A051378 -id:A051378 - cp's OEIS frontend

A383864 The sum of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or a unitary divisor of the p-adic valuation of n.

1, 3, 4, 7, 6, 12, 8, 11, 13, 18, 12, 28, 14, 24, 24, 19, 18, 39, 20, 42, 32, 36, 24, 44, 31, 42, 31, 56, 30, 72, 32, 35, 48, 54, 48, 91, 38, 60, 56, 66, 42, 96, 44, 84, 78, 72, 48, 76, 57, 93, 72, 98, 54, 93, 72, 88, 80, 90, 60, 168, 62, 96, 104, 79, 84, 144

Offset: 1

Amiram Eldar, May 12 2025

First differs from A383866 at n = 256.
The sum of divisors d of n such that each is a unitary divisor of an exponential unitary divisor of n (see A361255).
Analogous to the sum of (1+e)-divisors (A051378) as exponential unitary divisors (A361255, A322857) are analogous to exponential divisors (A322791, A051377).
The number of these divisors is A383863(n).

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

Cf. A051377, A209061, A322791, A322857, A361255, A383863, A383866.

f[p_, e_] := 1 + DivisorSum[e, p^# &, CoprimeQ[#, e/#] &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + sumdiv(f[i, 2], d, if(gcd(d, f[i, 2]/d) == 1, f[i, 1]^d)));}

Multiplicative with a(p^e) = 1 + Sum_{d|e, gcd(d, e/d) = 1} p^d.
a(n) <= A051378(n), with equality if and only if n is an exponentially squarefree number (A209061).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 1.52168352620962354041..., and f(x) = (1-x) * (1 + Sum_{k>=1} (1 + Sum{d|k, gcd(d, k/d)=1} x^(2*k-d))).

A383866 The sum of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or an infinitary divisor of the p-adic valuation of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 11, 13, 18, 12, 28, 14, 24, 24, 19, 18, 39, 20, 42, 32, 36, 24, 44, 31, 42, 31, 56, 30, 72, 32, 35, 48, 54, 48, 91, 38, 60, 56, 66, 42, 96, 44, 84, 78, 72, 48, 76, 57, 93, 72, 98, 54, 93, 72, 88, 80, 90, 60, 168, 62, 96, 104, 79, 84, 144

Offset: 1

Amiram Eldar, May 13 2025

First differs from A383864 at n = 256.
The sum of divisors d of n such that each is a unitary divisor of an exponential infinitary divisor of n (see A383760).	
Analogous to the sum of (1+e)-divisors (A051378) as exponential infinitary divisors (A383760, A361175) are analogous to exponential divisors (A322791, A051377).
The number of these divisors is A383865(n).

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

Cf. A036537, A051377, A051378, A077609, A322791, A361175, A383760, A383864, A383865.

infdivs[n_] := If[n == 1, {1}, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]];  (* Michael De Vlieger at A077609 *)
f[p_, e_] := 1 + Total[p^infdivs[e]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
infdivs(n) = {d = divisors(n); f = factor(n); idiv = []; for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
a(n) = {my(f = factor(n), d); prod(i = 1, #f~, d = infdivs(f[i, 2]); 1 + sum(j = 1, #d, f[i, 1]^d[j]));}

Multiplicative with a(p^e) = 1 + Sum_{d infinitary divisor of e} p^d.
a(n) <= A051378(n), with equality if and only if all the exponents in the prime factorization of n are in A036537.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 1.52187097260174705015..., and f(x) = (1-x) * (1 + Sum_{k>=1} (1 + Sum{d infinitary divisor of k} x^(2*k-d))).

A087745 Numbers A001317 repeated.

Original entry on oeis.org

1, 1, 3, 3, 5, 5, 15, 15, 17, 17, 51, 51, 85, 85, 255, 255, 257, 257, 771, 771, 1285, 1285, 3855, 3855, 4369, 4369, 13107, 13107, 21845, 21845, 65535, 65535, 65537, 65537, 196611, 196611, 327685, 327685, 983055, 983055, 1114129, 1114129

Offset: 0

Philippe Deléham, Oct 02 2003

Triangles A049444, A049459, A051338, A051379, A051523 (Mitrinovic's triangles) mod 2 converted to decimal.

Sequence [1, 3, 5, 15, 17, 51, 85, 255, 257, ...] = A001317.

Cf. A001317, A049444, A049459, A051338, A051378, A051523.

def A087745(n): return sum((bool(~(m:=n>>1)&m-k)^1)<>1)+1)) # Chai Wah Wu, May 02 2023

Definition corrected and edited by Omar E. Pol, Dec 24 2008

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);

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]

A383867 The sum of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or a squarefree divisor of the p-adic valuation of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 11, 13, 18, 12, 28, 14, 24, 24, 7, 18, 39, 20, 42, 32, 36, 24, 44, 31, 42, 31, 56, 30, 72, 32, 35, 48, 54, 48, 91, 38, 60, 56, 66, 42, 96, 44, 84, 78, 72, 48, 28, 57, 93, 72, 98, 54, 93, 72, 88, 80, 90, 60, 168, 62, 96, 104, 79, 84, 144, 68

Offset: 1

Amiram Eldar, May 13 2025

Analogous to the sum of (1+e)-divisors (A051378) as exponential squarefree exponential divisors (A383761, A361174) are analogous to exponential divisors (A322791, A051377).

The number of these divisors is A383863(n).

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

Cf. A051378, A209061, A322791, A361174, A383761, A383863.

f[p_, e_] := 1 + DivisorSum[e, p^# &, SquareFreeQ[#] &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + sumdiv(f[i, 2], d, if(issquarefree(d), f[i, 1]^d)));}

Multiplicative with a(p^e) = 1 + Sum_{d squarefree divisor of e} p^d.
a(n) <= A051378(n), with equality if and only if n is an exponentially squarefree number (A209061).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 1.47709589136345836345..., and f(x) = (1-x) * (1 + Sum_{k>=1} (1 + Sum{d|k, d squarefree} x^(2*k-d))).

cp's OEIS Frontend

A383864 The sum of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or a unitary divisor of the p-adic valuation of n.

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A383866 The sum of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or an infinitary divisor of the p-adic valuation of n.

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A087745 Numbers A001317 repeated.

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A274116 (1+e)-sigma amicable numbers.

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A274118 (1+e)-sigma betrothed numbers.

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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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A383867 The sum of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or a squarefree divisor of the p-adic valuation of n.

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