A256393 -id:A256393 - cp's OEIS frontend

A257244 First differences of A256393: a(n) = A256393(n+1) - A256393(n).

2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 3, 3, 3, 3, 3, 5, 5, 7, 7, 7, 7, 7, 3, 3, 3, 19, 7, 7, 3, 5, 5, 5, 3, 7, 5, 5, 5, 19, 11, 11, 3, 13, 13, 13, 3, 23, 13, 13, 5, 11, 11, 11, 3, 61, 7, 31, 3, 13, 13, 19, 3, 43, 13, 13, 3, 7, 5, 5, 5, 61, 11, 31, 23, 23, 3, 127, 7, 7, 3, 151, 7, 19, 3, 181, 7, 13, 3, 43, 31, 31

Offset: 1

Antti Karttunen, Apr 19 2015

nonn

Sequence gives the differences between the successive terms of A256393, each difference being equal to their unique shared prime factor.

Antti Karttunen, Table of n, a(n) for n = 1..4095

Cf. A257245, A257246 (bisections), A257247 (positions where they coincide).

Cf. A256393.

a257244 n = a257244_list !! (n-1)
a257244_list = zipWith gcd a256393_list $ tail a256393_list
-- Reinhard Zumkeller, May 06 2015

f[n_] := Block[{pf = First /@ FactorInteger@ n},
  If[EvenQ@ n, Max@ pf, Min@ pf]]; s = {2}; lmt = 94; For[k = 2,
k <= lmt, k++, AppendTo[s, s[[k - 1]] + f@ s[[k - 1]]]]; Abs[
Subtract @@@ Partition[s, 2, 1]] (* Michael De Vlieger, Apr 19 2015 *)

a(n) = A256393(n+1) - A256393(n).

a(n) = gcd(A256393(n+1), A256393(n)).

A061228 a(1) = 2, a(n) = smallest number greater than n that is not coprime to n.

Original entry on oeis.org

2, 4, 6, 6, 10, 8, 14, 10, 12, 12, 22, 14, 26, 16, 18, 18, 34, 20, 38, 22, 24, 24, 46, 26, 30, 28, 30, 30, 58, 32, 62, 34, 36, 36, 40, 38, 74, 40, 42, 42, 82, 44, 86, 46, 48, 48, 94, 50, 56, 52, 54, 54, 106, 56, 60, 58, 60, 60, 118, 62, 122, 64, 66, 66, 70, 68, 134, 70, 72, 72

Offset: 1

Amarnath Murthy, Apr 23 2001

nonn

			a(9) = 12 as 10 and 11 are coprime to 9.
a(11) = 22 as 11 is a prime.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A020639, A046666, A064413, A087349.

Cf. A070229, A256393.

a061228 n = n + a020639 n  -- Reinhard Zumkeller, May 06 2015

for n from 1 to 150 do if n=1 then printf(`%d,`,2); fi: for k from n+1 to 2*n do if igcd(n,k)>1 then printf(`%d,`,k); break; fi: od: od:
# alternative:
2, seq(t + min(numtheory:-factorset(t)), t = 2..1000); # Robert Israel, Oct 21 2015

Table[n+First@(First/@FactorInteger[n]),{n,200}] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2011 *)
nxt[{n_,a_}]:=Module[{c=n+2},While[CoprimeQ[n+1,c],c++];{n+1,c}]; NestList[nxt,{1,2},70][[;;,2]] (* Harvey P. Dale, May 21 2025 *)

a(n) = n + if(n == 1, 1, factor(n)[1,1]); \\ Amiram Eldar, Apr 10 2025

a(n) = A020639(n) + n.
a(2m) = 2m+2, a(p) = 2p if p is a prime.
a(n) = n + the smallest divisor of n that is larger than 1, for n >= 2.
a(p^k) = p^k + p if p is prime. - Robert Israel, Oct 21 2015
a(n) = A087349(n-1) + 1 for n >= 2. - Amiram Eldar, Apr 10 2025

More terms from James Sellers, Apr 24 2001

A070229 Next m>n such that m is divisible by lpf(n), lpf=A006530 largest prime factor.

Original entry on oeis.org

2, 4, 6, 6, 10, 9, 14, 10, 12, 15, 22, 15, 26, 21, 20, 18, 34, 21, 38, 25, 28, 33, 46, 27, 30, 39, 30, 35, 58, 35, 62, 34, 44, 51, 42, 39, 74, 57, 52, 45, 82, 49, 86, 55, 50, 69, 94, 51, 56, 55, 68, 65, 106, 57, 66, 63, 76, 87, 118, 65, 122

Offset: 1

Reinhard Zumkeller, May 07 2002

Or, for n>1, n + (largest prime divisor of n). [Anne Robinson, daughter of Herman P. Robinson, Oct 08 1981]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
H. P. Robinson, Letter to N. J. A. Sloane, Oct 1981

Cf. A036441.
Cf. A006530, A061228, A256393.
Iterations: A076271 (start=1), A036441 (start=2), A123581 (start=3).

a070229 n = n + a006530 n
-- Reinhard Zumkeller, Nov 08 2015, Nov 14 2011

Table[n + Last@(First/@FactorInteger[n]), {n, 70}] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2011 *)

vector(100, n, if(n>1, vecmax(factor(n)[, 1]), 1) + n) \\ Altug Alkan, Nov 13 2015

a(n) = (n/lpf(n)+1)*lpf(n).

a(n) = n+lpf(n) where lpf=A006530 largest prime factor. For example, a(14)=14+7=21 (instead of ((14/7)+1)*7). - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Jun 14 2007

A257245 Odd bisection of A257244: a(n) = A257244(2n - 1).

Original entry on oeis.org

2, 3, 3, 3, 3, 5, 5, 3, 3, 5, 7, 7, 7, 3, 19, 7, 5, 5, 7, 5, 19, 11, 13, 13, 23, 13, 11, 11, 61, 31, 13, 19, 43, 13, 7, 5, 61, 31, 23, 127, 7, 151, 19, 181, 13, 43, 31, 233, 13, 13, 7, 11, 19, 31, 59, 31, 31, 163, 41, 41, 113, 11, 19, 19, 409, 41, 41, 41, 19, 41, 113, 29, 167, 13, 17, 11, 179, 19, 19, 13, 5

Offset: 1

Antti Karttunen, Apr 19 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048

Cf. A006530, A256393, A257244.

Cf. also A257246 (the other bisection), A257247 (gives the positions where both bisections have the same value).

a(n) = A257244(2n - 1).
a(n) = A006530(A256393(2n - 1)). [Equally, the largest prime factor of A256393(2n-1).]
Other identities and observations:
For all n >= 1, a(n) >= A257246(n).

A257246 Even bisection of A257244: a(n) = A257244(2n).

Original entry on oeis.org

2, 3, 3, 3, 3, 5, 3, 3, 3, 5, 7, 7, 3, 3, 7, 3, 5, 3, 5, 5, 11, 3, 13, 3, 13, 5, 11, 3, 7, 3, 13, 3, 13, 3, 5, 5, 11, 23, 3, 7, 3, 7, 3, 7, 3, 31, 3, 7, 13, 3, 7, 5, 7, 3, 31, 31, 3, 13, 41, 5, 3, 7, 19, 3, 7, 41, 41, 3, 19, 7, 29, 5, 3, 13, 11, 5, 3, 19, 3, 5, 5

Offset: 1

Antti Karttunen, Apr 19 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2047
Ratio a(n)/A257245(n) plotted with OEIS Plot2-script

Cf. A020639, A256393, A257244.

Cf. also A257245 (the other bisection), A257247 (gives the positions where both bisections have the same value).

a(n) = A257244(2n).
a(n) = A020639(A256393(2n)). [Equally, the smallest prime factor of A256393(2n).]
Other identities and observations:
For all n >= 1, a(n) <= A257245(n).

A257247 Numbers n such that A257245(n) = A257246(n), or equally, where A257244(2n) = A257244(2n-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 17, 20, 23, 27, 31, 36, 49, 51, 56, 59, 63, 66, 67, 69, 74, 78, 81, 88, 89, 91, 93, 97, 99, 101, 102, 114, 116, 120, 123, 126, 134, 141, 144, 146, 148, 152, 161, 163, 172, 173, 179, 192, 194, 195, 202, 207, 214, 219, 223, 227, 235, 240, 242, 246, 250, 252, 256

Offset: 1

Antti Karttunen, Apr 19 2015

nonn

The sequence gives numbers n such that A257244(2n) = A257244(2n-1), in other words, where the difference A256393(2n+1) - A256393(2n) is equal to the difference A256393(2n) - A256393(2n-1). This is equal to the condition that A256393(2n-1) and A256393(2n) share the same unique prime factor that A256393(2n) and A256393(2n+1) also share.

Also, numbers n such that the largest prime factor (A006530) of A256393(2n - 1) is equal to the smallest prime factor (A020639) of A256393(2n).

			Terms A256393[3949..3951] = (6299579580, 6299580499, 6299581418) factorize as 2*2*3*5*7*19*859*919, 919*919*7459, 2*13*919*263647. We see that gcd(A256393(3949),A256393(3950)) = gcd(A256393(3950),A256393(3951)) = 919, thus 3950/2 = 1975 is included in the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..289

Cf. A006530, A020639, A256393, A257244, A257245, A257246.

A276269 a(n) = n + the largest (if n is even), or the smallest (if n is odd) prime factor of n if n > 1; and a(1)=1.

Original entry on oeis.org

1, 4, 6, 6, 10, 9, 14, 10, 12, 15, 22, 15, 26, 21, 18, 18, 34, 21, 38, 25, 24, 33, 46, 27, 30, 39, 30, 35, 58, 35, 62, 34, 36, 51, 40, 39, 74, 57, 42, 45, 82, 49, 86, 55, 48, 69, 94, 51, 56, 55, 54, 65, 106, 57, 60, 63, 60, 87, 118, 65, 122, 93, 66, 66, 70, 77, 134

Offset: 1

Michel Marcus, Aug 26 2016

nonn

This is the underlying sequence for sequence A256393.

a(n) is never prime, since for n>1 it is alternatively divisible by A006530(n) then by A020639(n). This explains the existence of the sequences noted [p] in A256393 comments. There are also such [c] sequences where c is composite like for instance for c=8.

Cf. A006530 (Gpf), A020639 (Lpf), A256393.

a(n) = if (n==1, 1, my(f = factor(n)); if (n % 2, n + f[1,1], n + f[#f~, 1]));

cp's OEIS Frontend

A257244 First differences of A256393: a(n) = A256393(n+1) - A256393(n).

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A061228 a(1) = 2, a(n) = smallest number greater than n that is not coprime to n.

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A070229 Next m>n such that m is divisible by lpf(n), lpf=A006530 largest prime factor.

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A257245 Odd bisection of A257244: a(n) = A257244(2n - 1).

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A257246 Even bisection of A257244: a(n) = A257244(2n).

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A257247 Numbers n such that A257245(n) = A257246(n), or equally, where A257244(2n) = A257244(2n-1).

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A276269 a(n) = n + the largest (if n is even), or the smallest (if n is odd) prime factor of n if n > 1; and a(1)=1.

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PARI