A331972 -id:A331972 - cp's OEIS frontend

A331974 Infinitary highly touchable numbers: numbers m > 1 such that a record number of numbers k have m as the sum of the proper infinitary divisors of k.

2, 6, 8, 17, 21, 37, 49, 55, 67, 79, 85, 91, 121, 151, 175, 181, 211, 295, 301, 361, 391, 421, 481, 511, 571, 631, 781, 841, 991, 1051, 1231, 1261, 1471, 1561, 1651, 1681, 1891, 2101, 2311, 2731, 3151, 3361, 3571, 3991, 4201, 4291, 4411, 4621, 5251, 5461, 6091

Offset: 1

Amiram Eldar, Feb 03 2020

nonn

The corresponding record values are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...

The infinitary version of A238895.

			a(1) = 2 since it is the first number which is not the sum of proper infinitary divisors of any number.
a(2) = 6 since it is the least number which is the sum of proper infinitary divisors of one number: 6 = A126168(6).
a(3) = 8 since it is the least number which is the sum of proper infinitary divisors of 2 numbers: 8 = A126168(10) = A126168(12).

Amiram Eldar, Table of n, a(n) for n = 1..76 (terms below 30000)

Cf. A049417, A077609, A126168, A238895, A324277, A325177, A331972, A331973.

fun[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ (fun @@@ FactorInteger[n]); is[n_] := isigma[n] - n; m = 300; v = Table[0, {m}]; Do[i = is[k]; If[2 <= i <= m, v[[i]]++], {k, 1, m^2}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, k]], {k, 2, m}]; s

A372741 Coreful highly touchable numbers: numbers m > 0 such that a record number of numbers k have m as the sum of the aliquot coreful divisors (A336563) of k.

Original entry on oeis.org

1, 2, 6, 30, 210, 930, 2310, 2730, 30030, 71610, 84630

Offset: 1

Amiram Eldar, May 12 2024

A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).
Indices of records of A372739.
The corresponding record values are 0, 1, 3, 6, 8, 9, 11, 12, 15, 16, 17, ... .
a(12) > 2*10^5.

			a(1) = 1 since it is the least number that is not the sum of aliquot coreful divisors of any number.
a(2) = 2 since it is the least number that is the sum of aliquot coreful divisors of one number: 2 = A336563(4).
a(3) = 6 since it is the least number that is the sum of aliquot coreful divisors of 3 numbers: 6 = A336563(8) = A336563(12) = A336563(18), and there is no number between 2 and 6 that is the sum of aliquot coreful divisors of exactly 2 numbers.

Cf. A307958, A336563, A372739, A372740.

Similar sequences: A238895, A325177, A331972, A331974.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; seq[m_] := Module[{v = Table[0, {m}], vm = -1, w = {}, i}, Do[i = s[k]; If[1 <= i <= m, v[[i]]++], {k, 1, m^2}]; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[w, k]], {k, 1, m}]; w]; seq[1000]

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 = vector(nmax), vmax = -1, i); for(k = 1, nmax^2, i = s(k); if(i > 0 && i <= nmax, v[i]++)); for(k = 1, nmax, if(v[k] > vmax, vmax = v[k]; print1(k, ", ")));}

cp's OEIS Frontend

A331974 Infinitary highly touchable numbers: numbers m > 1 such that a record number of numbers k have m as the sum of the proper infinitary divisors of k.

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