A219790 Smallest prime not neighboring a prime(n)-smooth number.
11, 29, 43, 67, 103, 137, 173, 173, 173, 283, 283, 283, 283, 283, 317, 653, 653, 653, 653, 653, 653, 653, 653, 653, 653, 653, 653, 653, 787, 787, 787, 907, 907, 907, 907, 1433, 1433, 1433, 1433, 1433, 1447, 1543, 1543, 1867, 1867, 1867, 1867, 1867, 1867
Offset: 1
Examples
a(2) = 29, the smallest prime not neighboring a 3-smooth number, since 3 is the 2nd prime; i.e., not of the form 2^j*3^k +/- 1. 43-1 = 2*3*7, 43+1 = 2*2*11, so neither are 5-smooth. a(3) = 43, the smallest prime not neighboring a 5-smooth number, since 5 is the 3rd prime, and 43-1 = 42 = 2 * 3 * 7 is not 5 smooth, and 43+1 = 44 = 2^2 * 11 is not 5 smooth. - corrected by _Jason Kimberley_, Nov 29 2012 a(4) = 67, the smallest prime not neighboring a 7-smooth number, since 7 is the 4th prime, and 67-1 = 66 = 2 * 3 * 11 is not 7 smooth, and 67+1 = 68 = 2^2 * 17 is not 7 smooth. - corrected by _Jason Kimberley_, Nov 29 2012 a(5) = 103, the smallest prime not neighboring a 11-smooth number, since 11 is the 5th prime, and 103-1 = 102 = 2 * 3 * 17 is not 11 smooth, and 103+1 = 104 = 2^3 * 13 is not 11 smooth. a(6) = 137, the smallest prime not neighboring a 13-smooth number, since 13 is the 6th prime, and 137-1 = 136 = 2^3 * 17 is not 13 smooth, and 137+1 = 138 = 2 * 3 * 23 is not 13 smooth.
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Programs
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PARI
a(n)=my(p=prime(n));forprime(q=6*p-1,,if(vecmax(factor(q-1)[,1])>p && vecmax(factor(q+1)[,1])>p,return(q))) \\ Charles R Greathouse IV, Nov 28 2012
Formula
a(n) > 6p for n > 1, where p is the n-th prime. - Charles R Greathouse IV, Nov 28 2012
Extensions
a(3) and a(4) corrected by Charles R Greathouse IV, Nov 28 2012
a(1) and a(7)-a(53) from Charles R Greathouse IV, Nov 28 2012