A147645 -id:A147645 - cp's OEIS frontend

A154402 Inverse Moebius transform of Fredholm-Rueppel sequence, cf. A036987.

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

Offset: 1

Vladeta Jovovic, Jan 08 2009

Number of ways to write n as a sum a_1 + ... + a_k where the a_i are positive integers and a_i = 2 * a_{i-1}, cf. A000929.

Number of divisors of n of the form 2^k - 1 (A000225) for k >= 1. - Jeffrey Shallit, Jan 23 2017

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 10000 terms from Robert Israel)

Cf. A000225, A000929, A001511, A036987, A065442, A147645, A161790 (positions of 1's), A305426, A353786.

Cf. also A305436.

N:= 200: # to get a(1)..a(N)
A:= Vector(N):
for k from 1 do
   t:= 2^k-1;
   if t > N then break fi;
   R:= [seq(i,i=t..N,t)];
   A[R]:= map(`+`,A[R],1)
od:
convert(A,list); # Robert Israel, Jan 23 2017

Table[DivisorSum[n, 1 &, IntegerQ@ Log2[# + 1] &], {n, 105}] (* Michael De Vlieger, Jun 11 2018 *)

A209229(n) = (n && !bitand(n,n-1));
A036987(n) = A209229(1+n);
A154402(n) = sumdiv(n,d,A036987(d)); \\ Antti Karttunen, Jun 11 2018

A154402(n) = { my(m=1,s=0); while(m<=n, s += !(n%m); m += (m+1)); (s); }; \\ Antti Karttunen, May 12 2022

G.f.: Sum_{k>0} x^(2^k-1)/(1-x^(2^k-1)).
From Antti Karttunen, Jun 11 2018: (Start)
a(n) = Sum_{d|n} A036987(d).
a(n) = A305426(n) + A036987(n). (End)
a(n) = A147645(n) + A353786(n). - Antti Karttunen, May 12 2022
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A065442 = 1.606695... . - Amiram Eldar, Dec 31 2023

A080225 Number of perfect divisors of n.

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

Offset: 1

Reinhard Zumkeller, Feb 07 2003

nonn

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

			Divisors of n = 84: {1,2,3,4,6,7,12,14,21,24,28,42}, two of them are perfect: 6 = A000396(1) and 28 = A000396(2), therefore a(84) = 2.

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

Cf. A000203, A000396, A080224, A080226, A147645, A335118.

a080225 n = length [d | d <- takeWhile (<= n) a000396_list, mod n d == 0]
-- Reinhard Zumkeller, Jan 20 2012

a[n_] := DivisorSum[n, 1 &, DivisorSigma[-1, #] == 2 &]; Array[a, 100] (* Amiram Eldar, Dec 31 2023 *)

a(n) = sumdiv(n, d, sigma(d, -1) == 2); \\ Amiram Eldar, Dec 31 2023

A080224(n) + a(n) + A080226(n) = A000005(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A335118 = 0.2045201... . - Amiram Eldar, Dec 31 2023

A239930 Number of distinct quarter-squares dividing n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jun 19 2014

nonn

For more information about the quarter-squares see A002620.

			For n = 12 the quarter-squares <= 12 are [0, 0, 1, 2, 4, 6, 9, 12]. There are five quarter-squares that divide 12; they are [1, 2, 4, 6, 12], so a(12) = 5.

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

Cf. A000005, A001221, A001511, A002620, A005086, A006519, A007862, A013661, A027750, A046951, A147645, A236103.

Cf. A240025.

a239930 = sum . map a240025 . a027750_row
-- Reinhard Zumkeller, Jul 05 2014

isA002620 := proc(n)
    local k,qsq ;
    for k from 0 do
        qsq := floor(k^2/4) ;
        if n = qsq then
            return true;
        elif qsq > n then
            return false;
        end if;
    end do:
end proc:
A239930 := proc(n)
    local a,d ;
    a :=0 ;
    for d in numtheory[divisors](n) do
        if isA002620(d) then
            a:= a+1 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Jul 03 2014

qsQ[n_] := AnyTrue[Range[Ceiling[2 Sqrt[n]]], n == Floor[#^2/4]&]; a[n_] := DivisorSum[n, Boole[qsQ[#]]&]; Array[a, 110] (* Jean-François Alcover, Feb 12 2018 *)

a(n) = sumdiv(n, d, issquare(d) + issquare(4*d + 1)); \\ Amiram Eldar, Dec 31 2023

a(n) = Sum_{k=1..A000005(n)} A240025(A027750(n,k)). - Reinhard Zumkeller, Jul 05 2014

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) + 1 = A013661 + 1 = 2.644934... . - Amiram Eldar, Dec 31 2023

A353786 Number of distinct nonprime numbers of the form 2^k - 1 that divide n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Antti Karttunen, May 12 2022

nonn

			Divisors of 255 are [1, 3, 5, 15, 17, 51, 85, 255], of these of the form 2^k - 1 (A000225) are 1, 3, 15 and 255, but only three of them are counted (because 3 is a prime), therefore a(255) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000225, A000265, A147645, A154402.

Cf. A065442, A135972, A173898.

a[n_] := DivisorSum[n, 1 &, !PrimeQ[#] && # + 1 == 2^IntegerExponent[# + 1, 2] &]; Array[a, 120] (* Amiram Eldar, May 12 2022 *)

A353786(n) = { my(m=1,s=0); while(m<=n, s += (!isprime(m))*!(n%m); m += (m+1)); (s); };

a(n) = A154402(n) - A147645(n).
a(n) = a(2*n) = a(A000265(n)).
For all primes p, a(p) = 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=2} 1/A135972(n) = A065442 - A173898 = 1.0902409734... . - Amiram Eldar, Dec 31 2023

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

A244964 Number of distinct generalized pentagonal numbers dividing n.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 3, 1, 3, 3, 2, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 3, 1, 4, 1, 2, 1, 2, 4, 3, 1, 2, 1, 4, 1, 3, 1, 3, 3, 2, 1, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 2, 1, 5, 1, 2, 2, 2, 2, 3, 1, 2, 1, 6, 1, 3, 1, 2, 3, 2, 3, 3, 1, 4, 1, 2, 1, 4, 2, 2, 1, 3, 1, 4, 2, 3, 1, 2, 2, 3, 1, 3, 1, 4, 1, 3, 1, 3, 5

Offset: 1

Omar E. Pol, Jul 10 2014

nonn

For more information about the generalized pentagonal numbers see A001318.

			For n = 10 the generalized pentagonal numbers <= 10 are [0, 1, 2, 5, 7]. There are three generalized pentagonal numbers that divide 10; they are [1, 2, 5], so a(10) = 3.

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

Cf. A000005, A001221, A001318, A001511, A005086, A006519, A007862, A027750, A046951, A080995, A147645, A175003, A236103, A238442, A239930.

a[n_] := DivisorSum[n, 1 &, IntegerQ[Sqrt[24*# + 1]] &]; Array[a, 100] (* Amiram Eldar, Dec 31 2023 *)

a(n) = sumdiv(n, d, issquare(24*d + 1)); \\ Amiram Eldar, Dec 31 2023

From Amiram Eldar, Dec 31 2023: (Start)
a(n) = Sum_{d|n} A080995(d).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 6 - 2*Pi/sqrt(3) = 2.372401... . (End)

cp's OEIS Frontend

A154402 Inverse Moebius transform of Fredholm-Rueppel sequence, cf. A036987.

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A080225 Number of perfect divisors of n.

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A239930 Number of distinct quarter-squares dividing n.

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A353786 Number of distinct nonprime numbers of the form 2^k - 1 that divide n.

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A147648 Number of distinct even superperfect numbers dividing n.

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A244964 Number of distinct generalized pentagonal numbers dividing n.

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