A049603 -id:A049603 - cp's OEIS frontend

A049599 Number of (1+e)-divisors of n: if n = Product p(i)^r(i), d = Product p(i)^s(i) and s(i) = 0 or s(i) divides r(i), then d is a (1+e)-divisor of n.

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 3, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4, 6, 2, 12, 4, 6, 4, 4, 4, 6, 2, 6, 6, 9, 2, 8, 2

Offset: 1

Yasutoshi Kohmoto

A divisor of n is a (1+e)-divisor if and only if it is a unitary divisor of an exponential divisor of n (see A077610 and A322791). - Amiram Eldar, Feb 26 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A049603, A051378.

Cf. A124010, A000005, A049419, A077610, A322791.

a049599 = product . map ((+ 1) . a000005 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012

a[n_] := Times @@ (DivisorSigma[0, #] + 1 &)  /@ FactorInteger[n][[All, 2]]; a[1] = 1; Table[a[n], {n, 1, 103}] (* Jean-François Alcover, Oct 10 2011 *)

a(n) = vecprod(apply(x->numdiv(x)+1, factor(n)[, 2])); \\ Amiram Eldar, Aug 13 2023

If n = Product p(i)^r(i) then a(n) = Product (tau(r(i))+1), where tau(n) = number of divisors of n, cf. A000005. - Vladeta Jovovic, Apr 29 2001

More terms from Naohiro Nomoto, Apr 12 2001

A333927 Recursive perfect numbers: numbers k such that A333926(k) = 2*k.

Original entry on oeis.org

6, 28, 264, 1104, 3360, 75840, 151062912, 606557952, 2171581440

Offset: 1

Amiram Eldar, Apr 10 2020

Since a recursive divisor is also a (1+e)-divisor (see A049599), then the first 6 terms and other terms of this sequence coincide with those of A049603.

			264 is a term since the sum of its recursive divisors is 1 + 2 + 3 + 6 + 8 + 11 + 22 + 24 + 33 + 66 + 88 + 264 = 528 = 2 * 264.

Cf. A049603, A282446, A333926.

Analogous sequences: A000396, A002827 (unitary), A007357 (infinitary), A054979 (exponential), A064591 (nonunitary).

recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], recDivSum[#] == 2*# &]

A274116 (1+e)-sigma amicable numbers.

Original entry on oeis.org

220, 284, 366, 378, 2620, 2924, 3864, 4584, 5020, 5564, 16104, 16536, 16632, 16728, 26448, 28752, 29760, 30912, 43524, 53692, 63020, 67344, 69552, 69615, 76084, 87633, 100485, 122265, 124155, 139815, 142290, 142310, 168730, 179118, 196248, 196724, 198990, 202444

Offset: 1

Paolo P. Lava, Jun 10 2016

The first time a pair ordered by its first element is not adjacent is x = 16104, y = 16632 which correspond to a(11) and a(13), respectively.

			(1+e)-sigma(366) = 378 and (1+e)-sigma(378) = 366.

Paolo P. Lava, Table of n, a(n) for n = 1..500

Cf. A049603, A051378, A274118.

with(numtheory): T:=proc(n) local a,d,p,e,s,sp; a:=1;
for d in ifactors(n)[2] do p:=op(1,d); e:= op(2,d); sp:=1;
for s in divisors(e) do sp:=sp+p^s; od: a:=a*sp; od: a; end:
P:=proc(q) local n,x,y; for n from 1 to q do x:=T(n)-n; y:=T(x)-x;
if n=y and x<>y then print(n); fi; od; end: P(10^10);

A274118 (1+e)-sigma betrothed numbers.

Original entry on oeis.org

108, 140, 195, 1050, 1925, 8892, 16587, 312620, 549219, 587460, 1057595, 1279950, 2576945, 5088650, 6446325, 7460004, 7875450, 10925915, 13922100, 16381925, 22559060, 26502315, 29864120, 30809415, 31213899, 41137620, 84854315, 89446860, 102019644, 114859884

Offset: 1

Paolo P. Lava, Jun 10 2016

nonn

Members of a pair (m,n) such that (1+e)-sigma(m)=(1+e)-sigma(n)=m+n+1, where (1+e)-sigma = A051378.

So far, 108 is the only fixed point of the transform n -> (1+e)sigma(n)-n-1.

			(1+e)-sigma(140) - 140 - 1 = 336 - 140 - 1 = 195 and (1+e)-sigma(195) - 195 - 1 = 336 - 195 - 1 = 140.

Paolo P. Lava, Table of n, a(n) for n = 1..50

Cf. A049603, A051378, A274116.

with(numtheory): T:=proc(n) local a,d,p,e,s,sp; a:=1;
for d in ifactors(n)[2] do p:=op(1,d); e:= op(2,d); sp:=1;
for s in divisors(e) do sp:=sp+p^s; od: a:=a*sp; od: a; end:
P:=proc(q) local n,x,y; for n from 2 to q do x:=T(n)-n-1; y:=T(x)-x-1;
if n=y then print(n); fi; od; end: P(10^10);

A349284 Numbers k such that A051378(k) > 2k and A333926(k) <= 2k.

Original entry on oeis.org

126720, 134400, 149760, 188160, 195840, 456960, 510720, 549120, 618240, 718080, 748800, 779520, 802560, 833280, 940800, 979200, 994560, 1094400, 1102080, 1155840, 1263360, 1324800, 1393920, 1424640, 1585920, 1639680, 1670400, 1785600, 1800960, 1908480, 1946880

Offset: 1

Amiram Eldar, Nov 13 2021

nonn

(1+e)-abundant numbers are numbers k such that A051378(k) > 2*k, i.e., numbers k whose sum of (1+e)-divisors exceeds 2*k.

Since all the recursive divisors (see A282446) of a number are also its (1+e)-divisors, the sequence of (1+e)-abundant numbers includes all the recursive abundant numbers (A333928). The first 21387 (1+e)-abundant numbers are also recursive abundant numbers. Therefore, this sequence includes only the (1+e)-abundant numbers that are not recursive abundant numbers.

			126720 is a term since A051378(126720) = 261144 > 2*126720 = 253440 and A333926(126720) = 246168 < 253440.

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

Cf. A049599, A049603, A051378, A282446, A333926, A333928.

oesigma[1] = 1; oesigma[n_] := Times @@ (1 + Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recsigma[1] = 1; recsigma[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^6], oesigma[#] > 2*# && recsigma[#] <= 2*# &]

A349285 (1+e)-weird numbers: (1+e)-abundant numbers k such that no subset of the aliquot (1+e)-divisors of k sums to k.

Original entry on oeis.org

70, 836, 4030, 5830, 10430, 10570, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17570, 17990, 18410, 18830, 18970, 19390, 19670, 19810, 20510, 21490, 21770, 21910, 22190, 23170, 23590, 24290

Offset: 1

Amiram Eldar, Nov 13 2021

nonn

The (1+e)-abundant numbers are numbers k such that A051378(k) > 2*k (union of A333928 and A349284).
Is there any number besides 836 which is in this sequence but not in A348631? - R. J. Mathar, Nov 16 2021
The next term after 836 that is not in A348631 is a(89) = 45356. - Amiram Eldar, Nov 21 2021

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

Cf. A049603, A051378, A333928, A349284.

Similar sequences: A006037, A064114, A292986, A306984, A321146, A327948, A348525.

divQ[n_, m_] := (n > 0 && (m == 0 || Divisible[n, m])); oeDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[divQ, {f[[;; , 2]], IntegerExponent[d, f[[;; , 1]]]}]]; oeDivs[1] = {1}; oeDivs[n_] := Module[{d = Divisors[n]}, Select[d, oeDivQ[n, #] &]]; oesigma[1] = 1; oesigma[n_] := Total@oeDivs[n]; oeAbundantQ[n_] := oesigma[n] > 2*n; oeWeirdQ[n_] := oeAbundantQ[n] && Module[{d = Most[oeDivs[n]]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 0]; Select[Range[12000], oeWeirdQ]

cp's OEIS Frontend

A049599 Number of (1+e)-divisors of n: if n = Product p(i)^r(i), d = Product p(i)^s(i) and s(i) = 0 or s(i) divides r(i), then d is a (1+e)-divisor of n.

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A333927 Recursive perfect numbers: numbers k such that A333926(k) = 2*k.

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A274116 (1+e)-sigma amicable numbers.

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A274118 (1+e)-sigma betrothed numbers.

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A349284 Numbers k such that A051378(k) > 2*k and A333926(k) <= 2*k.

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A349285 (1+e)-weird numbers: (1+e)-abundant numbers k such that no subset of the aliquot (1+e)-divisors of k sums to k.

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A349284 Numbers k such that A051378(k) > 2k and A333926(k) <= 2k.