A328052 -id:A328052 - cp's OEIS frontend

A328051 Numbers m such that sigma(m)/(d(m)*sopf(m)) is an integer, where d is the number of divisors (A000005) and sopf the sum of prime factors without repetition (A008472).

20, 35, 42, 54, 140, 189, 195, 207, 209, 276, 378, 464, 470, 500, 506, 510, 527, 540, 608, 660, 672, 741, 846, 864, 875, 899, 923, 945, 989, 1029, 1120, 1276, 1316, 1323, 1334, 1349, 1365, 1519, 1539, 1564, 1595, 1715, 1725, 1736, 1755, 1815, 1880, 1887, 1914, 2058

Offset: 1

Michel Marcus, Oct 03 2019

nonn

This sequence is motivated by the short fate of A134382.

			For n=20, sigma(20)/(d(20)*sopf(20)) = 42/(6*7) = 1, an integer, so 20 is a term.

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

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

[k: k in [2..2100]|IsIntegral(DivisorSigma(1,k)/(#Divisors(k)*(&+PrimeDivisors(k))))]; // Marius A. Burtea, Oct 03 2019

f[p_, e_] := (p^(e + 1) - 1)/((e + 1)*(p - 1)); Select[Range[2, 2100], IntegerQ[ Times @@ (f @@@ (fct = FactorInteger[#])) / Plus @@ (fct[[;; , 1]])] &] (* Amiram Eldar, Oct 03 2019 *)

sopf(f) = sum(j=1, #f~, f[j, 1]); \\ A008472
isok(m) = if (m>1, my(f=factor(m)); (sigma(f) % (numdiv(f)*sopf(f))) == 0);

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])), ", "););}

cp's OEIS Frontend

A328051 Numbers m such that sigma(m)/(d(m)*sopf(m)) is an integer, where d is the number of divisors (A000005) and sopf the sum of prime factors without repetition (A008472).

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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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PARI