A113496 -id:A113496 - cp's OEIS frontend

A113484 a(n) = smallest composite integer > n and coprime to n.

4, 9, 4, 9, 6, 25, 8, 9, 10, 21, 12, 25, 14, 15, 16, 21, 18, 25, 20, 21, 22, 25, 24, 25, 26, 27, 28, 33, 30, 49, 32, 33, 34, 35, 36, 49, 38, 39, 40, 49, 42, 55, 44, 45, 46, 49, 48, 49, 50, 51, 52, 55, 54, 55, 56, 57, 58, 63, 60, 77, 62, 63, 64, 65, 66, 85, 68, 69, 70, 81, 72, 77

Offset: 1

Leroy Quet, Jan 11 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A113496.

f[n_] := Block[{k = n + 1}, While[PrimeQ@k || GCD[k, n] > 1, k++ ]; k]; Array[f, 72] (* Robert G. Wilson v *)

a(n) = my(k = n+1); while(isprime(k) || (gcd(n,k) != 1), k++); k; \\ Michel Marcus, Mar 13 2017

from sympy import isprime, gcd
def A113484(n):
    k = n+1
    while gcd(k,n) != 1 or isprime(k):
        k += 1
    return k # Chai Wah Wu, Mar 28 2021

a(p) = p+1 for prime p > 2. - Michel Marcus, Mar 13 2017

More terms from Robert G. Wilson v, Jan 13 2006

A342175 a(n) is the difference between the n-th composite number and the smallest larger composite to which it is relatively prime.

Original entry on oeis.org

5, 19, 1, 1, 11, 13, 1, 1, 5, 7, 1, 1, 3, 1, 1, 1, 1, 5, 19, 1, 1, 1, 1, 13, 1, 1, 9, 13, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 17, 1, 1, 1, 1, 19, 1, 1, 11, 5, 1, 1, 1, 1, 7, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 19, 1, 1, 11, 13, 1, 1, 5, 7, 1, 1, 3, 1

Offset: 1

William C. Laursen, Mar 04 2021

nonn

Conjecture: The only nonprime terms are squares (based on checking the first 2 million terms). - Ivan N. Ianakiev, Mar 28 2021

The conjecture above is false (see A353203 for counterexamples). - Ivan N. Ianakiev, Jul 04 2022

			The first composite number is 4, and the smallest larger composite to which it is coprime is 9, so a(1) = 9 - 4 = 5.
The second composite number is 6, and the smallest larger composite to which it is coprime is 25, so a(2) = 25 - 6 = 19.

Cf. A002808, A113496, A353203.

Table[Block[{k = 1}, While[Nand[GCD[#, k] == 1, CompositeQ[# + k]], k++]; k] &@ FixedPoint[n + PrimePi@ # + 1 &, n + PrimePi@ n + 1], {n, 83}] (* Michael De Vlieger, Mar 19 2021 *)

lista(nn) = {forcomposite(c=1, nn, my(x=c+1); while (isprime(x) || (gcd(x,c) != 1), x++); print1(x - c, ", "););} \\ Michel Marcus, Mar 04 2021

from sympy import isprime, gcd, composite
def A342175(n):
    m = composite(n)
    k = m+1
    while gcd(k,m) != 1 or isprime(k):
        k += 1
    return k-m # Chai Wah Wu, Mar 28 2021

a(n) = A113496(n) - A002808(n). - Jon E. Schoenfield, Mar 04 2021

A353203 Let b be a composite number, c be the smallest composite number greater than b and coprime to b, and d = c-b. This sequence contains all b such that d is neither a prime nor a square.

Original entry on oeis.org

67613590, 72808450, 125918320, 153469030, 190281850, 229119880, 328315900, 339204910, 360203140, 395961280, 447304000, 450075340, 692309530, 844334920, 861327610, 909001390, 1029358330, 1166831380, 1178236510, 1321005400, 1344348610, 1366379080, 2035500610, 2045710810, 2156564410

Offset: 1

Fausto Morales Díaz and Ivan N. Ianakiev, Apr 30 2022

nonn

Other such terms are 18806843674476 and 18806855958880.

a(n) is even. Proof: If a(n) = b is odd then c = a(n) + 1 where gcd(b, c) = 1 and d = c-b = 1 which is a square. Contradiction. - David A. Corneth, May 01 2022

			If b = 6, then c = 25 and d = c-b = 19 (prime), so 6 is not in the sequence.
If b = 67613590, then c = 67613611, and d = c-b = 21 (neither prime nor square), so 67613590 is in the sequence.

Cf. A002808, A113496, A342175.

c[n_]:=Module[{k=n+1},While[GCD[n,k]!=1||PrimeQ[k],k++];k];
Select[Range[10^8],CompositeQ[#]&&CompositeQ[c[#]-#]&&!IntegerQ[Sqrt[c[#]-#]]&]

cp's OEIS Frontend

A113484 a(n) = smallest composite integer > n and coprime to n.

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A342175 a(n) is the difference between the n-th composite number and the smallest larger composite to which it is relatively prime.

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Author

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Programs

Mathematica

PARI

Python

Formula

A353203 Let b be a composite number, c be the smallest composite number greater than b and coprime to b, and d = c-b. This sequence contains all b such that d is neither a prime nor a square.

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Author

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Programs

Mathematica