A090196 -id:A090196 - cp's OEIS frontend

A244579 Numbers k with the property that the number of parts in the symmetric representation of sigma(k) equals the number of divisors of k.

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 79, 81, 83, 85, 87, 89, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131, 133, 137, 139, 141, 145

Offset: 1

Omar E. Pol, Jul 02 2014

nonn

Numbers n such that A243982(n) = 0.
First differs from A151991 at a(25).
Let n = 2^m * q with m >= 0 and q odd. Let c_n denote the count of regions in the symmetric representation of sigma(n), which is determined by the positions of 1's in the n-th row of A237048. The maximum of c_n occurs when odd and even positions of 1's alternate implying that all regions have width 1, denoted by w_n = 1. When m > 0 then sigma_0(n) > sigma_0(q) and c_n = sigma_0(n) is impossible. Therefore, exactly those odd n with w_n = 1 are in this sequence. Furthermore, since the 1's in A237048 represent the odd divisors of n, their odd-even alternation expresses the property 2*f < g for any two adjacent divisors f < g of odd number n; in other words, this sequence is also the complement of A090196 relative to the odd numbers. This last property permits computations of elements in this sequence faster than with function a244579, which is based on Dyck paths. - Hartmut F. W. Hoft, Oct 11 2015
From Hartmut F. W. Hoft, Dec 06 2016: (Start)
Also, integers n such that for any pair a < b of divisors of n the inequality 2*a < b holds (hence n is odd).
Let 1 = d_1 < ... < d_k = n be all (odd) divisors of n. The property 2*d_i < d_(i+1), for 1 <= i < k, is equivalent for the 1's in the n-th row of A249223 to be in positions 1 = d_1 < 2 < d_2 < 2*d_2 < ... < d_i <2*d_i < d_(i+1) < ... where 2*d_i represents the odd divisor e_i with d_i * e_i = n. In other words, the odd divisors are the number of parts in the symmetric representation of sigma(n). The rightmost 1 in the n-th row occurs in an odd (even) position when k is odd (even).
As a consequence this sequence is also the complement of A090196 in the set of odd numbers. (End)

			9 is in the sequence because the parts of the symmetric representation of sigma(9) are [5, 3, 5] and the divisors of 9 are [1, 3, 9] and in both cases there is the same number of elements: A237271(9) = A000005(9) = 3.
See the link for a diagram of the symmetric representations of sigma for sequence data listed above. The symmetric representations of sigma(a(35)) = sigma(81) = sigma(3^4) consists of 5 regions whose areas are [41, 15, 9, 15, 41] and computed as 41 = (3^4+3^0)/2, 15 = (3^3+3^1)/2, and 9 = 3^2 for the central area. Observe also that the 81st row in triangle A237048 is [ 1 1 1 0 0 1 0 0 1 0 0 0 ] with the 1's in positions 1, 2, 3, 6, and 9. This is the largest count for the symmetric regions of sigma shown in the diagram. - _Hartmut F. W. Hoft_, Oct 11 2015

Hartmut F. W. Hoft, Illustration of the symmetric representations of sigma for sequence data

Cf. A000005, A001227, A005279, A071561, A071562, A090196, A000203, A196020, A236104, A237048, A237270, A237271, A237593, A238443, A238524, A239657, A239929, A239663, A240062, A241558, A241559, A243982, A245092.

(* Function a237270[] is defined in A237270 *)
a244579[m_, n_] := Select[Range[m,n], Length[a237270[#]] == Length[Divisors[#]]&]
a244579[1, 150] (* data *)
(* Hartmut F. W. Hoft, Sep 19 2014 *)
(* alternative function using the divisor property *)
divisorPairsQ[n_] := Module[{d=Divisors[n]}, Select[2*Most[d] - Rest[d], # >= 0&] == {}]
a244579Alt[m_?OddQ, n_] := Select[Range[m, n, 2], divisorPairsQ]
a244579Alt[1, 145] (* data *)
(* Hartmut F. W. Hoft, Oct 11 2015 *)

A237271(a(k)) = A000005(a(k)).

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

A346641 Numbers k with at least one partition into two parts (s,t), s<=t such that t | s*k but no proper divisor of k has this property.

Original entry on oeis.org

2, 15, 35, 63, 77, 91, 99, 117, 143, 153, 187, 209, 221, 247, 299, 323, 325, 357, 391, 399, 425, 437, 475, 483, 493, 513, 527, 551, 575, 589, 609, 621, 651, 667, 703, 713, 725, 759, 775, 777, 783, 837, 851, 861, 899, 925, 943, 957, 989, 999, 1023, 1025, 1073, 1075

Offset: 1

David A. Corneth, Jan 22 2022

nonn

Primitive subsequence of A350803.

Odd terms form primitive subsequence of A090196. - Bernard Schott, Jan 23 2022

			k = 15 = 6 + 9 = s + t is in the sequence (t = 9 | 6*15 = 90 = s*k) but no proper divisor of 15 has this property.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A350803.

Subsequence: A082663.

upto(n) = { my(v = vector(n, i, -1)); for(i = 1, n, if(v[i] == -1, if(isA350803(i), v[i] = 1; for(j = 2, n\i, v[i*j] = 0; ) ) ) ); select(x->x==1, v, 1) }
isA350803(n) = { for(i = 1, n\2, if((n*i)%(n-i) == 0, return(1) ) ); return(0) }

cp's OEIS Frontend

A244969 Duplicate of A090196.

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A244579 Numbers k with the property that the number of parts in the symmetric representation of sigma(k) equals the number of divisors of k.

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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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A346641 Numbers k with at least one partition into two parts (s,t), s<=t such that t | s*k but no proper divisor of k has this property.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI