A080186
Primes p such that 7 is the largest of all prime factors of the numbers between p and the next prime (cf. A052248).
Original entry on oeis.org
13, 41, 419, 881, 1049, 2267, 2687, 3359, 3527, 5879, 6299, 7349, 7559, 8231, 8819, 10499, 18521, 26249, 26879, 28349, 29399, 30869, 33599, 35279, 49391, 81647, 100799, 102059, 131249, 131711, 134399, 158759, 170099, 183707, 197567, 241919
Offset: 1
13 is a term since 14 = 2*7, 15 = 3*5, 16 = 2^4 are the numbers between 13 and the next prime 17; 419 is a term since 420 = 2^2*3*5*7 is the only number between 419 and the next prime 421.
-
lpf7Q[n_]:=Max[Flatten[Transpose[FactorInteger[#]][[1]]&/@Range[ n+1, NextPrime[ n]-1]]]==7; Select[Prime[Range[22000]],lpf7Q] (* Harvey P. Dale, Sep 25 2015 *)
-
{forprime(p=2,250000,q=nextprime(p+1); m=0; j=p+1; while(j
A320885
7-smooth but not 5-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0.
Original entry on oeis.org
28, 126, 175, 280, 336, 378, 441, 560, 630, 672, 1225, 1470, 1680, 1701, 1792, 2016, 2520, 2835, 3136, 3969, 4200, 5250, 5600, 6860, 7840, 7875, 8400, 8960, 9072, 9408, 11025, 11340, 12096, 13125, 15120, 17640, 19845, 20160, 21000, 23520, 24696, 27440, 30625, 32928, 35000
Offset: 1
Cf.
A080194 (greatest prime factor = 7).
Cf.
A180045 (numbers (ab+1)(ac+1), a>b>c>0),
A320883 (subsequence of 3-smooth terms),
A320884 (subsequence of 5-smooth terms).
-
Reap[For[k = 7, k <= 35000, k = k+7, If[FactorInteger[k][[-1, 1]] == 7, If[ Reduce[k == (a b + 1)(a c + 1) && a > b > c > 0, {a, b, c}, Integers] =!= False, Print[k]; Sow[k]]]]][[2, 1]] (* Jean-François Alcover, Dec 07 2018 *)
-
is_A320885(n)={vecmax(factor(n,7)[,1])==7 && is_A180045(n)}
A320885=select( is_A180045, A080194_list(1e20)) \\ Only initial terms, not the complete sequence. For more efficiency, use is_A180045 or a dedicated implementation inside the nested loops in A080194_list().
A219785
Primes not neighboring an 11-smooth number.
Original entry on oeis.org
103, 137, 157, 173, 227, 229, 233, 277, 283, 311, 313, 317, 347, 367, 373, 389, 409, 443, 457, 467, 509, 521, 523, 547, 557, 563, 569, 571, 607, 613, 619, 643, 653, 677, 683, 691, 709, 733, 739, 743, 761, 773, 787, 797, 821, 823, 827, 829, 853, 857, 859, 877
Offset: 1
103 is in the sequence because it is prime and the closest 11-smooth numbers are 100 and 105, which differ from 103 by 3 and -2 respectively, neither being -1 or +1.
137 is in the sequence because it is prime and neither 137 - 1 = 136 = 2^3 * 17 nor 137 + 1 = 138 = 2 * 3 * 23 are 11-smooth.
-
mx = 2^10; t11 = Select[Sort[Flatten[Table[2^i 3^j 5^k 7^l 11^m, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx]}, {k, 0, Log[5, mx]}, {l, 0, Log[7, mx]}, {m, 0, Log[11, mx]}]]], # <= mx &]; Complement[Prime[Range[PrimePi[mx]]], Union[Select[t11 + 1, PrimeQ], Select[t11 - 1, PrimeQ]]] (* T. D. Noe, Nov 27 2012 *)
A219790
Smallest prime not neighboring a prime(n)-smooth number.
Original entry on oeis.org
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
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.
-
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
Comments