A370681 -id:A370681 - cp's OEIS frontend

A370683 Nonsquarefree numbers k such that A370681(k) = A071324(k).

4, 68425, 78045, 4460155, 28268625, 114468171, 177972505, 554353635, 554821905, 555758445, 556226715, 556382805, 558099795, 558724155, 560128965, 560909415, 561377685, 562470315, 562782495, 562938585, 563406855, 564187305, 564811665, 565279935, 565592115, 566060385

Offset: 1

Amiram Eldar, Feb 26 2024

nonn

For every squarefree number k, A370681(k) = A071324(k), since all the divisors of a squarefree number are unitary.

This sequence is infinite: if p >= 7103 is prime then 78045*p is a term. Terms a(8)-a(536) are of this form.

			4 is a term since its divisors are 1, 2 and 4, and its unitary divisors are 1 and 4, and 4 - 2 + 1 = 4 - 1.

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

Cf. A005117, A013929, A071324, A370681.

q[n_] := Module[{d = Reverse[Divisors[n]], u}, u = Select[d, CoprimeQ[#, n/#] &]; Total[(-1)^(Range[Length[d]] + 1)*d] == Total[(-1)^(Range[Length[u]] + 1)*u]]; Select[Range[10^5], ! SquareFreeQ[#] && q[#] &]

iseq(n) = my(d = Vecrev(divisors(n)), u = select(x->(gcd(x, n/x) == 1), d)); sum(i=1, #d, (-1)^(i+1)*d[i]) == sum(i=1, #u, (-1)^(i+1)*u[i]);
is(n) = !issquarefree(n) && iseq(n)

A370682 Numbers k such that A370681(k) = A370681(k+1).

Original entry on oeis.org

1, 5, 11, 19, 71, 203, 271, 979, 991, 2755, 5976, 6407, 7004, 8560, 9143, 33552, 46583, 75856, 87955, 90287, 101955, 243271, 265587, 269648, 457135, 510192, 512783, 551172, 559656, 1804555, 1917639, 2902175, 4339952, 6783820, 9842748, 10463980, 17428447, 21677503

Offset: 1

Amiram Eldar, Feb 26 2024

nonn

			5 is a term since A370681(5) = A370681(6) = 4.

Amiram Eldar, Table of n, a(n) for n = 1..120

Cf. A064125, A333261, A370681.

s[n_] := Module[{d = Reverse[Select[Divisors[n], CoprimeQ[#, n/#] &]]}, Total[(-1)^(Range[Length[d]] + 1)*d]]; Select[Range[10^5], s[#] == s[# + 1] &]

s(n) = {my(d = Vecrev(select(x->(gcd(x, n/x) == 1), divisors(n)))); sum(i=1, #d, (-1)^(i+1)*d[i]);}
lista(nmax) = {my(s1 = s(1), s2); for(n = 2, nmax, s2 = s(n); if(s1 == s2, print1(n-1,", ")); s1=s2);}

cp's OEIS Frontend

A370683 Nonsquarefree numbers k such that A370681(k) = A071324(k).

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