A057709 -id:A057709 - cp's OEIS frontend

A361419 Numbers k such that there is a unique number m for which the sum of the aliquot infinitary divisors of m (A126168) is k.

0, 6, 7, 9, 11, 18, 32, 44, 56, 62, 72, 82, 94, 96, 98, 102, 104, 110, 116, 122, 132, 136, 138, 146, 150, 152, 178, 180, 182, 210, 222, 226, 230, 236, 238, 242, 248, 252, 264, 272, 284, 292, 296, 304, 322, 332, 338, 342, 350, 356, 360, 374, 382, 390, 392, 404

Offset: 1

Amiram Eldar, Mar 11 2023

nonn

Numbers k such that A331973(k) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A126168, A324277, A331974, A331973, A361420.

Similar sequences: A057709, A357324.

f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; is[1] = 0; is[n_] := Times @@ f @@@ FactorInteger[n] - n;
seq[max_] := Module[{v = Table[0, {max}], i}, Do[i = is[k] + 1; If[i <= max, v[[i]]++], {k, 1, max^2}]; -1 + Position[v, 1] // Flatten];
seq[500]

s(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) + 1, 1))) - n; }
lista(nmax) = {my(v = vector(nmax+1)); for(k=1, nmax^2, i = s(k) + 1; if(i <= nmax+1, v[i] += 1)); for(i = 1, nmax+1, if(v[i] == 1, print1(i-1, ", "))); }

a(n) = A126168(A361420(n)).

A366110 a(n) is the difference between the maximum and minimum number whose proper divisors sum to n, or 0 if there is no such number.

Original entry on oeis.org

0, 0, 0, 0, 19, 0, 39, 0, 0, 0, 0, 8, 147, 17, 14, 16, 0, 12, 327, 73, 18, 28, 0, 48, 0, 64, 0, 72, 0, 189, 903, 202, 0, 160, 0, 168, 0, 0, 37, 328, 1651, 387, 1767, 280, 34, 364, 0, 476, 54, 448, 0, 432, 2767, 677, 0, 604, 0, 432, 0, 528, 3603, 753, 66, 826, 0, 768, 0, 720, 0

Offset: 2

Michel Marcus, Oct 28 2023

nonn

A152454 is the irregular triangle in which row n lists the numbers whose proper divisors sum to n.

			A152454 begins as []; [4]; [9]; []; [6, 25]; [8]; [10, 49]...
so sequence begins 0, 0, 0, 0, 19, 0, 39, ...

Michel Marcus, Table of n, a(n) for n = 2..10001

Cf. A001065, A152454.
Cf. A070015, A135244.
Cf. A005114, A057709.

lista(nn) = my(v = vector(nn, k, [])); forcomposite (i=1, nn^2, my(x=sigma(i)-i); if (x <=  nn, v[x] = concat(v[x], i));); vector(nn-1, k, k++; if (#v[k], vecmax(v[k]) - vecmin(v[k])));

a(n) = A135244(n) - A070015(n).

a(A005114(n)) = a(A057709(n)) = 0.

A358199 a(n) is the least integer whose sum of the i-th powers of the proper divisors is a prime for 1 <= i <= n, or -1 if no such number exists.

Original entry on oeis.org

4, 4, 981, 8829, 8829, 122029105, 2282761881

Offset: 1

Jean-Marc Rebert, Nov 02 2022

			4 is a term since the strict divisors of 4 are {1, 2}, 1^1 + 2^1 = 3 and 1^2 + 2^2 = 5 are prime and no number < 4 has this property.

Cf. A001065 (s1), A180852, A357324, A057709, A000203, A001157, A001158.

Subsequence of A037020.

card(n)=my(c=1,s=0);s=sigma(n)-n;while(isprime(s),c++;s=sigma(n,c)-n^c);c--
a(n)=my(x=0);for(k=1,+oo,x=card(k);if(x>=n,return(k)))

from itertools import count
from math import prod
from sympy import isprime, factorint
def A358199(n):
    for m in count(2):
        f = factorint(m).items()
        if all(map(isprime,(prod((p**((e+1)*i)-1)//(p**i-1) for p,e in f) - m**i for i in range(1,n+1)))):
            return m # Chai Wah Wu, Nov 15 2022

cp's OEIS Frontend

A361419 Numbers k such that there is a unique number m for which the sum of the aliquot infinitary divisors of m (A126168) is k.

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A366110 a(n) is the difference between the maximum and minimum number whose proper divisors sum to n, or 0 if there is no such number.

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A358199 a(n) is the least integer whose sum of the i-th powers of the proper divisors is a prime for 1 <= i <= n, or -1 if no such number exists.

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