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(60) = 7, since all primes smaller than 7 divide 60 but 7 does not. a(90) = a(120) = a(150) = a(180) = 7 because 90,120,150,180 all have same squarefree kernel = 30 = A002110(3), and 7 is the smallest prime which does not divide 30. - _David James Sycamore_, Dec 04 2024
a053669 n = head $ dropWhile ((== 0) . (mod n)) a000040_list -- Reinhard Zumkeller, Nov 11 2012
f:= proc(n) local p; p:= 2; while n mod p = 0 do p:= nextprime(p) od: p end proc: map(f, [$1..100]); # Robert Israel, May 18 2016
Table[k := 1; While[Not[GCD[n, Prime[k]] == 1], k++ ]; Prime[k], {n, 1, 60}] (* Stefan Steinerberger, Apr 01 2006 *)
With[{prs=Prime[Range[10]]},Flatten[Table[Select[prs,!Divisible[ n,#]&,1],{n,110}]]] (* Harvey P. Dale, May 03 2012 *)
a(n)=forprime(p=2,,if(n%p,return(p))) \\ Charles R Greathouse IV, Nov 20 2012
from sympy import nextprime
def a(n):
p = 2
while True:
if n%p: return p
else: p=nextprime(p) # Indranil Ghosh, May 12 2017
# using standard library functions only
import math
def a(n):
k = 2
while math.gcd(n,k) > 1: k += 1
return k # Ely Golden, Nov 26 2020
(define (A053669 n) (let loop ((i 1)) (cond ((zero? (modulo n (A000040 i))) (loop (+ i 1))) (else (A000040 i))))) ;; Antti Karttunen, Jan 26 2014
For n = 12, the numbers 1 <= k < n such that the prime divisors p of k also divide n are {2, 3, 4, 6, 8, 9}; {2, 3, 4, 6} divide n = 12, thus row n = 12 is {8, 9}.
n: k
1: 0
2: 0
3: 0
4: 0
5: 0
6: 0
7: 0
8: 6
9: 6
10: 6
11: 0
12: 10
13: 0
14: 6 10 12
15: 6 10 12
16: 6 10 12 14
17: 0
18: 10 14 15
19: 0
20: 6 12 14 15 18
Table[With[{r = First /@ FactorInteger@ n}, Select[Range@ n, Function[m, And[! SubsetQ[r, First /@ FactorInteger@ m], 1 < GCD[m, n] < n]]]], {n, 30}] /. {} -> {0} // Flatten (* Michael De Vlieger, May 03 2016 *)
From _David James Sycamore_, Feb 28 2025: (Start)
Using my formula above: n = 4235 = 5*7*11^2, so a(n) = 2*5 = 10.
For n = odd prime p, a(n) = 2*p.
For n = 2, a(n) = min{2^2, 2*3} = 4.
For n = 4, a(n) = min{2^3, 2*3} = 6. (For all n = 2^k, k >= 2, a(n) = 6.)
For n = 120 = 2^3*3*5, a(n) = min{16, 9, 25, 14} = 9.
For n = 5040 = 2^4*3^2*5*7, a(n) = min{32, 27, 25, 49, 22} = 22.
For n = 3603600 = 2^4*3^2*5^2*7*11*13, a(n) = min{32,27,125,49,121,169,34} = 27. (End)
Table[k = 3; Until[1 < GCD[k, n] < k, k++]; k, {n, 2, 120}] (* Michael De Vlieger, Feb 20 2025 *)
a(n)=for(k=4,2*n,if(gcd(n,k)>1 && n%k, return(k))) \\ Charles R Greathouse IV, Apr 05 2013
a(n)=my(f=factor(n),b);forprime(p=2,,if(n%p,b=p*f[1,1];break));for(i=1,#f[,1],b=min(b,f[i,1]^(f[i,2]+1)));b \\ Charles R Greathouse IV, Apr 05 2013
All m < 10 are not in the sequence since they either divide 6^e with integer e >= 0 or are coprime to 6. 10 is in the sequence since gcd(6, 10) = 2 and 10 does not divide 6^e with integer e >= 0. 11 is not in the sequence since 11 is coprime to 6. 12 is not in the sequence since 12 | 6^2.
With[{nn = 123, k = 6}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]
a(6) = A096014(6) = 10 since for 6, among the next composites {8, 9, 10, ...}, 10 is the first that is divisible by at least one prime p = 2 | 6, and at least one prime 5 that is coprime to 6. Since A020639(6) = 2 and A053669(6) = 5, a(6) and A096014(6) are identical.
a(12) = 14 since 14 is both the next composite after 12, and divisible by at least one prime divisor 2 of 12 and one prime q = 7 that is coprime to 12. This differs from A096014(12) = 10 because A053669(12) = 5, and 2 * 5 = 10.
Table[k = n + 2; While[Or[CoprimeQ[k, n], PowerMod[n, k, k] == 0], k++]; k, {n, 2, 66}] (* Michael De Vlieger, Sep 20 2017 *)
rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; {0}~Join~Table[If[Nor[PrimePowerQ[n], n == 6], k = n - 2; While[Or[CoprimeQ[n, k], Divisible[n, rad[k]]], k--]; k, 0], {n, 2, 120}]
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