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A284229 a(n) is the least k such that A073802(k) = n. 0
1, 10, 12, 6, 336, 24, 5952, 168, 792, 496, 666624, 270, 10924032, 6720, 7344, 120, 3757637632, 4284, 45091651584, 2160, 79488, 1820672, 11544784011264, 672, 298080, 29331456, 106200, 13440, 53620880789471232, 10080, 1501384662105194496, 6552, 7022592, 7515275264 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Composite numbers that have just 1 as divisor that satisfies the condition for which sigma(k) / d_i is an integer are the Duffinian numbers (A003624).

Alternative definition: Least k such that d(gcd(k,sigma(k)) = n. - Giovanni Resta, Mar 23 2017

LINKS

Table of n, a(n) for n=1..34.

EXAMPLE

The divisors of 12 are 1, 2, 3, 4, 6, 12 and sigma(12) = 28. Then:

1) 28 / 1 = 28;

2) 28 / 2 = 14;

3) 28 / 4 = 7;

and 12 is the least number to have this property. Therefore a(3) = 12.

MAPLE

with(numtheory): P:=proc(q) local k, n; for k from 1 to q do

for n from 1 to q do if tau(gcd(n, sigma(n)))=k then

print(n); break; fi; od; od; end: P(10^9);

MATHEMATICA

TakeWhile[#, # > 0 &] &@ Table[If[KeyExistsQ[#, n], First@ Lookup[#, n], -1], {n, Max@ Keys@ #}] &@ KeySort@ PositionIndex@ Table[DivisorSum[k, 1 &, IntegerQ[DivisorSigma[1, k]/#] &], {k, 10^6}] (* per Name, Version 10, or *)

TakeWhile[#, # > 0 &] &@ Table[If[KeyExistsQ[#, n], First@ Lookup[#, n], -1], {n, Max@ Keys@ #}] &@ KeySort@ PositionIndex@ Table[DivisorSigma[0, GCD[k, DivisorSigma[1, k]]], {k, 10^7}] (* faster, Version 10, Michael De Vlieger, Mar 24 2017 *)

PROG

(PARI) nb(n) = my(s = sigma(n)); sumdiv(n, d, (s % d) == 0);

a(n) = k=1; while(nb(k) != n, k++); k; \\ Michel Marcus, Mar 24 2017

CROSSREFS

Cf. A000203, A003624, A073802.

Sequence in context: A350444 A342356 A338290 * A078285 A303151 A103618

Adjacent sequences:  A284226 A284227 A284228 * A284230 A284231 A284232

KEYWORD

nonn

AUTHOR

Paolo P. Lava, Mar 23 2017

EXTENSIONS

a(13), a(17), a(19) and from a(22) to a(34) from Giovanni Resta, Mar 23 2017

Name proposed by Michel Marcus, Mar 24 2017

STATUS

approved

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Last modified July 2 10:39 EDT 2022. Contains 355004 sequences. (Running on oeis4.)