A058080 -id:A058080 - cp's OEIS frontend

A086473 Numbers in A058080 having only one factor pair (both distinct from 1) that add up to a sum belonging to A014092.

18, 24, 28, 50, 52, 54, 76, 92, 96, 98, 100, 112, 124, 140, 144, 148, 152, 160, 172, 176, 188, 192, 208, 212, 216, 220, 228, 232, 242, 244, 260, 266, 268, 280, 288, 290, 292, 304, 308, 316, 332, 336, 338, 344, 356, 374, 384, 388, 392, 400, 412, 434, 436, 448

Offset: 1

Lekraj Beedassy, Sep 09 2003

nonn

Related to the "Impossible Problem" of Martin Gardner.

			A058080(3)=18 is in the sequence because 18=2*9=3*6 and only 2+9=11 belongs to A014092. However, A058080(7)=30 is not in the sequence because 30=2*15=3*10=5*6 with both 2+15 and 5+6 belonging to A014092.

M. Criton, "Le Problème Impossible", Jeux Mathématiques, Tangente Hors Série No. 20 Pole Paris 2004.

N. Hobson, Nick's Mathematical Miscellany, Two Logicians
Dr. Rob, The Math Forum, Two Mathematicians Problem

Cf. A058080, A014092.

Corrected and extended by Ray Chandler, Sep 21 2003

A000430 Primes and squares of primes.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223

Offset: 1

R. Muller

Also numbers n such that the product of proper divisors is < n.
See A050216 for lengths of blocks of consecutive primes. - Reinhard Zumkeller, Sep 23 2011
Numbers q > 1 such that d(q) < 4. Numbers k such that the number of ways of writing k = m + t is equal to the number of ways of writing k = r*s, where m|t and r|s. - Juri-Stepan Gerasimov, Oct 14 2017
Called multiplicatively deficient numbers by Chau (2004). - Amiram Eldar, Jun 29 2022

F. Smarandache, Definitions solved and unsolved problems, conjectures and theorems in number theory and geometry, edited by M. Perez, Xiquan Publishing House 2000
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.

T. D. Noe, Table of n, a(n) for n = 1..1000
William Chau, The tau, sigma, rho functions, and some related numbers, Pi Mu Epsilon Journal, Vol. 11, No. 10 (Spring 2004), pp. 519-534; entire issue.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.

Union of A000040 and A001248.

Cf. A007422, A010051, A010055, A032741, A046951, A050216, A056595, A058080, A064911, A084110, A084114, A293575.

a000430 n = a000430_list !! (n-1)
a000430_list = m a000040_list a001248_list where
   m (x:xs) (y:ys) | x < y = x : m xs (y:ys)
                   | x > y = y : m (x:xs) ys
-- Reinhard Zumkeller, Sep 23 2011

nn = 223; t = Union[Prime[Range[PrimePi[nn]]], Prime[Range[PrimePi[Sqrt[nn]]]]^2] (* T. D. Noe, Apr 11 2011 *)
Module[{upto=250,prs},prs=Prime[Range[PrimePi[upto]]];Select[Join[ prs,prs^2], #<=upto&]]//Sort (* Harvey P. Dale, Oct 08 2016 *)

is(n)=isprime(n) || (issquare(n,&n) && isprime(n)) \\ Charles R Greathouse IV, Sep 04 2013

from math import isqrt
from sympy import primepi
def A000430(n):
    def f(x): return n+x-primepi(x)-primepi(isqrt(x))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return int(m) # Chai Wah Wu, Aug 09 2024

A084114(a(n)) = 0, see also A084110. - Reinhard Zumkeller, May 12 2003
A109810(a(n)) = 2. - Reinhard Zumkeller, May 24 2010
A010051(a(n)) + A010055(a(n))*A064911(a(n)) = 1;
A056595(a(n)) = 1. - Reinhard Zumkeller, Aug 15 2011
A032741(a(n)) = A046951(a(n)); A293575(a(n)) = 0. - Juri-Stepan Gerasimov, Oct 14 2017
The number of terms not exceeding x is N(x) ~ (x + 2*sqrt(x))/log(x) (Chau, 2004). - Amiram Eldar, Jun 29 2022

A007964 Numbers k such that product of proper divisors of k is <= k; i.e., product of divisors of k is <= k^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106

Offset: 1

R. Muller

Numbers which are the product of up to two primes (not necessarily distinct) or the cube of a prime. Alternatively, numbers having prime decomposition p*q, where q either is distinct from p or equals p^k for 0 <= k <= 2.
Corresponds to numbers having at most four divisors. (For numbers with exactly four divisors see A030513.) - Lekraj Beedassy, Sep 23 2003
For n > 3: numbers that can occur as fourth divisors; union of A000040, A001248, A006881 and A030078. - Reinhard Zumkeller, May 15 2006

Liu Hongyan and Zhang Wenpeng, On the simple numbers and the mean value properties, Smarandache Notions (Book Series, Vol. 14), American Research Press, 2004; pp. 171-175.

Amiram Eldar, Table of n, a(n) for n = 1..10000
F. Smarandache, Only Problems, Not Solutions! (listing the sequence as the "simple numbers" in Unsolved Problem: 23 p.23).

Cf. A007955, A058080.

Select[Range[100], DivisorSigma[0, #] < 5 &] (* Amiram Eldar, Apr 30 2020 *)

is(n)=numdiv(n)<5 \\ Charles R Greathouse IV, Sep 16 2015

Description corrected by Henry Bottomley, Nov 24 2000

A080256 Sum of numbers of distinct and of all prime factors of n.

Original entry on oeis.org

0, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 5, 2, 5, 4, 4, 2, 6, 3, 4, 4, 5, 2, 6, 2, 6, 4, 4, 4, 6, 2, 4, 4, 6, 2, 6, 2, 5, 5, 4, 2, 7, 3, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 7, 2, 4, 5, 7, 4, 6, 2, 5, 4, 6, 2, 7, 2, 4, 5, 5, 4, 6, 2, 7, 5, 4, 2, 7, 4, 4, 4, 6, 2, 7, 4, 5, 4, 4, 4, 8, 2, 5, 5, 6, 2, 6, 2, 6, 6

Offset: 1

Reinhard Zumkeller, Feb 10 2003

a(n) = 2 iff n is prime, A000040; a(n) > 2 iff n is composite, A002808; a(n) <= 3 iff n is prime or square of prime, A000430; a(n) = 3 iff n is square of prime, A001248; a(A080257(n)) > 3;

a(n) <= 4 iff product of proper divisors <= n^2, A007964; a(n) = 4 iff n has four divisors, A030513; a(n) > 4 iff product of proper divisors > n^2, A058080; a(A064598(n)) <= 5; a(A080258(n)) = 5.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000040, A000430, A002808, A001248, A007964, A058080, A064598, A080257, A080258.

Cf. A001221, A001222, A046660, A077761, A083342.

f[n_] := Plus @@ (Last /@ FactorInteger[n] + 1); Table[ f[n], {n, 105}] (* Robert G. Wilson v, Aug 03 2005 *)

a(n) = {my(f = factor(n)); omega(f) + bigomega(f);} \\ Amiram Eldar, Sep 28 2023

a(n) = Omega(n) + omega(n) = A001221(n) + A001222(n).
Additive with a(p^e) = e + 1.
Sum_{k=1..n} a(k) = 2 * n * log(log(n)) + c * n + O(n/log(n)), where c = A077761 + A083342 = 1.29615109474508069537... . - Amiram Eldar, Sep 28 2023

A080258 Either 4th power of a prime, or product of a prime and the square of a different prime.

Original entry on oeis.org

12, 16, 18, 20, 28, 44, 45, 50, 52, 63, 68, 75, 76, 81, 92, 98, 99, 116, 117, 124, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 242, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 356, 363, 369, 387, 388, 404, 412, 423, 425, 428, 436

Offset: 1

Reinhard Zumkeller, Feb 10 2003

nonn

Union of A030514 and A054753.

Numbers that equal the product of the proper divisors of their proper divisors. - Scott R. Shannon, Jul 04 2021

			81=3*3*3*3 and 50=2*5*5 are terms.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A007964, A058080, A001221, A001222, A002033, A006881, A030078.

Module[{nn=20,p4,sdp},p4=Prime[Range[nn]]^4;sdp=#[[1]] #[[2]]^2&/@ Select[ Tuples[Prime[Range[nn]],2],#[[1]]!=#[[2]]&];Take[Join[p4,sdp]// Union,40]] (* Harvey P. Dale, Jan 31 2021 *)

is(n)=my(f=factor(n)[,2]~); f==[4] || f==[2,1] || f==[1,2] \\ Charles R Greathouse IV, Oct 16 2015

A080256(a(n)) = 5.

A002033(a(n)) = 8. - Juri-Stepan Gerasimov, Sep 26 2009

A327642 Number of partitions of n into divisors of n that are at most sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 4, 1, 5, 4, 6, 1, 19, 1, 8, 6, 25, 1, 37, 1, 36, 8, 12, 1, 169, 6, 14, 10, 64, 1, 247, 1, 81, 12, 18, 8, 1072, 1, 20, 14, 405, 1, 512, 1, 144, 82, 24, 1, 2825, 8, 146, 18, 196, 1, 1000, 12, 743, 20, 30, 1, 19858, 1, 32, 112, 969, 14, 1728, 1, 324, 24, 1105

Offset: 0

Ilya Gutkovskiy, Sep 20 2019

a(n) > n if n is in A058080 Union {0}, and, a(n) <= n if n is in A007964; indeed, a(n) = n only for n = 1. - Bernard Schott, Sep 22 2019

			The divisors of 6 are 1, 2, 3, 6 and sqrt(6) = 2.449..., so the possible partitions are 1+1+1+1+1+1 = 1+1+1+1+2 = 1+1+2+2 = 2+2+2; thus a(6) = 4. - _Bernard Schott_, Sep 22 2019

David A. Corneth, Table of n, a(n) for n = 0..9999 (first 5001 terms from Robert Israel)

Cf. A018818, A210442, A211110, A293813.

[1] cat [#RestrictedPartitions(n,{d:d in Divisors(n)| d le Sqrt(n)}):n in [1..70]]; // Marius A. Burtea, Sep 20 2019

f:= proc(n) local x, t, S;
    S:= 1;
    for t in numtheory:-divisors(n) do
      if t^2 <= n then
        S:= series(S/(1-x^t),x,n+1);
      fi
    od;
    coeff(S,x,n);
end proc:
map(f, [$0..100]); # Robert Israel, Sep 22 2019

a[n_] := SeriesCoefficient[Product[1/(1 - Boole[d <= Sqrt[n]] x^d), {d, Divisors[n]}], {x, 0, n}]; Table[a[n], {n, 0, 70}]

a(n) = [x^n] Product_{d|n, d <= sqrt(n)} 1 / (1 - x^d).
a(p) = 1, where p is prime.
a(p*q) = q+1 if p <= q are primes. - Robert Israel, Sep 22 2019

A086860 Numbers in A086473 corresponding to the unique product of two numbers having the unique sum of A086533.

Original entry on oeis.org

52, 244, 1168, 1776, 4672, 4192, 2608, 724, 8128, 916, 1912, 3328, 15424, 9952, 3352, 3592, 53632, 80128, 36352, 51712, 65152, 5272, 20512, 72832, 22432, 111756, 133888, 84352, 6472, 48448, 26272, 172288, 107392, 37480, 187648, 242496

Offset: 1

Lekraj Beedassy, Sep 12 2003

nonn

Related to Martin Gardner's "Impossible Problem".

a(n) is thus a subsequence of A086473, itself a subsequence of A058080. Consider the mapping f:P->S defined thus: S is the sum of a factor pair (both different from 1) of P, where P is a(n). If S is A086533(n) (a subsequence of A014092), then both f and its inverse are injective (but not onto).

Cf. A086533.

Corrected by Ray Chandler, Oct 23 2003

A333326 Odd numbers k such that k is the product of 2 numbers greater than one, in two or more ways.

Original entry on oeis.org

45, 63, 75, 81, 99, 105, 117, 135, 147, 153, 165, 171, 175, 189, 195, 207, 225, 231, 243, 245, 255, 261, 273, 275, 279, 285, 297, 315, 325, 333, 345, 351, 357, 363, 369, 375, 385, 387, 399, 405, 423, 425, 429, 435, 441, 455, 459, 465, 475

Offset: 1

Dimitris Valianatos, Mar 15 2020

nonn

Odd numbers that are not 1, a prime, the square or cube of a prime, or the product of two primes. - Robert Israel, Mar 19 2020

			For n=2 the a(2) = 63 because 63 = 3 * 21 and 7 * 9.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005408, A058080, A226025.

select(t -> numtheory:-tau(t) >= 5, [seq(i,i=3..1000,2)]); # Robert Israel, Mar 19 2020

Select[2 * Range[250] + 1, DivisorSigma[0, #] > 4 &] (* Amiram Eldar, Mar 15 2020 *)

forcomposite(n=4, 475, if(n%2 == 0, next()); if(numdiv(n) >= 5, print1(n,", "))); \\ Joerg Arndt, Mar 15 2020

A375120 Number of complete binary unordered tree-factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 6, 1, 1, 1, 4, 1, 3, 1, 2, 2, 1, 1, 9, 1, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 9, 1, 1, 2, 6, 1, 3, 1, 2, 1, 3, 1, 15, 1, 1, 2, 2, 1, 3, 1, 9, 2, 1, 1, 9, 1, 1, 1

Offset: 1

Baron Kurt Hannsz, Jul 30 2024

nonn

For prime n, the factorization tree is a single vertex in just one way so that a(n) = 1.
For composite n, the two subtrees at n are a split of n into two factors n = d * (n/d), without order, so that a(n) = Sum_{d|n, 2 <= d <= n/d} a(d)*a(n/d).
a(1) = 1 is by convention, reckoning 1 as having a single empty factorization.
Greg Martin observed: if p is prime then a(p^k) equals the k-th 'half-Catalan number' A000992. - Peter Luschny, Nov 04 2024

			For n = 4, the a(4) = 1 sole factor tree is
     4     4 = 2*2
    / \
   2   2
For n=12, the a(12) = 2 factor trees are
    12          12
   /  \        /  \
  2    6      3    4
      / \         / \
     2   3       2   2
The tree structures are the same but the values are not the same and are therefore distinct factorizations.

Cf. A281119, A292505, A007964 (a(n)=1), A058080 (a(n)>1), A000992.

@cached_function
def a(n):
    if is_prime(n) or n == 1: return 1
    T = [t for t in divisors(n) if 1 < t <= n/t]
    return sum(a(d)*a(n//d) for d in T)
print([a(n) for n in range(1, 88)])  # Peter Luschny, Nov 03 2024

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A086473 Numbers in A058080 having only one factor pair (both distinct from 1) that add up to a sum belonging to A014092.

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A000430 Primes and squares of primes.

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A007964 Numbers k such that product of proper divisors of k is <= k; i.e., product of divisors of k is <= k^2.

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A080256 Sum of numbers of distinct and of all prime factors of n.

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A080258 Either 4th power of a prime, or product of a prime and the square of a different prime.

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A327642 Number of partitions of n into divisors of n that are at most sqrt(n).

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A086860 Numbers in A086473 corresponding to the unique product of two numbers having the unique sum of A086533.

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