Johan Lindgren - cp's OEIS frontend

A360477 Numbers whose product of distinct prime factors is greater than or equal to the sum of its prime factors (with repetition).

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Johan Lindgren, Feb 08 2023

nonn

Numbers k where A007947(k) >= A001414(k).
Similar to A359870 but also includes the primes (A000040).
All primes are terms since in that case the product of distinct prime factors and the sum of prime factors are equal.

			45 = 3^2*5 is a term since its product of distinct prime factors 3 * 5 = 15 is greater than its sum of prime factors with multiplicity 3 + 3 + 5 = 11.
48 = 2^4*3 is not a term since its product of distinct prime factors 2 * 3 = 6 is less than its sum of prime factors with multiplicity 2 + 2 + 2 + 2 + 3 = 11.

Cf. A000040, A001414, A007947, A359869, A359870.

q[n_] := Module[{f = FactorInteger[n]}, Times @@ f[[;; , 1]] >= Plus @@ (f[[;; , 1]]*f[[;; , 2]])]; q[1] = True; Select[Range[100], q] (* Amiram Eldar, Feb 08 2023 *)

A359870 Numbers whose product of distinct prime factors is greater than the sum of its prime factors (with repetition).

Original entry on oeis.org

1, 6, 10, 14, 15, 20, 21, 22, 26, 28, 30, 33, 34, 35, 38, 39, 42, 44, 45, 46, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 99, 102, 104, 105, 106, 110, 111, 114, 115, 116, 117

Offset: 1

Johan Lindgren, Jan 16 2023

nonn

Numbers k where A007947(k) > A001414(k).
No term is prime since in that case the product of distinct prime factors and the sum of prime factors are equal.
Composite squarefree numbers (A120944) form a subsequence, so squarefree semiprimes (A006881) also. - Bernard Schott, Feb 22 2023

			45 = 3^2*5 is a term since its product of distinct prime factors 3 * 5 = 15 is greater than its sum of prime factors with multiplicity 3 + 3 + 5 = 11.
48 = 2^4*3 is not a term since its product of distinct prime factors 2 * 3 = 6 is less than its sum of prime factors with multiplicity 2 + 2 + 2 + 2 + 3 = 11.

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

Cf. A001414, A007947, A359869.

Cf. A006881, A120944.

filter:= proc(n) local F,t;
  F:= ifactors(n)[2];
  mul(t[1],t=F) > add(t[1]*t[2],t=F);
end proc:
select(f, [$1..1000]); # Robert Israel, Feb 07 2023

q[n_] := Module[{f = FactorInteger[n]}, Times @@ f[[;; , 1]] > Plus @@ (f[[;; , 1]]*f[[;; , 2]])]; q[1] = True; Select[Range[120], q] (* Amiram Eldar, Jan 16 2023 *)

isok(n)={my(f=factor(n)); vecprod(f[,1]) > sum(i=1, #f~, f[i,1]*f[i,2])} \\ Andrew Howroyd, Jan 16 2023

A359869 Numbers whose product of distinct prime factors is less than the sum of its prime factors (with repetition).

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 24, 25, 27, 32, 36, 40, 48, 49, 50, 54, 64, 72, 80, 81, 96, 98, 100, 108, 112, 121, 125, 128, 144, 160, 162, 169, 192, 196, 200, 216, 224, 225, 242, 243, 250, 256, 288, 289, 320, 324, 338, 343, 361, 375, 384, 392, 400, 405, 432, 448, 484, 486, 500

Offset: 1

Johan Lindgren, Jan 16 2023

nonn

Numbers n where A007947(n) < A001414(n).

			12 = 2^2*3 is a term since its product of distinct prime factors 2 * 3 = 6 is less than its sum of prime factors with multiplicity 2 + 2 + 3 = 7.
45 = 3^2*5 is not a term since its product of distinct prime factors 3 * 5 = 15 is greater than its sum of prime factors with multiplicity 3 + 3 + 5 = 11.
All prime numbers fail as terms since the product of distinct prime factors is equal to sum of prime factors.

Cf. A001414, A007947.

q[n_] := Module[{f = FactorInteger[n]}, Times @@ f[[;; , 1]] < Plus @@ (f[[;; , 1]]*f[[;; , 2]])]; Select[Range[500], q] (* Amiram Eldar, Jan 16 2023 *)

isok(n)={my(f=factor(n)); vecprod(f[, 1]) < sum(i=1, #f~, f[i, 1]*f[i, 2])} \\ Andrew Howroyd, Jan 16 2023

A353124 Numbers k which have a record number of non-divisors < k (i.e., A049820(k)).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 22, 23, 25, 27, 29, 31, 34, 35, 37, 41, 43, 46, 47, 49, 51, 53, 57, 58, 59, 61, 65, 67, 71, 73, 77, 79, 82, 83, 86, 87, 89, 93, 94, 95, 97, 101, 103, 106, 107, 109, 113, 118, 119, 121, 123, 125, 127, 131, 134, 137, 139, 142, 143

Offset: 1

Johan Lindgren, Apr 24 2022

nonn

From Jon E. Schoenfield, Apr 30 2022: (Start)
This sequence includes all noncomposite numbers, the squares of all odd primes, and the cube of every odd prime p such that p^3 - 2 is composite.
It also includes every number k of the form p*q, with p and q distinct primes, such that k-2 is composite and k-1 is neither a prime nor the square of a prime.
In general, it includes every number k such that tau(k-j) > tau(k) - j for each j in 1..tau(k)-1.
Terms with larger numbers of divisors occur less frequently. The first terms with 0, 1, 2, 3, and 4 distinct prime factors are 1, 3, 22, 2110, and 17585778, respectively (each of which is squarefree). What is the first term with 5 distinct prime factors?
(End)

Johan Lindgren, Table of n, a(n) for n = 1..10000

Cf. A049820, A173540.

s = {}; fm = -1; Do[f = n - DivisorSigma[0, n]; If[f > fm, fm = f; AppendTo[s, n]], {n, 1, 120}]; s (* Amiram Eldar, Apr 25 2022 *)

f(n) = n - numdiv(n); \\ A049820
lista(nn) = {my(m=-oo, list=List(), fn); for (n=1, nn, if ((fn=f(n)) > m, listput(list, n); m = fn;);); Vec(list);} \\ Michel Marcus, Apr 25 2022

A352418 Decimal expansion of the sum of the reciprocals of the highly composite numbers.

Original entry on oeis.org

2, 1, 3, 2, 8, 7, 2, 8, 2, 3, 5, 2, 5, 8, 1, 2, 0, 1, 0, 3, 4, 3, 8, 7, 8, 4, 1, 1, 8, 3, 0, 5, 1, 8, 7, 7, 9, 0, 7, 2, 8, 8, 8, 0, 6, 8, 9, 5, 9, 3, 3, 7, 0, 5, 4, 3, 2, 0, 3, 3, 0, 8, 7, 4, 3, 7, 3, 7, 8, 5, 9, 9, 2, 4, 0, 9, 3, 3, 7, 7, 0, 9, 2, 1, 7, 9, 9, 0, 3, 0, 6, 5, 3, 2, 5, 6, 1, 1, 9, 2, 0, 0, 2, 7, 7, 0, 4

Offset: 1

Johan Lindgren, Mar 15 2022

			1/1 + 1/2 + 1/4 + 1/6 + 1/12 + 1/24 + 1/36 + 1/48 + ... =
2.13287282352581201034387841183051877907288806895933...

Johan Lindgren, Table of n, a(n) for n = 1..1100

Cf. A002182.

Equals Sum_{n>=1} 1/A002182(n).

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User: Johan Lindgren

A360477 Numbers whose product of distinct prime factors is greater than or equal to the sum of its prime factors (with repetition).

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A359870 Numbers whose product of distinct prime factors is greater than the sum of its prime factors (with repetition).

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A359869 Numbers whose product of distinct prime factors is less than the sum of its prime factors (with repetition).

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PARI

A353124 Numbers k which have a record number of non-divisors < k (i.e., A049820(k)).

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Keywords

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Mathematica

PARI

A352418 Decimal expansion of the sum of the reciprocals of the highly composite numbers.

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