(p,3)+1)^3 - p), nb++);); nb;} \\ Michel Marcus, Jun 22 2019
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list(nmax) = {my(m = 0, c = 2, cc = c^3, n = 0, pow = 10); forprime(p = 1, , if(p > pow, print1(m, ", "); n++; if(n == nmax, break); pow *= 10); if(p > cc, c++; cc = c^3); if(isprime(cc - p), m++));} \\ Amiram Eldar, Jan 20 2025
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10 'cu less pr are prime
20 N=1:O=1:C=1
30 A=3:S=sqrt(N):if N>10^3 then print N,C-1:stop
40 B=N\A
50 if B*A=N then 100
60 A=A+2
70 if A<=S then 40
80 R=O^3:Q=R-N
90 if N1 print R;N;Q;C:N=N+2:C=C+1:goto 30
100 N=N+2:if N
A146758
Last prime subtrahend at 10^n in A146757.
Original entry on oeis.org
7, 97, 983, 9941, 99839, 999983, 9998239, 99999989, 999950881, 9999999929, 99999999833, 999999999989, 9999999999863, 99999999999971, 999999961946087, 9999999999999917, 99999999989350667, 999999999999999989
Offset: 1
A(2)=97 because 97 is the 15th and last prime difference under 10^2.
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A146758 := proc(n) local p: p:=10^n: do p:=prevprime(p): if(isprime(ceil(evalf(sqrt(p)))^2-p))then return p: fi: od: end: seq(A146758(n),n=1..14); # Nathaniel Johnston, Oct 01 2011
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10 'sq less pr are prime 20 N=1:O=1:C=1 30 A=3:S=sqrt(N):if N>10^3 then print N,C-1:stop 40 B=N\A 50 if B*A=N then 100 60 A=A+2 70 if A<=S then 40 80 R=O^2:Q=R-N 90 if N1 print R;N;Q;C:N=N+2:C=C+1:goto 30 100 N=N+2:if N
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