A079891 -id:A079891 - cp's OEIS frontend

A079890 Least number > n having one more prime factor than n, not necessarily distinct.

2, 4, 4, 8, 6, 8, 9, 16, 12, 12, 14, 16, 14, 18, 18, 32, 21, 24, 21, 24, 27, 27, 25, 32, 27, 27, 36, 36, 33, 36, 33, 64, 42, 42, 42, 48, 38, 42, 42, 48, 46, 54, 46, 54, 54, 50, 49, 64, 50, 54, 52, 54, 55, 72, 63, 72, 63, 63, 62, 72, 62, 63, 81, 128, 66, 81, 69, 81, 70, 81, 74, 96

Offset: 1

Reinhard Zumkeller, Jan 14 2003

A001222(a(n)) = A001222(n) + 1;
a(2^k) = 2^(k+1).
a(A076156(n)) = A076156(n)+1. - Reinhard Zumkeller, Feb 01 2008

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

Cf. A079891, A079892.

Cf. A165712.

a079890 n = head [x | x <- [n + 1 ..], a001222 x == 1 + a001222 n]
-- Reinhard Zumkeller, Aug 29 2013

lng[n_]:=Module[{x=n+1,pon=PrimeOmega[n]},While[PrimeOmega[x]-pon!=1, x++]; x]; Array[lng,80] (* Harvey P. Dale, Nov 09 2011 *)

A079893 a(n) = gcd(n, A079892(n)), where A079892(n) is the least number > n having one more distinct prime factor than n.

Original entry on oeis.org

1, 2, 3, 2, 1, 6, 1, 2, 1, 10, 1, 6, 1, 2, 15, 2, 1, 6, 1, 10, 3, 2, 1, 6, 1, 2, 1, 2, 1, 30, 1, 1, 3, 2, 7, 6, 1, 2, 3, 2, 1, 42, 1, 4, 15, 2, 1, 12, 1, 10, 3, 4, 1, 6, 5, 4, 3, 2, 1, 30, 1, 2, 3, 1, 1, 6, 1, 2, 1, 70, 1, 6, 1, 2, 3, 2, 1, 6, 1, 4, 1, 2, 1, 42, 5, 2, 3, 2, 1, 30, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2

Offset: 1

Reinhard Zumkeller, Jan 14 2003

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A079891, A079894.

A079893(n) = { my(x=1+omega(n)); for(k=1+n, oo, if(omega(k)==x, return(gcd(k,n)))); }; \\ Antti Karttunen, Mar 02 2023

A079894 a(n) = gcd(A079890(n), A079892(n)).

Original entry on oeis.org

2, 2, 2, 2, 6, 2, 1, 2, 2, 6, 2, 2, 14, 6, 6, 2, 3, 6, 1, 6, 3, 3, 1, 2, 1, 3, 4, 6, 33, 6, 33, 1, 42, 42, 42, 6, 38, 42, 42, 6, 2, 6, 2, 6, 6, 10, 1, 4, 50, 6, 4, 6, 1, 12, 3, 12, 3, 3, 62, 6, 62, 3, 3, 1, 66, 3, 1, 1, 70, 3, 2, 6, 74, 3, 3, 3, 78, 3, 2, 12, 2, 4, 85, 6, 2, 2, 2, 18, 91, 6, 2, 2, 2, 2

Offset: 1

Reinhard Zumkeller, Jan 14 2003

nonn

Cf. A079891, A079893.

A218621 a(n) = unique divisor d of n such that d + (n/d - 1)/2 is minimal and integral.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 1, 8, 3, 2, 1, 4, 1, 2, 3, 16, 1, 2, 1, 4, 3, 2, 1, 8, 5, 2, 3, 4, 1, 6, 1, 32, 3, 2, 5, 4, 1, 2, 3, 8, 1, 6, 1, 4, 5, 2, 1, 16, 7, 10, 3, 4, 1, 6, 5, 8, 3, 2, 1, 4, 1, 2, 7, 64, 5, 6, 1, 4, 3, 10, 1, 8, 1, 2, 5, 4, 7, 6, 1, 16, 9, 2, 1, 4, 5

Offset: 1

L. Edson Jeffery, Feb 18 2013

nonn

Differs from A079891 starting at a(18).
For integers M, k, with 0<=k<=M, consider a representation of n as n = T(M) - T(M-k) = M + (M-1) + ... + (M-k+1), in which k is maximal, where T(r) = r*(r+1)/2 is the r-th triangular number. Then k = A109814(n), and M = A212652(n) = a(n) + (n/a(n) - 1)/2 is minimal.
Conjecture. For n, p, v, j natural numbers, the conditions on a(n) seem to be the following:
1. If n is an odd prime, then a(n) = 1.
2. If n is odd and composite, then
     a(n) = max(p : p | n, p <= sqrt(n), p is a prime).
3. If n is equal to a power of 2, then a(n) = n.
4. If n = 2^j*v, with v odd, v>1 and j>1, then a(n) = 2^j.
5. If n = 2*v, with v odd and composite, then
     a(n) = 2*p, where p is the least prime such that p | v.
6. If n = 2*p, for p an odd prime, then a(n) = 2.

T. D. Noe, Table of n, a(n) for n = 1..2048

Cf. A000217, A109814, A118235, A141419, A209260, A212652.

Table[d = Divisors[n]; mn = Infinity; best = 0; Do[q = i + (n/i - 1)/2; If[IntegerQ[q] && q < mn, mn = q; best = i], {i, d}]; best, {n, 100}] (* T. D. Noe, Feb 21 2013 *)

cp's OEIS Frontend

A079890 Least number > n having one more prime factor than n, not necessarily distinct.

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A079893 a(n) = gcd(n, A079892(n)), where A079892(n) is the least number > n having one more distinct prime factor than n.

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A079894 a(n) = gcd(A079890(n), A079892(n)).

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A218621 a(n) = unique divisor d of n such that d + (n/d - 1)/2 is minimal and integral.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica