A082663 -id:A082663 - cp's OEIS frontend

A082683 Smaller of the two prime numbers whose product is A082663(n).

3, 5, 7, 7, 11, 11, 11, 13, 13, 13, 17, 17, 19, 17, 17, 19, 19, 23, 19, 23, 23, 29, 23, 23, 29, 31, 29, 29, 31, 31, 29, 31, 37, 29, 37, 31, 37, 41, 31, 31, 41, 37, 43, 41, 37, 37, 43, 41, 37, 47, 41, 43, 43, 37, 37, 41, 47, 47, 43, 41, 41

Offset: 1

Naohiro Nomoto, May 19 2003

Cf. A082684.

A082684 Larger of the two prime numbers whose product is A082663(n).

Original entry on oeis.org

5, 7, 11, 13, 13, 17, 19, 17, 19, 23, 19, 23, 23, 29, 31, 29, 31, 29, 37, 31, 37, 31, 41, 43, 37, 37, 41, 43, 41, 43, 47, 47, 41, 53, 43, 53, 47, 43, 59, 61, 47, 53, 47, 53, 59, 61, 53, 59, 67, 53, 61, 59, 61, 71, 73, 67, 59, 61, 67, 71

Offset: 1

Naohiro Nomoto, May 19 2003

Cf. A082683.

A229964 Number of pairs of integers q1, q2 with 1 < q1 < q2 < n such that if we randomly pick an integer in {1, ..., n}, the event of being divisible by q1 is independent of being divisible by q2.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 3, 2, 1, 0, 4, 0, 5, 1, 3, 0, 8, 0, 4, 3, 4, 0, 10, 0, 7, 3, 5, 2, 9, 0, 6, 4, 9, 0, 13, 0, 12, 6, 6, 0, 16, 0, 9, 6, 9, 0, 14, 1, 12, 3, 8, 0, 25, 0, 12, 10, 11, 4, 17, 0, 12, 7, 17, 0, 25, 0, 14, 12, 14, 2, 21, 0, 21, 5

Offset: 1

Eric M. Schmidt, Oct 04 2013

nonn

			If n = 12, then q1 = 2 and q2 = 5 satisfy the condition as the probability of an integer in {1, ..., 12} being divisible by 2 is 1/2, by 5 is 1/6, and by both 2 and 5 is 1/12.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Rosemary Sullivan and Neil Watling, Independent Divisibility Pairs on the Set of Integers from 1 to N, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A65, 2013.

The n such that a(n) = m for various m are given by: m=0, A166684; m=1, A229965; m=2, A082663; m=3, A229966; m=4, A229967.

from math import lcm
from sympy import divisors
def A229964(n): return sum(1 for d in divisors(n,generator=True) for e in range(d+1,n) if 1Chai Wah Wu, Aug 09 2024

def A229964(n) : return sum(sum(dprob(q1, n) * dprob(q2, n) == dprob(lcm(q1,q2), n) for q2 in range(q1+1, n)) for q1 in n.divisors() if q1 not in [1,n])
def dprob(q, n) : return (n // q)/n

A090196 Odd integers with two divisors a, b such that a < b <= 2a.

Original entry on oeis.org

15, 35, 45, 63, 75, 77, 91, 99, 105, 117, 135, 143, 153, 165, 175, 187, 189, 195, 209, 221, 225, 231, 245, 247, 255, 273, 285, 297, 299, 315, 323, 325, 345, 351, 357, 375, 385, 391, 399, 405, 425, 429, 435, 437, 441, 455, 459, 465, 475, 483, 493, 495, 513, 525, 527, 539, 551, 555

Offset: 1

Steven Finch, Jan 22 2004

nonn

Clearly all even integers have two such divisors a, b. Consider the set S of all integers satisfying this property. Maier & Tenenbaum proved Erdős' conjecture that S has asymptotic density 1.
A244579 and the present sequences are complements in the sequence of odd numbers. - Hartmut F. W. Hoft, Dec 10 2016
From Omar E. Pol, Jan 10 2017: (Start)
Odd numbers k with the property that the number of parts in the symmetric representation of sigma(k) is not equal to the number of divisors of k.
Odd numbers that are not in A244579.
All terms are composites. (End)
The subsequence of semiprimes is A082663. - Bernard Schott, Apr 17 2022

R. R. Hall and G. Tenenbaum, Divisors, Cambridge Univ. Press, 1988, pp. 95-99.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
H. Maier and G. Tenenbaum, On the set of divisors of an integer, Invent. Math. 76 (1984) 121-128.

Cf. A000005, A000203, A071562, A082663, A090196, A196020, A236104, A237270, A237271, A237590, A237593, A244579.

Select[Range[1, 999, 2], (Divisors[#] /. {_, a_, _, b_, _} /; a < b <= 2a -> True) === True&] (* Jean-François Alcover, Nov 05 2016 *)

is(n)=my(d=divisors(n));for(i=2,#d\2+1,if(d[i]<2*d[i-1], return(n%2))); 0 \\ Charles R Greathouse IV, Jun 20 2013

a(n) ~ 2n. - Charles R Greathouse IV, Jun 20 2013

Corrected by Charles R Greathouse IV, Jul 23 2012

Corrected by Jean-François Alcover and Charles R Greathouse IV, Jun 20 2013

A186777 Solutions x of the equation A064380(x)-A000010(x)=1 in integers x>=2.

Original entry on oeis.org

4, 6, 10, 15, 35, 77, 91, 143, 187, 209, 221, 247, 299, 323, 391, 437, 493, 527, 551, 589, 667, 703, 713, 851, 899, 943, 989, 1073, 1147, 1189, 1247, 1271, 1333, 1363, 1457, 1517, 1537, 1591, 1643, 1739, 1763, 1829, 1891, 1927, 1961, 2021, 2173, 2183, 2257, 2279

Offset: 1

Vladimir Shevelev, Feb 26 2011

nonn

The defining equation has infinitely many solutions.

Equals {4, 6, 10} UNION A082663. - Eric M. Schmidt, Oct 04 2013

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric M. Schmidt, The Near Equivalence of A082663 and A186777

Cf. A000010, A064380, A082663, A176472, A176509.

More terms from Amiram Eldar, Sep 14 2019

A229966 Numbers n such that A229964(n) = 3.

Original entry on oeis.org

12, 14, 22, 27, 33, 57, 85, 161, 203, 533, 689, 901, 1121, 1633, 2581, 4181, 5513, 5633, 7439, 10561, 18023, 18881, 20833, 21389, 23941, 25043, 28421, 32033, 37733, 48641, 58241, 64643, 66901, 77423, 80033, 84001, 90133, 106439, 116821, 119201, 149189, 155041

Offset: 1

Eric M. Schmidt, Oct 04 2013

nonn

Equals {12, 14, 22, 27, 57} UNION {pq | p, q prime, q = 3p+2 or (p >= 5 and q = 4p+1)}.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Rosemary Sullivan and Neil Watling, Independent Divisibility Pairs on the Set of Integers from 1 to N, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A65, 2013.

Cf. A166684, A082663, A229965, A229967, A023208, A023212.

[p * (3*p+2) for p in prime_range(10000) if (3*p+2).is_prime()] + [p * (4*p+1) for p in prime_range(5, 10000) if (4*p+1).is_prime()] + [12, 14, 22, 27, 57]

A229967 Numbers n such that A229964(n) = 4.

Original entry on oeis.org

18, 26, 28, 39, 65, 115, 119, 133, 319, 341, 377, 403, 481, 517, 629, 697, 731, 779, 799, 817, 893, 1007, 1207, 1219, 1357, 1403, 1541, 1769, 1943, 2059, 2077, 2117, 2201, 2263, 2291, 2407, 2449, 2573, 2759, 2923, 3071, 3293, 3589, 3649, 3737, 3811, 3827, 3959

Offset: 1

Eric M. Schmidt, Oct 04 2013

nonn

Equals {18, 26, 28, 39} UNION {pq | p, q prime, p >= 5 and (2p+3 <= q <= 3p-2 or (p == 2 (mod 3) and q = 4p+3))}.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Rosemary Sullivan and Neil Watling, Independent Divisibility Pairs on the Set of Integers from 1 to N, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A65, 2013.

Cf. A166684, A082663, A229965, A229966.

sum(([p * q for q in prime_range(2*p+3, 3*p-1)] for p in prime_range(5, 10000)), []) + [p * (4*p + 3) for p in prime_range(5, 10000) if (4*p+3).is_prime() and p%3==2] + [18, 26, 28, 39]

A350803 Numbers k with at least one partition into two parts (s,t), s<=t such that t | s*k.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 86, 88, 90, 91, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116

Offset: 1

Wesley Ivan Hurt, Jan 16 2022

nonn

From Bernard Schott, Jan 22 2022: (Start)
A299174 is a subsequence because, if k = 2*u, we have s=t=u, s<=t, and u | u*k.
A082663 is another subsequence because, if k = p*q with p < q < 2p, then with s = k-p^2 = p*(q-p) and t = p^2, we have s <= t and p^2 | p*(q-p) * (pq).
It seems that A090196 is the subsequence of odd terms. (End)
gcd(s, t) > 1 where s and t and k > 2 are as in name. - David A. Corneth, Jan 22 2022
Numbers k such that k^2 has at least one divisor d with k/2 <= d < k. - Robert Israel, Jan 08 2025

			15 is in the sequence since 15 = 6+9 where 9 | 6*15 = 90.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions

Cf. A338021, A350804 (exactly one).
Subsequences: A082663, A299174.
Cf. A090196.

filter:= proc(n) nops(select(t -> t >= n/2 and t < n, numtheory:-divisors(n^2)))>=1 end proc:
select(filter, [$1..300]); # Robert Israel, Jan 08 2025

f(n) = sum(s=1, n\2, !((s*n)%(n-s))); \\ A338021
isok(k) = f(k) >= 1; \\ Michel Marcus, Jan 17 2022

A229965 Numbers n such that A229964(n) = 1.

Original entry on oeis.org

6, 8, 10, 16, 21, 55, 253, 1081, 1711, 3403, 5671, 13861, 15931, 25651, 34453, 60031, 64261, 73153, 108811, 114481, 126253, 158203, 171991, 258121, 351541, 371953, 392941, 482653, 518671, 703891, 822403, 853471, 869221, 933661, 1034641, 1104841, 1159003

Offset: 1

Eric M. Schmidt, Oct 04 2013

nonn

Equals {6, 8, 16} UNION A156592.

Rosemary Sullivan and Neil Watling, Independent Divisibility Pairs on the Set of Integers from 1 to N, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A65, 2013.

Cf. A166684, A082663, A229966, A229967.

A338486 Numbers n whose symmetric representation of sigma(n) consists of 3 regions with maximum width 2.

Original entry on oeis.org

15, 35, 45, 70, 77, 91, 110, 130, 135, 143, 154, 170, 182, 187, 190, 209, 221, 225, 238, 247, 266, 286, 299, 322, 323, 350, 374, 391, 405, 418, 437, 442, 493, 494, 506, 527, 550, 551, 572, 589, 598, 638, 646, 650, 667, 682, 703, 713, 748, 754, 782, 806, 814, 836, 850

Offset: 1

Hartmut F. W. Hoft, Oct 30 2020

nonn

This sequence is a subsequence of A279102. The definition of the sequence excludes squares of primes, A001248, since the 3 regions of their symmetric representation of sigma have width 1 (first column in the irregular triangle of A247687).
Table of numbers in this sequence arranged by the number of prime factors, counting multiplicities:
     2     3     4      5      6      7  ...
  ------------------------------------------
    15    45   135    405   1215   3645
    35    70   225   1125   5625    ...
    77   110   350   1750   8750 744795
    91   130   550   2584    ...    ...
   143   154   572   2750  85455
   187   170   650   3128    ...
   209   182   748   3250
   221   190   836   3496
   247   238   850   3944
   299   266   884   4216
   ...   ...   ...    ...
              1035   9585
               ...    ...
The numbers in the first row of the table above are b(k) = 5*3^k, k>=1, (see A005030) so that infinitely many odd numbers occur outside of the first column. The central region of the symmetric representation of sigma(b(k)) contains 2*k-1 separate contiguous sections consisting of sequences of entire legs of width 2, k>=1 (see Lemma 2 in the link).
Conjecture: The combined extent of these sections in sigma(b(k)) is 2*3^(k-1) - 1 = A048473(k-1), k>=1.
Since each number n in the first column and first row has a prime factor of odd exponent a contiguous section of the symmetric representation of sigma(n) centered on the diagonal has width 2. For odd numbers n not in the first row or column in which all prime factors have even powers, such as 225 and 5625 in the second row, a contiguous section of the symmetric representation of sigma(n) centered on the diagonal has width 1 (see Lemma 1 in the link).
For each k>=3 and every prime p such that b(k-1) < 2*p < 4*b(k-2), the odd number p*b(k-1) is in the column of b(k). The two inequalities are equivalent to b(k-1) <= row(p*b(k-1)) < 2*b(k-1) ensuring that the symmetric representation of sigma(p*b(k-1)) consists of 3 regions.
45 is the only odd number in its column (see Lemma 3 in the link).
Since the factors of n = p*q satisfy 2 < p < q < 2*p the first column in the table above is a subsequence of A082663 and of A087718 (see Lemma 4 in the link). Each of the two outer regions consists of a single leg of width 1 and length (1 + p*q)/2. The center region of size p+q consists of two subparts (see A196020 & A280851) of width 1 of sizes 2*p-q and 2*q-p, respectively (see Lemma 5 in the link). The table below arranges the first column in the table above according to the length 2*p-q of their single contiguous extent of width 2 in the center region:
     1     3      5      7      9     11     13     15  ...
------------------------------------------------------
    15    35    187    247    143    391   2257    323
    91    77    493    589    221   1363   3139    437
   703   209    943   2479    551   2911   6649    713
  1891   299   1537   3397    851   3901    ...   1247
  2701   527   4183   8509   1643   6313          1457
  ...    ...    ...    ...    ...    ...          ....
A129521: p*q satisfies 2*p - q = 1 (excluding A129521(1)=6)
A226755: p*q satisfies 2*p - q = 3 (excluding A226755(1)=9)
Sequences with larger differences 2*p - q are not in OEIS.

			a(6) = 91 = 7*13 is in the sequence and in the 2-column of the first table since 1 < 2 < 7 < 13 = row(91) representing the 4 odd divisors 1 - 91 - 7 - 13 (see A237048) results in the following pattern for the widths of the legs (see A249223): 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2 for 3 regions with width not exceeding 2. It also is in the 1-column of the second table since it has a single area of width 2 which is 1 unit long.
a(29) = 405 = 5*3^4 is in the sequence and in the 5-column of the first table since 1 < 2 < 3 < 5 < 6 < 9 < 10 < 15 < 18 < 27 = row(405) representing the 10 odd divisors 1 - 405 - 3 - 5 - 135 - 9 - 81 - 15 - 45 - 27 results in the following pattern for the widths of the legs: 1, 0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2 for 3 regions with width not exceeding 2, and 7 = 2*4 - 1 sections of width 2 in the central region.
a(35) = 506 = 2*11*23 is in the sequence since positions 1 < 4 < 11 < 23 < row(506) = 31 representing the 4 odd divisors 1 - 253 - 11 - 23 results in the following pattern for the widths of the legs: 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2 for 3 regions with width not exceeding 2, with the two outer regions consisting of 3 legs of width 1, and a single area of width 2 in the central region.

Hartmut F. W. Hoft, proofs for quoted lemmas

Cf. A001248, A005030, A048473, A082663, A087718, A129521, A196020, A226755, A235791, A237048, A237270, A237271, A237591, A237593, A247687, A249223, A279102, A280107, A280851.

(* Functions path and a237270 are defined in A237270 *)
maxDiagonalLength[n_] := Max[Map[#[[1]]-#[[2]]&, Transpose[{Drop[Drop[path[n], 1], -1], path[n-1]}]]]
a338486[m_, n_] := Module[{r, list={}, k}, For[k=m, k<=n, k++, r=a237270[k]; If[Length[r]== 3 && maxDiagonalLength[k]==2,AppendTo[list, k]]]; list]
a338486[1, 850]

cp's OEIS Frontend

A082683 Smaller of the two prime numbers whose product is A082663(n).

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A082684 Larger of the two prime numbers whose product is A082663(n).

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A229964 Number of pairs of integers q1, q2 with 1 < q1 < q2 < n such that if we randomly pick an integer in {1, ..., n}, the event of being divisible by q1 is independent of being divisible by q2.

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A090196 Odd integers with two divisors a, b such that a < b <= 2a.

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A186777 Solutions x of the equation A064380(x)-A000010(x)=1 in integers x>=2.

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A229966 Numbers n such that A229964(n) = 3.

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A229967 Numbers n such that A229964(n) = 4.

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A350803 Numbers k with at least one partition into two parts (s,t), s<=t such that t | s*k.

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A229965 Numbers n such that A229964(n) = 1.

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A338486 Numbers n whose symmetric representation of sigma(n) consists of 3 regions with maximum width 2.