A353039 -id:A353039 - cp's OEIS frontend

A361387 Infinitary arithmetic numbers k whose mean infinitary divisor is an infinitary divisor of k.

1, 6, 60, 270, 420, 630, 2970, 5460, 8190, 36720, 136500, 172900, 204750, 245700, 491400, 790398, 791700, 819000, 1037400, 1138320, 1187550, 1228500, 1801800, 2457000, 3767400, 4176900, 4504500, 5405400, 6397300, 6688500, 6741630, 7698600, 8353800, 10032750, 10228680

Offset: 1

Amiram Eldar, Mar 10 2023

nonn

Also, infinitary harmonic numbers k whose harmonic mean of the infinitary divisors of k is an infinitary divisor of k.

			6 is a term since the arithmetic mean of its infinitary divisors, {1, 2, 3, 6}, is 3, and 3 is also an infinitary divisor of 6.
60 is a term since the arithmetic mean of its infinitary divisors, {1, 3, 4, 5, 12, 15, 20, 60}, is 15, and 15 is also an infinitary divisor of 60.

Amiram Eldar, Table of n, a(n) for n = 1..135

Subsequence of A063947 and A361386.
Similar sequence: A007340, A353039.
Cf. A037445, A049417, A077609.

idivs[1] = {1}; idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])]; Select[Range[10^5], IntegerQ[(r = Mean[(i = idivs[#])])] && MemberQ[i, r] &]

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); } \\ Michel Marcus at A077609
is(n) = {my(f = factor(n), b, r); r = prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], (f[i, 1]^(2^(#b-k))+1)/2, 1))); denominator(r) == 1 && n%r==0 && isidiv(r, f); }

A361787 Bi-unitary arithmetic numbers k whose mean bi-unitary divisor is a bi-unitary divisor of k.

Original entry on oeis.org

1, 6, 60, 270, 420, 630, 672, 2970, 5460, 8190, 10080, 22848, 30240, 99792, 136500, 172900, 204750, 208656, 245700, 249480, 312480, 332640, 342720, 385560, 491400, 695520, 708288, 791700, 819000, 861840, 1028160, 1037400, 1187550, 1228500, 1421280, 1528800, 1571328

Offset: 1

Amiram Eldar, Mar 24 2023

nonn

Also, bi-unitary harmonic numbers k whose harmonic mean of the bi-unitary divisors of k is a bi-unitary divisor of k.

			6 is a term since the arithmetic mean of its bi-unitary divisors, {1, 2, 3, 6}, is 3, and 3 is also a bi-unitary divisor of 6.
60 is a term since the arithmetic mean of its bi-unitary divisors, {1, 3, 4, 5, 12, 15, 20, 60}, is 15, and 15 is also a bi-unitary divisor of 60.

Amiram Eldar, Table of n, a(n) for n = 1..236

Subsequence of A286325 and A361786.
Similar sequence: A007340, A353039, A361387.
Cf. A188999, A222266, A286324.

biudivQ[f_, d_] := AllTrue[f, OddQ[Last[#]] || IntegerExponent[d, First[#]] != Last[#]/2 &]; biuDivs[n_] := Module[{d = Divisors[n], f = FactorInteger[n]}, Select[d, biudivQ[f, #] &]]; Select[Range[10^5], IntegerQ[(r = Mean[(i = biuDivs[#])])] && MemberQ[i, r] &]

isbdiv(f, d) = {for (i=1, #f~, if(f[i, 2]%2 == 0 && valuation(d, f[i, 1]) == f[i, 2]/2, return(0))); 1;}
is(n) = {my(f = factor(n), r, p, e); r = prod(i=1, #f~, p = f[i, 1]; e = f[i, 2]; if(e%2, (p^(e+1)-1)/((e + 1)*(p-1)), ((p^(e+1)-1)/(p-1)-p^(e/2))/e)); denominator(r) == 1 && n%r==0 && isbdiv(f, r); }

cp's OEIS Frontend

A361387 Infinitary arithmetic numbers k whose mean infinitary divisor is an infinitary divisor of k.

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