A084969 -id:A084969 - cp's OEIS frontend

A251758 Let n>=2 be a positive integer with divisors 1 = d_1 < d_2 < ... < d_k = n, and s = d_1d_2 + d_2d_3 + ... + d_(k-1)*d_k. The sequence lists the values a(n) = floor(n^2/s).

2, 3, 1, 5, 1, 7, 1, 2, 1, 11, 1, 13, 1, 2, 1, 17, 1, 19, 1, 2, 1, 23, 1, 4, 1, 2, 1, 29, 1, 31, 1, 2, 1, 4, 1, 37, 1, 2, 1, 41, 1, 43, 1, 2, 1, 47, 1, 6, 1, 2, 1, 53, 1, 4, 1, 2, 1, 59, 1, 61, 1, 2, 1, 4, 1, 67, 1, 2, 1, 71, 1, 73, 1, 2, 1, 6, 1, 79, 1, 2, 1

Offset: 2

Michel Lagneau, Dec 08 2014

nonn

s is always less than n^2 and if n is a prime number then s divides n^2.
For n >= 2, the sequence has the following properties:
a(n) = n if n is prime.
a(n) = 1 if n is in A005843 and > 2;
a(n) <= 2 if n is in A016945 and > 3;
a(n) <= 4 if n is in A084967 and > 5;
a(n) <= 6 if n is in A084968 and > 7;
a(n) = 8: <= 35336848261, ...;
a(n) <= 10 if n is in A084969 and > 11;
a(n) <= 12 if n is in A084970 and > 13;
a(n) = 14: 6678671, ...;
This is different from A250480 (a(n) = n for all prime n, and a(n) = A020639(n) - 1 for all composite n), which thus satisfies the above conditions exactly, while with this sequence A020639(n)-1 gives only the guaranteed upper limit for a(n) at composite n. Note that the first different term does not occur until at n = 2431 = 11*13*17, for which a(n) = 9. (See the example below.)
Conjecture: Terms x, where a(x)=n, x=p#k/p#j, p#i is the i-th primorial, k>j is suitable large k and j is the number of primes less than n. As an example, n=9, x = p#7/p#4 = 2431. For n=10, x = p#6/p#4 = 143 although 121 = 11^2 is the least x where a(x)=10 (see formula section). For n=8, x = p#12/p#4, p#13/p#4, p#14/p#4, p#15/p#4, p#16/p#4, etc. But is p#12/p#4 the least such x? - Robert G. Wilson v, Dec 18 2014
n^2/s is only an integer iff n is prime. - Robert G. Wilson v, Dec 18 2014
First occurrence of n >= 1: 4, 2, 3, 25, 5, 49, 7, ??? <= 35336848261, 2431, 121, 11, 169, 13, 6678671, 7429, 289, 17, 361, 19, 31367009, 20677, 529, 23, ..., . - Robert G. Wilson v, Dec 18 2014

			For n = 2431 = 11*13*17, we have (as the eight divisors of 2431 are [1, 11, 13, 17, 143, 187, 221, 2431]) a(n) = floor((2431*2431) / ((1*11)+(11*13)+(13*17)+(17*143)+(143*187)+(187*221)+(221*2431))) = floor(5909761/608125) = floor(9.718) = 9.

Michel Lagneau, Table of n, a(n) for n = 2..10000
International Mathematical Olympiad, Problems, IMO-2002, Problem 4.

Cf. A000040 (prime numbers), A005843 (even numbers), A016945 (6n+3), A084967 (GCD( 5k, 6) =1), A084968 (GCD( 7k, 30) =1), A084969 (GCD( 11k, 30) =1), A084970 (Numbers whose smallest prime factor is 13).
Cf. also A020639 (the smallest prime divisor), A055396 (its index) and arrays A083140 and A083221 (Sieve of Eratosthenes).
Differs from A250480 for the first time at n = 2431, where a(2431) = 9, while A250480(2431) = 10.
Cf. A078730 (sum of products of two successive divisors of n).

with(numtheory):nn:=100:
for n from 2 to nn do:
   x:=divisors(n):n0:=nops(x):s:=sum('x[i]*x[i+1]','i'=1..n0-1):
   z:=floor(n^2/s):printf(`%d, `,z):
od:

f[n_] := Floor[ n^2/Plus @@ Times @@@ Partition[ Divisors@ n, 2, 1]]; Array[f, 81, 2] (* Robert G. Wilson v, Dec 18 2014 *)

a(n) <= A250480(n), and especially, for all composite n, a(n) < A020639(n). [Cf. the Comments section above.] - Antti Karttunen, Dec 09 2014
From Robert G. Wilson v, Dec 18 2014: (Start)
a(n) = floor(n^2/A078730(n));
a(n) = n iff n is prime. (End)

Comments section edited by Antti Karttunen, Dec 09 2014

Instances of n for which a(n) = 8 and 14 found by Robert G. Wilson v, Dec 18 2014

A376839 a(1) = 1. For n > 1 if A007947(a(n-1)) is in A002110, a(n) is the smallest prime not already a term. Otherwise a(n) is the least novel multiple of the smallest non divisor prime of a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 10, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 23, 26, 27, 28, 30, 29, 32, 31, 34, 33, 36, 37, 38, 39, 40, 42, 25, 44, 45, 46, 48, 41, 50, 51, 52, 54, 43, 56, 57, 58, 60, 47, 62, 63, 64, 53, 66, 35, 68, 69, 70, 72, 59, 74, 75

Offset: 1

David James Sycamore, Oct 06 2024

nonn

A non divisor prime of a(n-1) is any prime p < Gpf(a(n-1)) which does not divide a(n-1). A007947(a(n-1)) is in A002110 iff a(n-1) is a term in A055932. Sequence is conjectured to be a permutation of the natural numbers (A000027) with primes in order.

Scatterplot shows trajectories of numbers whose smallest prime factor is prime p, e.g., for p = 5, numbers in A084967, p = 7, those in A084968, p = 11 those in A084969, etc. - Michael De Vlieger, Oct 09 2024

			a(1) = 1 = A002110(0) so a(2) = 2 (smallest prime not already a term).
a(2) = 2 = A002110(1) so a(3) = 3.
a(3) = 3 not a term in A002110 so a(4) is least novel multiple of 2, the least non divisor prime of 3. Therefore a(4) = 4 since 2 has occurred earlier.
a(39) = 42, not a term in A002110 so a(40) = 25, the least novel multiple of 5, the smallest non divisor prime of 42.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue or purple, with the latter additionally representing powerful numbers that are not prime powers.

Cf. A000040, A000027, A002110, A007947, A055932, A084967, A084968, A084969, A372368, A376865.

nn = 120; c[] := False; m[] := 1; f[x_] := FactorInteger[x][[All, 1]];
  Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2]; j = 2; v = 3;
  Do[If[Or[IntegerQ@ Log2[j], And[EvenQ[j], Union@ Differences@ PrimePi[#] == {1}]],
     k = v; While[c[k*m[k]], m[k]++]; k *= m[k],
     k = 2; While[Divisible[j, k], k = NextPrime[k]];
     While[c[k*m[k]], m[k]++]; k *= m[k]] &[f[j]];
  Set[{a[n], c[k], j}, {k, True, k}];
  If[k == v, While[c[v], v = NextPrime[v]]], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Oct 09 2024 *)

More terms from Michael De Vlieger, Oct 09 2024

cp's OEIS Frontend

A251758 Let n>=2 be a positive integer with divisors 1 = d_1 < d_2 < ... < d_k = n, and s = d_1d_2 + d_2d_3 + ... + d_(k-1)*d_k. The sequence lists the values a(n) = floor(n^2/s).

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A376839 a(1) = 1. For n > 1 if A007947(a(n-1)) is in A002110, a(n) is the smallest prime not already a term. Otherwise a(n) is the least novel multiple of the smallest non divisor prime of a(n-1).

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cp's OEIS Frontend

A251758 Let n>=2 be a positive integer with divisors 1 = d_1 < d_2 < ... < d_k = n, and s = d_1*d_2 + d_2*d_3 + ... + d_(k-1)*d_k. The sequence lists the values a(n) = floor(n^2/s).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A376839 a(1) = 1. For n > 1 if A007947(a(n-1)) is in A002110, a(n) is the smallest prime not already a term. Otherwise a(n) is the least novel multiple of the smallest non divisor prime of a(n-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A251758 Let n>=2 be a positive integer with divisors 1 = d_1 < d_2 < ... < d_k = n, and s = d_1d_2 + d_2d_3 + ... + d_(k-1)*d_k. The sequence lists the values a(n) = floor(n^2/s).