A357324 -id:A357324 - cp's OEIS frontend

A357325 a(n) is the unique number m such that A034460(m) = A357324(n).

6, 15, 21, 35, 250, 138, 4192, 10048, 6112, 748, 20736, 5968, 802, 12256, 41728, 3592, 498, 53632, 8656, 80128, 2284, 2308, 36352, 2372, 10288, 5272, 11728, 84352, 1594, 630, 6472, 48448, 6616, 50368, 1426, 1762, 102016, 172288, 32416, 8872, 2328, 9544, 19408

Offset: 1

Amiram Eldar, Sep 24 2022

nonn

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

The unitary version of A357313.

Cf. A034460, A057709, A357324.

us[1] = 0; us[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; m = 1500; v = s = Table[0, {m}]; Do[u = us[k]; If[2 <= u <= m, v[[u]]++; s[[u]] = k], {k, 1, m^2}]; s[[Position[v, 1] // Flatten]]

A034460(a(n)) = A357324(n).

A372742 Numbers k such that there is a unique number m for which the sum of the aliquot coreful divisors of m (A336563) is k.

Original entry on oeis.org

2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 23, 24, 29, 31, 36, 37, 41, 43, 47, 53, 56, 59, 61, 67, 71, 73, 79, 80, 83, 84, 89, 96, 97, 98, 101, 103, 107, 109, 112, 113, 127, 131, 135, 137, 139, 140, 149, 150, 151, 156, 157, 163, 167, 173, 179, 181, 191, 193, 197, 198

Offset: 1

Amiram Eldar, May 12 2024

nonn

A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).
Numbers k such that A372739(k) = 1.
The corresponding values of m are in A372743.
Includes all prime numbers.

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

Cf. A336563, A372739, A372743.
A000040 is a subsequence.
Similar sequences: A057709, A357324, A361419.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; seq[max_] := Module[{v = Table[0, {max}], i}, Do[i = s[k]; If[1 <= i <= max, v[[i]]++], {k, 1, max^2}]; Position[v, 1] // Flatten]; seq[200]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] -1) - 1) - n;}
lista(nmax) = {my(v = vector(nmax), i); for(k=1, nmax^2, i = s(k); if(i > 0 && i <= nmax, v[i]++)); for(k = 1, nmax, if(v[k] == 1, print1(k, ", ")));}

a(n) = A336563(A372743(n)).

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.

Original entry on oeis.org

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)).

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

A357325 a(n) is the unique number m such that A034460(m) = A357324(n).

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A372742 Numbers k such that there is a unique number m for which the sum of the aliquot coreful divisors of m (A336563) is k.

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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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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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