A080225 -id:A080225 - cp's OEIS frontend

A080224 Number of abundant divisors of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 1, 0, 1, 0

Offset: 1

Reinhard Zumkeller, Feb 07 2003

nonn

Number of divisors d of n with sigma(d)>2*d (sigma = A000203)

a(n)>0 iff n is abundant: a(A005101(n))>0, a(A000396(n))=0 and a(A005100(n))=0; a(A091191(n))=1; a(A091192(n))>1; a(A091193(n))=n and a(m)<>n for m < A091193(n). - Reinhard Zumkeller, Dec 27 2003

			Divisors of n=24: {1,2,3,4,6,8,12,24}, two of them are abundant: 12=A005101(1) and 24=A005101(4), therefore a(24)=2.

R. Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Abundant Number.

Cf. A000005, A000203, A005101, A080225, A080226, A187795, A294890, A294929, A294930, A294937.

Cf. also A294904.

A080224 := proc(n)
    a := 0 ;
    for d in numtheory[divisors](n) do
        if numtheory[sigma](d) > 2*d then
            a := a+1 ;
        end if;
    end do:
    a;
end proc:
seq(A080224(n),n=1..80) ; # R. J. Mathar, Feb 22 2021

Table[Count[Divisors[n],?(DivisorSigma[1,#]>2#&)],{n,110}] (* _Harvey P. Dale, Jun 14 2013 *)

a(n) = sumdiv(n, d, sigma(d)>2*d)  \\ Michel Marcus, Mar 09 2013

a(n) + A080225(n) + A080226(n) = A000005(n).
From Antti Karttunen, Nov 14 2017: (Start)
a(n) = Sum_{d|n} A294937(d).
a(n) = A294929(n) + A294937(n).
a(n) = 1 iff A294930(n) = 1.
(End)

A080226 Number of deficient divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 4, 2, 4, 2, 4, 4, 5, 2, 4, 2, 5, 4, 4, 2, 5, 3, 4, 4, 5, 2, 6, 2, 6, 4, 4, 4, 5, 2, 4, 4, 6, 2, 6, 2, 6, 6, 4, 2, 6, 3, 6, 4, 6, 2, 5, 4, 6, 4, 4, 2, 7, 2, 4, 6, 7, 4, 6, 2, 6, 4, 7, 2, 6, 2, 4, 6, 6, 4, 6, 2, 7, 5, 4, 2, 7, 4, 4, 4, 7, 2, 8, 4, 6, 4, 4, 4, 7, 2, 6, 6, 7, 2, 6, 2, 7, 8

Offset: 1

Reinhard Zumkeller, Feb 07 2003

nonn

Number of divisors d of n with sigma(d)<2*d (sigma = A000203).

			All 4 divisors of n=21 are deficient: 1=A005100(1), 3=A005100(3), 7=A005100(6) and 21=A005100(17), therefore a(21)=4.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Deficient Number.

Cf. A000203, A005100, A187793, A294926, A294934.

a[n_] := Sum[If[DivisorSigma[1, d] < 2d, 1, 0], {d, Divisors[n]}];
Array[a, 105] (* Jean-François Alcover, Dec 02 2021 *)

A080226(n) = sumdiv(n, d, (sigma(d)<(2*d))); \\ Antti Karttunen, Nov 14 2017

A080224(n) + A080225(n) + a(n) = A000005(n).

a(n) = Sum_{d|n} A294934(d) = A294926(n) + A294934(n). - Antti Karttunen, Nov 14 2017

A341620 Number of nondeficient divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 6, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0, 5, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 5, 0, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 6, 0, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 8

Offset: 1

Antti Karttunen, Feb 21 2021

nonn

Number of nondeficient numbers (A023196) dividing n.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000005, A006039 (positions of ones), A023196, A080224, A080225, A080226, A294927, A294936, A337690, A341619.

Differs from a derived sequence A341624 for the first time at n=120, where a(120)=8, while A341624(120)=1.

a[n_] := DivisorSum[n, 1 &, DivisorSigma[1, #] >= 2*# &]; Array[a, 120] (* Amiram Eldar, Feb 22 2021 *)

A294936(n) = (sigma(n, -1)>=2); \\ From A294936.
A341620(n) = sumdiv(n,d,A294936(d));

A341620(n) = sumdiv(n,d,(sigma(d)>=(2*d)));

a(n) = Sum_{d|n} A294936(d).
a(n) = A294927(n) + A294936(n).
a(n) = A080224(n) + A080225(n) = A000005(n) - A080226(n).
a(n) >= A337690(n) for all n.
a(n) = 1 iff A341619(n) = 1.

A378662 Number of divisors d of n such that sigma(d) <= 2*d < A003961(d), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 2, 0, 1, 1, 3, 0, 2, 0, 2, 1, 0, 0, 3, 0, 0, 2, 3, 0, 3, 0, 4, 0, 0, 1, 3, 0, 0, 1, 3, 0, 3, 0, 2, 3, 0, 0, 4, 1, 2, 0, 2, 0, 3, 0, 4, 1, 0, 0, 4, 0, 0, 3, 5, 0, 1, 0, 2, 1, 3, 0, 4, 0, 0, 2, 2, 0, 2, 0, 4, 3, 0, 0, 5, 0, 0, 0, 3, 0, 5, 1, 2, 0, 0, 0, 5, 0, 3, 2, 3, 0, 1, 0, 3, 4

Offset: 1

Antti Karttunen, Dec 06 2024

nonn

Number of terms of A341614 that divide n.

Claim: a(n) > 0 if and only if A003961(n) > 2*n [i.e., n is in A246282]. That a(n) must be zero when n is in A246281 is obvious, as is also that a(n) > 0 when n is a term of A341614 [as then A378664(n) = n], but that a(n) > 0 for all abundant numbers (A005101) is slightly less clear. So the claim boils down to this: All abundant numbers have at least one (by necessity a proper) divisor d|n such that it is in A341614, i.e., sigma(d) <= 2*d < A003961(d), i.e., that for abundant numbers n, A337345(n) is always strictly greater than A080224(n). Equivalently, of the all nonabundant divisors d of an abundant number, at least one is primeshift-abundant, i.e., A003961(d) > 2*d. This has been proved Dec 11 2024 by Jianing Song in A337372. The claim given in A378658 also follows from that proof.

Inverse Möbius transform of A341612.
Cf. A000203, A003961, A005101, A080224, A080225, A337345, A341614, A378663, A378664.
Cf. A246281 (positions of 0's), A246282 (of terms > 0).
Cf. also A337372, A378658.

Table[Length@ Select[Divisors[n], DivisorSigma[1, #] <= 2 # < (Times @@ Map[Power @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi[p] + 1], e}] - Boole[# == 1]) &], {n, 105}] (* Michael De Vlieger, Dec 06 2024 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A341612(n) = ((sigma(n)<=(2*n))&&((2*n)<A003961(n)));
A378662(n) = sumdiv(n,d,A341612(d));

a(n) = Sum_{d|n} A341612(d).
a(n) = A337345(n) - A080224(n).
a(n) = A080225(n) + A378663(n).

A147645 Number of distinct Mersenne primes dividing n.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1, 0, 1, 1, 0

Offset: 1

Omar E. Pol, Nov 09 2008

a(n) = m first occurs at n = A098918(m). - Robert Israel, Feb 03 2020

			a(21)=2 because 1, 3, 7 and 21 are divisors of 21. Then 21 has two divisors that are Mersenne primes (A000668): 3 and 7.

Antti Karttunen, Table of n, a(n) for n = 1..131072 (terms 1..10000 from Robert Israel)

Cf. A000265, A000668, A001221, A080225, A098918, A154402, A173898, A353786.

Coincides with A331410 on A054784.

N:= 100: # for a(1)..a(N)
V:= Vector(N):
for i from 1 do
m:= numtheory:-mersenne([i]);
if m > N then break fi;
for j from m by m to N do
    V[j]:= V[j]+1
od od:
convert(V,list); # Robert Israel, Feb 03 2020

A147645(n) = { my(m=3,s=0); while(m<=n, s += (isprime(m)*!(n%m)); m += (m+1)); (s); }; \\ Antti Karttunen, May 12 2022

From Antti Karttunen, May 12 2022: (Start)
a(n) = A154402(n) - A353786(n)
a(n) = a(2*n) = a(A000265(n)).
a(n) <= A331410(n). (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A173898 = 0.516454... . - Amiram Eldar, Dec 31 2023

A097796 Number of partitions of n into perfect numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0

Offset: 1

Reinhard Zumkeller, Aug 25 2004

nonn

a(2*n) = A097795(n).

a(A204878(n)) = 0; a(A204879(n)) > 0.

			a(90)=2: 90 = 15*6 = 15*A000396(1) = 3*28 + 1*6 = 3*A000396(2) + 1*A000396(1).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Number
Eric Weisstein's World of Mathematics, Partition
Wikipedia, Perfect number

Cf. A000396, A000041, A058696, A080225.

a097796 = p a000396_list where
   p _ 0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Jan 20 2012

f[x_] := Product[-(1/(-1 + x^i)), {i, {6, 28, 496, 8128, 33550336}}]; CoefficientList[Series[f[x], {x, 0, 1000}], x] (* Ben Branman, Jan 07 2012 *)

A097795 Number of partitions of 2*n into perfect numbers.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3

Offset: 1

Reinhard Zumkeller, Aug 25 2004

nonn

a(n) = A097796(2*n).

			a(45)=2: 2*45 = 90 = 15*6 = 15*A000396(1) = 3*28 + 1*6 = 3*A000396(2) + 1*A000396(1).

Eric Weisstein's World of Mathematics, Perfect Number
Eric Weisstein's World of Mathematics, Partition

Cf. A000396, A000041, A058696, A080225.

A378663 Number of divisors d of n such that sigma(d) < 2*d < A003961(d), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 1, 0, 1, 0, 1, 1, 3, 0, 1, 0, 2, 1, 0, 0, 2, 0, 0, 2, 2, 0, 2, 0, 4, 0, 0, 1, 2, 0, 0, 1, 3, 0, 2, 0, 2, 3, 0, 0, 3, 1, 2, 0, 2, 0, 2, 0, 3, 1, 0, 0, 3, 0, 0, 3, 5, 0, 0, 0, 2, 1, 3, 0, 3, 0, 0, 2, 2, 0, 1, 0, 4, 3, 0, 0, 3, 0, 0, 0, 3, 0, 4, 1, 2, 0, 0, 0, 4, 0, 3, 2, 3, 0, 0, 0, 3, 4

Offset: 1

Antti Karttunen, Dec 06 2024

nonn

Number of terms of A341615 that divide n.

Inverse Möbius transform of A341613.

Cf. A000203, A003961, A080225, A337345, A341615, A341620, A378662.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A341613(n) = ((sigma(n)<(2*n))&&((2*n)<A003961(n)));
A378663(n) = sumdiv(n,d,A341613(d));

a(n) = Sum_{d|n} A341613(d).
a(n) = A337345(n) - A341620(n).
a(n) = A378662(n) - A080225(n).

A147648 Number of distinct even superperfect numbers dividing n.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2

Offset: 1

Omar E. Pol, Nov 09 2008

Also, numbers of distinct superperfect numbers dividing n, if there are no odd superperfect numbers.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001221, A019279, A061652, A080225, A147645.

Array[DivisorSum[#, 1 &, And[EvenQ@ #, Nest[DivisorSigma[1, #] &, #, 2] == 2 #] &] &, 105] (* Michael De Vlieger, Nov 06 2018 *)

A147648(n) = sumdiv(n,d,(!(d%2)&&(sigma(sigma(d))==(2*d)))); \\ Antti Karttunen, Nov 06 2018

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=1} 1/A061652(n) = 0.828388215042... . - Amiram Eldar, Jan 01 2024

cp's OEIS Frontend

A080224 Number of abundant divisors of n.

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A080226 Number of deficient divisors of n.

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A341620 Number of nondeficient divisors of n.

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A378662 Number of divisors d of n such that sigma(d) <= 2*d < A003961(d), where A003961 is fully multiplicative with a(p) = nextprime(p).

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A147645 Number of distinct Mersenne primes dividing n.

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A097796 Number of partitions of n into perfect numbers.

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A097795 Number of partitions of 2*n into perfect numbers.

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