A331972 -id:A331972

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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.

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

The infinitary version of A238895.

LINKS

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

EXAMPLE

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).

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Amiram Eldar, Feb 03 2020

STATUS

approved

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.

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..11.

EXAMPLE

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.

MATHEMATICA

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]

PROG

(PARI) 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, ", "))); }

CROSSREFS

Cf. A307958, A336563, A372739, A372740.

Similar sequences: A238895, A325177, A331972, A331974.

KEYWORD

nonn,more

AUTHOR

Amiram Eldar, May 12 2024

STATUS

approved

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