A328051 -id:A328051 - cp's OEIS frontend

A328052 Value of sigma(m)/(d(m)*sopf(m)) for the integers m that make this expression an integer.

1, 1, 1, 3, 2, 4, 2, 2, 2, 2, 5, 3, 2, 13, 3, 3, 3, 7, 5, 4, 7, 4, 3, 21, 13, 4, 3, 8, 4, 20, 9, 5, 4, 19, 5, 4, 6, 8, 11, 6, 6, 25, 8, 6, 10, 14, 5, 6, 6, 25, 4, 4, 6, 7, 9, 10, 5, 8, 26, 6, 9, 8, 10, 6, 6, 6, 7, 12, 6, 9, 26, 19, 6, 7, 10, 13, 9, 9, 15, 9, 7, 123

Offset: 1

Michel Marcus, Oct 03 2019

nonn

			For A328052(1)=20, sigma(20)/(d(20)*sopf(20)) = 42/(6*7) = 1, so a(1) = 1.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A008472, A134382, A328051.

[a: k in [2..5000]|IsIntegral(a) where a is DivisorSigma(1,k)/(#Divisors(k)*(&+PrimeDivisors(k)))]; // Marius A. Burtea, Oct 03 2019

f[p_, e_]  := (p^(e + 1) - 1)/((e + 1)*(p - 1)); r[n_] := Times @@ (f @@@ (fct = FactorInteger[n])) / Plus @@ (fct[[;; , 1]]); Select[r /@ Range[2, 4500], IntegerQ] (* Amiram Eldar, Oct 03 2019 *)

lista(nn) = {for (n=2, nn, my(f=factor(n)); if (denominator(q = sigma(f)/(numdiv(f)*sopf(f))) == 1, print1(q, ", ")););}

a(n) = A000203(A328051(n))/(A000005(A328051(n))*A008472(A328051(n))). - Felix Fröhlich, Oct 03 2019

A328174 a(n) is the least integer k such that sigma(k)/(d(k)*sopf(k)) = n where d=A000005, sigma=A000203 and sopf=A008472.

Original entry on oeis.org

20, 140, 54, 189, 378, 1365, 540, 945, 1120, 1755, 1539, 3465, 500, 1815, 4256, 6384, 14645, 5280, 1323, 1029, 864, 23871, 34579, 12903, 1715, 2673, 11934, 5589, 106805, 12285, 5600, 11625, 21070, 41915, 4459, 16905, 61320, 6615, 11178, 5145, 110839, 19656, 109225

Offset: 1

Michel Marcus, Oct 06 2019

nonn

Cf. A000005, A000203, A008472, A134382, A328051, A328052, A328175.

sopf(f) = sum(j=1, #f~, f[j, 1]); \\ A008472
isok(k, n) = my(fk=factor(k)); n*numdiv(fk)*sopf(fk) == sigma(fk);
a(n) = {my(k=1); while (!isok(k, n), k++); k;}

A328175 a(n) is the largest integer k such that sigma(k)/(d(k)*sopf(k)) = n where d=A000005, sigma=A000203 and sopf=A008472.

Original entry on oeis.org

42, 470, 923, 2159, 12924, 3735, 4316, 8786, 23939, 24412, 76502, 26768, 28612, 47849, 145620, 36002, 118204, 189143, 116999, 105657, 109559, 252474, 142687, 236860, 504899, 265682, 388798, 1558808, 154559, 345687, 709564, 544829, 383086, 652049, 361905, 1193075

Offset: 1

Michel Marcus, Oct 06 2019

nonn

Cf. A000005, A000203, A008472, A134382, A328051, A328052, A328174.

sopf(f) = sum(j=1, #f~, f[j, 1]); \\ A008472
lista(nn) = {/* nn should be > 10^7 */ my(nmax = 43, v = vector(nmax, k, List())); for (n=2, nn, my(f=factor(n), q); if (denominator(q=sigma(f)/(numdiv(f)*sopf(f))) == 1, if (q <= nmax, listput(v[q], n)););); for (i=1, nmax, if (#v[i] == 0, break); print1(vecmax(Vec(v[i])), ", "););}

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A328052 Value of sigma(m)/(d(m)*sopf(m)) for the integers m that make this expression an integer.

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A328174 a(n) is the least integer k such that sigma(k)/(d(k)*sopf(k)) = n where d=A000005, sigma=A000203 and sopf=A008472.

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A328175 a(n) is the largest integer k such that sigma(k)/(d(k)*sopf(k)) = n where d=A000005, sigma=A000203 and sopf=A008472.

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