A227993 -id:A227993 - cp's OEIS frontend

A138808 Number of integer pairs (x,y), x > 0, y > 0, such that x <= p, y <= q for any factorization n = p*q.

1, 3, 5, 8, 9, 14, 13, 20, 21, 26, 21, 35, 25, 38, 41, 48, 33, 57, 37, 64, 61, 62, 45, 84, 65, 74, 81, 96, 57, 109, 61, 112, 101, 98, 101, 138, 73, 110, 121, 151, 81, 160, 85, 160, 161, 134, 93, 196, 133, 185, 161, 192, 105, 216, 173, 223, 181, 170, 117, 258

Offset: 1

Jonas Wallgren, May 16 2008

nonn

Conjecture: the row sums of the plane partitions A010766 are upper bounds. - R. J. Mathar, Aug 06 2008
a(n) is divisible by n iff n=1 or n belongs to A227993. - Rémy Sigrist, Mar 06 2017
a(n) >= 2*n - 1, with equality iff n is not composite. - Rémy Sigrist, Mar 12 2017

			a(8) = these 20 marked *'s:
-|12345678
-+--------
1|********
2|****
3|**
4|**
5|*
6|*
7|*
8|*

Paul Tek, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms

Cf. A227993.

a(n) = my(ar=0, pw=0); fordiv(n, w, ar=ar+(w-pw)*n/w; pw=w); return (ar) \\ Paul Tek, Mar 21 2015

a(n) = n*(m - Sum_{k=1..m-1} d(k)/d(k+1)), where d(1) < d(2) < ... < d(m) denote the divisors of n. - Rémy Sigrist, Mar 06 2017

More terms from Paul Tek, Mar 21 2015

Typo in name corrected by Rémy Sigrist, Mar 05 2017

A255586 Composite k such that Sum_{i=1..t-1} d(i+1)/d(i) is an integer, where d(1), ..., d(t) are the divisors of k in ascending order.

Original entry on oeis.org

4, 8, 9, 16, 18, 25, 27, 32, 48, 49, 50, 64, 81, 98, 108, 121, 125, 128, 162, 169, 242, 243, 256, 289, 338, 343, 361, 375, 512, 529, 578, 625, 722, 729, 841, 961, 1024, 1029, 1058, 1250, 1331, 1369, 1458, 1681, 1682, 1849, 1920, 1922, 2048, 2187, 2197, 2209

Offset: 1

Michel Lagneau, Feb 27 2015

nonn

The sequence is infinite because the powers of 2 (A000079) are in the sequence.
The prime powers with even exponents (A056798) are in the sequence.
The cubes of primes (A030078) are in the sequence.
The numbers of the form 2p^2 (A079704) with p prime are in the sequence.
The corresponding integers are 4, 6, 6, 8, 9, 10, 9, 10, 14, 14, 11, 12, 12, 13, 17, 22, 15, 14, 16, 26, 17, 15, 16, 34, 19, ...

			18 is in the sequence because the divisors of 18 are {1, 2, 3, 6, 9, 18} and 2/1 + 3/2 + 6/3 + 9/6 + 18/9 = 9 is an integer.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000079, A030078, A079704, A227993, A255585.

lst={};Do[s=0;Do[s=s+Divisors[n][[i+1]]/Divisors[n][[i]],{i,1,Length[Divisors[n]]-1}];If[IntegerQ[s]&&!PrimeQ[n],AppendTo[lst,n]],{n,2300}];lst
Select[Range[2210],CompositeQ[#]&&IntegerQ[Total[#[[2]]/#[[1]]&/@Partition[ Divisors[ #],2,1]]]&] (* Harvey P. Dale, Jul 09 2019 *)

A255576 Integers k such that Sum_{i=1..t-1} d(i)/d(i+1) is prime, where d(1), ..., d(t) are the divisors of k in ascending order.

Original entry on oeis.org

16, 64, 729, 1024, 1536, 6250, 9375, 16384, 19683, 39366, 1179648, 4194304, 6770688, 9765625, 14348907, 29229255, 39062500, 67108864, 125000000, 128472708, 335544320, 1337982976, 10460353203

Offset: 1

Michel Lagneau, Feb 25 2015

The corresponding primes are 2, 3, 2, 5, 13, 5, 5, 7, 3, 11, 41, 11, 89, 2, 5, 37, 19, 13, 53, 37, ...
a(n) is a power of 2 for n = 1, 2, 4, 8, 12, 18, ... with the corresponding primes 2, 3, 5, 7, 11, 13, ...
a(n) is a perfect square for n = 1, 2, 3, 4, 8, 12, 14, 17, 18, ... with the corresponding primes 2, 3, 2, 5, 7, 11, 2, 19, 13, ...

			64 is in the sequence because the divisors of 64 are {1, 2, 4, 8, 16, 32, 64} and 1/2 + 2/4 + 4/8 + 8/16 + 16/32 + 32/64 = 3 is prime.

Subsequence of A227993.

Do[s=0;Do[s=s+Divisors[n][[i]]/Divisors[n][[i+1]],{i,1,Length[Divisors[n]]-1}];If[PrimeQ[s]&&!PrimeQ[n],Print[n]],{n,10^6}]
Select[Range[40000],PrimeQ[Total[#[[1]]/#[[2]]&/@Partition[ Divisors[ #],2,1]]]&] (* The program generates the first 10 terms of the sequence. To generate more, increase the Range constant. *) (* Harvey P. Dale, Feb 06 2022 *)

from sympy import isprime, divisors
from fractions import Fraction
def ok(n):
    divs = divisors(n)
    f = sum(Fraction(dn, dd) for dn, dd in zip(divs[:-1], divs[1:]))
    return f.denominator == 1 and isprime(f.numerator)
print([k for k in range(1, 40000) if ok(k)]) # Michael S. Branicky, Feb 06 2022

a(20) inserted and a(22)-a(23) from Michael S. Branicky, Feb 06 2022 using A227993

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A138808 Number of integer pairs (x,y), x > 0, y > 0, such that x <= p, y <= q for any factorization n = p*q.

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A255586 Composite k such that Sum_{i=1..t-1} d(i+1)/d(i) is an integer, where d(1), ..., d(t) are the divisors of k in ascending order.

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A255576 Integers k such that Sum_{i=1..t-1} d(i)/d(i+1) is prime, where d(1), ..., d(t) are the divisors of k in ascending order.

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