A357325 -id:A357325 - cp's OEIS frontend

A357324 Numbers k such that there is a unique m for which the sum of the aliquot unitary divisors of m (A034460) is k.

6, 9, 11, 13, 128, 150, 164, 222, 224, 332, 338, 390, 404, 416, 420, 458, 510, 548, 558, 570, 576, 582, 584, 598, 660, 668, 750, 788, 800, 810, 818, 822, 836, 852, 878, 884, 926, 930, 1046, 1118, 1200, 1202, 1230, 1244, 1250, 1260, 1284, 1298, 1304, 1382, 1422, 1472, 1478

Offset: 1

Amiram Eldar, Sep 24 2022

nonn

Numbers k such that A324938(k) = 1.

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

The unitary version of A057709.

Cf. A034460, A063948, A324938, A357325.

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

a(n) = A034460(A357325(n)).

A361420 a(n) is the unique number m such that A126168(m) = A361419(n).

Original entry on oeis.org

1, 6, 8, 15, 21, 52, 58, 82, 106, 118, 268, 158, 356, 1264, 1296, 388, 202, 214, 226, 130, 508, 524, 1936, 160, 138, 298, 692, 2608, 358, 3088, 288, 446, 454, 466, 932, 478, 432, 348, 1792, 538, 562, 578, 586, 12032, 1268, 748, 20736, 1348, 694, 706, 26368, 544, 758

Offset: 1

Amiram Eldar, Mar 11 2023

nonn

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

Cf. A126168, A361419.

Similar sequences: A357313, A357325.

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 = s = Table[0, {max}], i}, Do[i = is[k] + 1; If[i <= max, v[[i]]++; s[[i]] = k], {k, 1, max^2}]; s[[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 = w = vector(nmax+1)); for(k=1, nmax^2, i = s(k) + 1; if(i <= nmax+1, v[i] += 1; w[i] = k)); for(i = 1, nmax+1, if(v[i] == 1, print1(w[i], ", "))); }

A126168(a(n)) = A361419(n).

A372743 a(n) is the unique number m such that A336563(m) = A372742(n).

Original entry on oeis.org

4, 9, 25, 49, 121, 27, 169, 289, 24, 361, 529, 54, 841, 961, 36, 1369, 1681, 1849, 2209, 2809, 343, 3481, 3721, 4489, 5041, 5329, 6241, 100, 6889, 189, 7921, 72, 9409, 112, 10201, 10609, 11449, 11881, 686, 12769, 16129, 17161, 225, 18769, 19321, 196, 22201, 160

Offset: 1

Amiram Eldar, May 12 2024

nonn

Includes all the squares of primes (A001248).

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

Cf. A336563, A372742.
A001248 is a subsequence.
Similar sequences: A357313, A357325, A361420.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; seq[max_] := Module[{v = w = Table[0, {max}], i}, Do[i = s[k]; If[1 <= i <= max, v[[i]]++; w[[i]] = k], {k, 1, max^2}]; w[[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 = w = vector(nmax), i); for(k = 1, nmax^2, i = s(k); if(i > 0 && i <= nmax, v[i]++; w[i] = k)); for(k = 1, nmax, if(v[k] == 1, print1(w[k], ", ")));}

A336563(a(n)) = A372742(n).

cp's OEIS Frontend

A357324 Numbers k such that there is a unique m for which the sum of the aliquot unitary divisors of m (A034460) is k.

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A361420 a(n) is the unique number m such that A126168(m) = A361419(n).

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A372743 a(n) is the unique number m such that A336563(m) = A372742(n).

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