A351903 -id:A351903 - cp's OEIS frontend

A373609 List of repeated terms in A351903.

45, 135, 225, 315, 405, 675, 945, 1035, 1125, 1155, 1215, 1305, 1365, 1485, 1575, 1755, 2079, 2275, 2475, 2565, 2835, 3105, 3375, 3465, 3825, 3915, 4095, 4275, 4550, 4725, 5175, 5265, 5355, 5625, 5775, 5985, 6050, 6237, 6370, 6525, 6615, 6650, 6825, 6885, 6975, 7245, 7315, 7425

Offset: 1

Hartmut F. W. Hoft, Jun 10 2024

nonn

It appears that all repeated terms in A351903 occur in pairs only, and either are odd or multiples of 10; true through a(4543) = 999999.

It is not known whether this sequence is infinite. A351903 is infinite, but not increasing.

			a(1) = 45 = A351903(23) = A351903(32) and A237270[45] = {23, 32, 23}.
a(2) = 135 = A351903(68) = A351903(104) and A237270[135] = {68, 104, 68}.
a(17) = 2079 = A351903(1040) = A351903(1064) and A237270[2079] = {1040, 348, 1064, 348, 1040} is the smallest number in this sequence whose symmetric representation of sigma has 5 parts.
a(137) = 22365 = A351903(11183) = A351903(11281) and A237270[22365] = {11183, 11281, 11281, 11183} is the smallest number in this sequence whose symmetric representation of sigma has 4 parts.

Cf. A000203, A237270, A351903.

See A237593 for more comprehensive cross-references re symmetric representation of sigma.

(* function a237270[ ] and its support functions are defined in A351903 *)
a373609[n_] := Module[{pL={}, rL={}, k, a, j, c}, For[k=1, k<=n, k++, a=a237270[k]; c=0; For[j=1, j<=Length[a], j++, If[!MemberQ[pL, a[[j]]], AppendTo[pL, a[[j]]]; c++]]; If[c>1, AppendTo[rL, k]]]; rL]
a373609[7425]

A351904 a(n) is the smallest number k such that the symmetric representation of sigma(k) has at least one subpart n.

Original entry on oeis.org

1, 3, 2, 7, 9, 11, 4, 15, 10, 19, 6, 14, 24, 27, 8, 31

Offset: 1

Omar E. Pol, Feb 25 2022

Conjecture: there are infinitely many pairs of the form a(x) = y; a(y) = x (see examples).

First differs from A351903 at a(11).

			For n = 11 we have that 6 is the smallest number k with at least one subpart 11 in the symmetric representation of sigma(k), so a(11) = 6.
The symmetric representation of sigma(6) in the first quadrant looks like this:
.
   _ _ _ _
  |_ _ _  |_ 1
        | |_|_ 11
        |_ _  |
            | |
            | |
            |_|
.
There are one subpart 11 and one subpart 1.
.
Some pairs of the form a(x) = y; a(y) = x:
   a(2) =  3;   a(3) =  2.
   a(4) =  7;   a(7) =  4.
   a(6) = 11;  a(11) =  6.
   a(8) = 15;  a(15) =  8.
  a(16) = 31;  a(31) = 16.
.

Row 1 of A352015.
Cf. A351903 (Analog for parts).
Cf. A000079, A000203, A000225, A001227 (number of subparts), A196020, A235791, A236104, A237270, A237271, A237591, A237593, A279387 (definition of subparts), A280850, A280851 (subparts), A296508, A296513, A347529, A351819.

cp's OEIS Frontend

A373609 List of repeated terms in A351903.

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