A323757 -id:A323757 - cp's OEIS frontend

A322609 Numbers k such that s(k) = 2*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

24, 54, 112, 150, 294, 726, 1014, 1734, 1984, 2166, 3174, 5046, 5766, 8214, 10086, 11094, 13254, 16854, 19900, 20886, 22326, 26934, 30246, 31974, 32512, 37446, 41334, 47526, 56454, 61206, 63654, 68694, 71286, 76614, 96774, 102966, 112614, 115926, 133206

Offset: 1

Amiram Eldar, Dec 20 2018

nonn

This sequence is infinite since 6*p^2 is included for all primes p. Terms that are not of the form 6*p^2: 112, 1984, 19900, 32512, 134201344, ...
Includes 4*k if k is an even perfect number: see A000396. - Robert Israel, Jan 06 2019
From Amiram Eldar, Oct 01 2022: (Start)
24 = 6*prime(1)^2 = 4*A000396(1) is the only term that is common to the two forms that are mentioned above.
19900 is the only term below 10^11 which is not of any of these two forms. Are there any other such terms?
All the known nonunitary perfect numbers (A064591) are also of the form 4*k, where k is an even perfect number.
Equivalently, numbers k such that A325314(k) = -k. (End)

			24 is a term since A162296(24) = 48 = 2*24.

Amiram Eldar, Table of n, a(n) for n = 1..12091 (terms below 10^11; terms 1..300 from Robert Israel)

Cf. A162296, A325314.
Subsequence of A005101 and A013929.
Numbers k such that A162296(k) = m*k: A005117 (m=0), A001248 (m=1), this sequence (m=2), A357493 (m=3), A357494 (m=4).
Similar sequences: A000396, A002827, A007357, A054979, A064591, A322486, A323757, A324707, A327633.

filter:= proc(n) convert(remove(numtheory:-issqrfree,numtheory:-divisors(n)),`+`)=2*n end proc:
select(filter, [$1..200000]); # Robert Israel, Jan 06 2019

s[1]=0; s[n_] := DivisorSigma[1,n] - Times@@(1+FactorInteger[n][[;;,1]]); Select[Range[10000], s[#] == 2# &]

s(n) = sumdiv(n, d, d*(1-moebius(d)^2)); \\ A162296
isok(n) = s(n) == 2*n; \\ Michel Marcus, Dec 20 2018

from sympy import divisors, factorint
A322609_list = [k for k in range(1,10**3) if sum(d for d in divisors(k,generator=True) if max(factorint(d).values(),default=1) >= 2) == 2*k] # Chai Wah Wu, Sep 19 2021

A379029 Modified exponential abundant numbers: numbers k such that A241405(k) > 2*k.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 114, 120, 138, 150, 168, 174, 186, 210, 222, 246, 258, 270, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690, 714, 726, 750, 762, 770, 780, 786, 798, 822

Offset: 1

Amiram Eldar, Dec 14 2024

All the squarefree abundant numbers (A087248) are terms since A241405(k) = A000203(k) for a squarefree number k.
If k is a term and m is coprime to k them k*m is also a term.
The numbers of terms that do no exceed 10^k, for k = 2, 3, ..., are 5, 67, 767, 7595, 76581, 764321, 7644328, 76468851, 764630276, ... . Apparently, the asymptotic density of this sequence exists and equals 0.07646... .

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

Cf. A000203, A241405, A323757, A323758, A323759, A379027.
Subsequence of A005101.
Subsequences: A034683, A087248, A379030, A379031.
Similar sequences: A064597, A129575, A129656, A292982, A348274, A348604.

f[p_, e_] := DivisorSum[e + 1, p^(# - 1) &]; mesigma[1] = 1; mesigma[n_] := Times @@ f @@@ FactorInteger[n]; meAbQ[n_] := mesigma[n] > 2*n; Select[Range[1000], meAbQ]

is(n) = {my(f=factor(n)); prod(i=1, #f~, sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1))) > 2*n;}

A323758 Lesser of modified exponential amicable pairs.

Original entry on oeis.org

114, 366, 1140, 3660, 3864, 5016, 11040, 16104, 16536, 44772, 57960, 67158, 68640, 142290, 142310, 180960, 196248, 198990, 240312, 248040, 308220, 322080, 326424, 339822, 348840, 352632, 366792, 462330, 669900, 671580, 785148, 815100, 817440, 849240, 912072

Offset: 1

Amiram Eldar, Jan 26 2019

nonn

A modified exponential amicable pair (m, n) has A241405(m) = A241405(n) = m + n + 1.

The larger counterparts are in A323759.

Amiram Eldar, Table of n, a(n) for n = 1..4725 (terms below 2*10^11)
David Moews, Perfect, amicable and sociable numbers.

Cf. A241405, A323757, A323759.

f[p_, e_] := DivisorSum[e+1, p^(#-1)&]; mesigma[1]=1; mesigma[n_] := Times @@ f @@@FactorInteger@ n; s={}; mes[n_] := mesigma[n]-n; Do[m=mes[n]; If[m>n && mes[m]==n, AppendTo[s, n]], {n,1,10000}]; s

f(n) = {my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1)));} \\ A241405
fm(n) = f(n) - n;
isok(n) = my(fn=fm(n)); (fn > n) && (fm(fn) == n); \\ Michel Marcus, Jan 30 2019

A323759 Larger of modified exponential amicable pairs.

Original entry on oeis.org

126, 378, 1260, 3780, 4584, 5544, 11424, 16632, 16728, 49308, 68760, 73962, 88608, 179118, 168730, 212160, 225096, 256338, 266568, 250920, 365700, 374304, 391656, 374418, 387720, 386568, 393528, 548550, 827700, 739620, 827652, 932100, 912288, 935400, 1052088

Offset: 1

Amiram Eldar, Jan 26 2019

nonn

A modified exponential amicable pair (m, n) has A241405(m) = A241405(n) = m + n + 1.

The lesser counterparts are in A323758.

Amiram Eldar, Table of n, a(n) for n = 1..4725 (terms whose lesser counterparts below 2*10^11)
David Moews, Perfect, amicable and sociable numbers.

Cf. A241405, A323757, A323758.

f[p_, e_] := DivisorSum[e+1, p^(#-1)&]; mesigma[1]=1; mesigma[n_] := Times @@ f @@@FactorInteger@ n; s={}; mes[n_] := mesigma[n]-n; Do[m=mes[n]; If[m>n && mes[m]==n, AppendTo[s, m]], {n,1,10000}]; s

f(n) = {my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1)));} \\ A241405
fm(n) = f(n) - n;
isok(n) = my(fn=fm(n)); (fn > 0) && (fn < n) && (fm(fn) == n); \\ Michel Marcus, Jan 30 2019

A349019 Modified e-perfect numbers: numbers k such that A348963(k) | k.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101

Offset: 1

Amiram Eldar, Nov 06 2021

nonn

First differs from A225354 at n = 25.
Not to be confused with modified exponential perfect numbers (A323757).
Sándor (2006) showed that the exponential harmonic numbers of type 2 (A348964) are terms in this sequence.
All the squarefree numbers are terms (A005117), since A348963(k) = 1 if k is squarefree.

			12 is a term since A348963(12) = 3 is a divisor of 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000
József Sándor, On exponentially harmonic numbers, Scientia Magna, Vol. 2, No. 3 (2006), pp. 44-47.
József Sándor, Selected Chapters of Geomety, Analysis and Number Theory, 2005, pp. 141-145.

A005117, A348964 and A349020 are subsequences.

Cf. A225354, A323757, A348963.

f[p_, e_] := p^e/DivisorSum[e, p^(e - #) &]; modEPerfQ[1] = True; modEPerfQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], modEPerfQ]

cp's OEIS Frontend

A322609 Numbers k such that s(k) = 2*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

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A379029 Modified exponential abundant numbers: numbers k such that A241405(k) > 2*k.

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A323758 Lesser of modified exponential amicable pairs.

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A323759 Larger of modified exponential amicable pairs.

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A349019 Modified e-perfect numbers: numbers k such that A348963(k) | k.

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