A000593 -id:A000593 - cp's OEIS frontend

A336361 Number of iterations of A000593 (sum of divisors of odd part of n) needed to reach a power of 2, or -1 if never reached.

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

Offset: 1

Antti Karttunen, Jul 30 2020

nonn

Also, for n > 1, one less than the number of iterations of A000593 to reach 1.

If there exists any hypothetical odd perfect numbers w, then the iteration will get stuck into a fixed point after encountering them, and we will have a(w) = a(2^k * w) = -1 by the escape clause.

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

Cf. A000265, A000593, A161942, A209229, A336362, A336363, A347249, A347250.
Cf. A054784 (positions of 0's and 1's in this sequence).
Cf. also A347240, A347241, A347242, A347243, A347244, A347245.

A336361(n) = if(!bitand(n,n-1),0,1+A336361(sigma(n>>valuation(n,2))));

If A209229(n) = 1 [when n is a power of 2], a(n) = 0, otherwise a(n) = 1+a(A000593(n)).

a(n) = a(2n) = a(A000265(n)).

A288417 a(n) = Sum_{d|n} A000593(n/d).

Original entry on oeis.org

1, 2, 5, 3, 7, 10, 9, 4, 18, 14, 13, 15, 15, 18, 35, 5, 19, 36, 21, 21, 45, 26, 25, 20, 38, 30, 58, 27, 31, 70, 33, 6, 65, 38, 63, 54, 39, 42, 75, 28, 43, 90, 45, 39, 126, 50, 49, 25, 66, 76, 95, 45, 55, 116, 91, 36, 105, 62, 61, 105, 63, 66, 162, 7, 105, 130, 69

Offset: 1

Seiichi Manyama, Jun 09 2017

Multiplicative because this sequence is the Dirichlet convolution of A000012 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 27 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000012.

Sum_{d|n} d^k*A000593(n/d): this sequence (k=0), A109386 (k=1), A288418 (k=2), A288419 (k=3), A288420 (k=4).

f[p_, e_] := Sum[(i + 1)*p^(e - i), {i, 0, e}]; f[2, e_] := e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2022 *)

a(n)={sumdiv(n, d, sigma(d>>valuation(d,2)))} \\ Andrew Howroyd, Jul 27 2018

L.g.f.: log(Product_{k>=1} (1 + x^k)^(sigma(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jun 19 2018
Multiplicative with a(2^e) = e+1 and a(p^e) = Sum_{i=0..e} (i+1)*p^(e-i) for e >= 0 and prime p > 2. - Werner Schulte, Jan 05 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/144 = 0.676452... . - Amiram Eldar, Oct 22 2022

A324186 Sum of odd divisors permuted by A163511: a(n) = A000593(A163511(n)).

Original entry on oeis.org

1, 1, 1, 4, 1, 13, 4, 6, 1, 40, 13, 31, 4, 24, 6, 8, 1, 121, 40, 156, 13, 124, 31, 57, 4, 78, 24, 48, 6, 32, 8, 12, 1, 364, 121, 781, 40, 624, 156, 400, 13, 403, 124, 342, 31, 228, 57, 133, 4, 240, 78, 248, 24, 192, 48, 96, 6, 104, 32, 72, 8, 48, 12, 14, 1, 1093, 364, 3906, 121, 3124, 781, 2801, 40, 2028, 624, 2400, 156, 1600, 400, 1464, 13, 1240, 403

Offset: 0

Antti Karttunen, Feb 17 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A000593, A038712, A054429, A163511, A324056, A324184.

A000593(n) = sigma(n>>valuation(n, 2)); \\ From A000593
A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A324186(n) = A000593(A163511(n));

a(n) = A000593(A163511(n)).

For n > 0, a(n) = A324056(A054429(n)).

A322819 a(n) = A000593(A122111(n)).

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 1, 6, 13, 4, 1, 6, 1, 4, 13, 8, 1, 24, 1, 6, 13, 4, 1, 8, 40, 4, 31, 6, 1, 24, 1, 12, 13, 4, 40, 32, 1, 4, 13, 8, 1, 24, 1, 6, 31, 4, 1, 12, 121, 78, 13, 6, 1, 48, 40, 8, 13, 4, 1, 32, 1, 4, 31, 14, 40, 24, 1, 6, 13, 78, 1, 48, 1, 4, 124, 6, 121, 24, 1, 12, 57, 4, 1, 32, 40, 4, 13, 8, 1, 48, 121, 6, 13, 4, 40, 14, 1, 240

Offset: 1

Antti Karttunen, Dec 27 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A000593, A122111, A322813.

A000593(n) = sigma(n>>valuation(n, 2));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A322819(n) = A000593(A122111(n));

a(n) = A000593(A122111(n)).

A347240 a(n) is the largest prime factor (A006530) of all terms encountered when iterating the map x -> A000593(x), when starting from x = n, but excluding the n itself. If n is a power of 2, then a(n) = 1. If 1 is never reached, then a(n) = -1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 2, 1, 13, 3, 3, 2, 7, 2, 3, 1, 13, 13, 5, 3, 2, 3, 3, 2, 31, 7, 5, 2, 5, 3, 2, 1, 3, 13, 3, 13, 19, 5, 7, 3, 7, 2, 11, 3, 13, 3, 3, 2, 19, 31, 13, 7, 5, 5, 13, 2, 5, 5, 5, 3, 31, 2, 13, 1, 7, 3, 17, 13, 3, 3, 13, 13, 37, 19, 31, 5, 3, 7, 5, 3, 19, 7, 7, 2, 5, 11, 5, 3, 13, 13, 7, 3, 2, 3, 5, 2, 19

Offset: 1

Antti Karttunen, Aug 28 2021

nonn

			For n = 17, the iteration proceeds as follows 17 -> 18 (= 2*3*3), 18 -> 13 (13 is a prime), 13 -> 14 (= 2*7), 14 -> 8 (= 2*2*2), 8 -> 1. The largest prime factor present after the initial step is 13, thus a(17) = 13.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000593, A006530, A336361, A347241, A347242, A347243, A347244.

Cf. also A161942.

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A000265(n) = (n >> valuation(n, 2));
A000593(n) = sigma(A000265(n));
A347240(n) = { my(m=1); while(n>1, n = A000593(n); m = max(m, A006530(n))); (m); };

a(n) = A347241(A000593(n)). - Antti Karttunen, Feb 10 2022

A288418 a(n) = Sum_{d|n} d^2*A000593(n/d).

Original entry on oeis.org

1, 5, 13, 21, 31, 65, 57, 85, 130, 155, 133, 273, 183, 285, 403, 341, 307, 650, 381, 651, 741, 665, 553, 1105, 806, 915, 1210, 1197, 871, 2015, 993, 1365, 1729, 1535, 1767, 2730, 1407, 1905, 2379, 2635, 1723, 3705, 1893, 2793, 4030, 2765, 2257, 4433, 2850, 4030

Offset: 1

Seiichi Manyama, Jun 09 2017

Multiplicative because this sequence is the Dirichlet convolution of A000290 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 27 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000290, A001001, A192065, A183699.

Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), this sequence (k=2), A288419 (k=3), A288420 (k=4).

a[n_] := DivisorSum[n, Function[d, d^2*DivisorSum[n/d, If[OddQ[#], #, 0]&]] ];
Array[a, 50] (* Jean-François Alcover, Jul 03 2017 *)
f[p_, e_] := (p^(e + 1) - 1)*(p^(e + 2) - 1)/((p - 1)*(p^2 - 1)); f[2, e_] := (4^(e + 1) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

a(n) = sumdiv(n, d, d^2*sigma((n/d)>>valuation(n/d, 2))); \\ Michel Marcus, Jul 03 2017; corrected Jun 12 2022

L.g.f.: log(Product_{k>=1} (1 + x^k)^sigma(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jun 19 2018
From Amiram Eldar, Nov 13 2022: (Start)
a(n) = A001001(n) for odd n.
Multiplicative with a(2^e) = (4^(e+1)-1)/3 and a(p^e) = (p^(e+1)-1)*(p^(e+2)-1)/((p-1)*(p^2-1)) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(2)*zeta(3)/4 = A183699 / 4 = 0.494326... . (End)

A288419 a(n) = Sum_{d|n} d^3*A000593(n/d).

Original entry on oeis.org

1, 9, 31, 73, 131, 279, 351, 585, 850, 1179, 1343, 2263, 2211, 3159, 4061, 4681, 4931, 7650, 6879, 9563, 10881, 12087, 12191, 18135, 16406, 19899, 22990, 25623, 24419, 36549, 29823, 37449, 41633, 44379, 45981, 62050, 50691, 61911, 68541, 76635, 68963, 97929

Offset: 1

Seiichi Manyama, Jun 09 2017

Multiplicative because this sequence is the Dirichlet convolution of A000578 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 27 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000578, A027847, A183700, A288414.

Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), A288418 (k=2), this sequence (k=3), A288420 (k=4).

f[p_, e_] := (p^(3*e+5) - (p^2+p+1)*p^(e+1) + p + 1)/((p^3-1)*(p^2-1)); f[2, e_] := (8^(e+1)-1)/7; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

a(n)={sumdiv(n, d, (n/d)^3*sigma(d>>valuation(d,2)))} \\ Andrew Howroyd, Jul 27 2018

From Amiram Eldar, Nov 13 2022: (Start)
a(n) = A027847(n) for odd n.
Multiplicative with a(2^e) = (8^(e+1)-1)/7 and a(p^e) = (p^(3*e+5) - (p^2+p+1)*p^(e+1) + p + 1)/((p^3-1)*(p^2-1)) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^4, where c = 7*Pi^4*zeta(3)/2880 = (7/32)*zeta(3)*zeta(4) = (7/32) * A183700 = 0.284596... . (End)

A288420 a(n) = Sum_{d|n} d^4*A000593(n/d).

Original entry on oeis.org

1, 17, 85, 273, 631, 1445, 2409, 4369, 6898, 10727, 14653, 23205, 28575, 40953, 53635, 69905, 83539, 117266, 130341, 172263, 204765, 249101, 279865, 371365, 394406, 485775, 558778, 657657, 707311, 911795, 923553, 1118481, 1245505, 1420163, 1520079, 1883154

Offset: 1

Seiichi Manyama, Jun 09 2017

Multiplicative because this sequence is the Dirichlet convolution of A000583 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 20 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000583, A013662, A013663, A027848, A288415.

Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), A288418 (k=2), A288419 (k=3), this sequence (k=4).

f[p_, e_] :=  (p^(4*e+7) - (p^3+p^2+p+1)*p^(e+1) + p^2 + p + 1)/(p^7 - (p^3+p^2+p+1)*p + p^2 + p + 1); f[2, e_] := (16^(e+1)-1)/15; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

From Amiram Eldar, Nov 13 2022: (Start)
a(n) = A027848(n) for odd n.
Multiplicative with a(2^e) = (16^(e+1)-1)/15 and a(p^e) = (p^(4*e+7) - (p^3+p^2+p+1)*p^(e+1) + p^2 + p + 1)/(p^7 - (p^3+p^2+p+1)*p + p^2 + p + 1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^5, where c = Pi^4*zeta(5)/480 = (3/16)*zeta(4)*zeta(5) = 0.210429... . (End)

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A347242 Numbers k such that when iterating the map x -> A000593(x), at some point before 1 is reached (after starting from x=k), a term is encountered whose largest prime factor is at least as large as A006530(k).

Original entry on oeis.org

9, 18, 25, 27, 36, 45, 49, 50, 54, 55, 63, 72, 75, 81, 90, 98, 99, 100, 108, 110, 117, 121, 125, 126, 135, 144, 147, 150, 162, 165, 169, 175, 180, 196, 198, 200, 216, 220, 225, 234, 242, 243, 245, 250, 252, 270, 275, 288, 289, 294, 300, 315, 324, 325, 330, 338, 343, 350, 360, 361, 363, 375, 385, 392, 396, 400

Offset: 1

Antti Karttunen, Aug 28 2021

nonn

Provided there do not exist any odd perfect numbers, these are numbers k for which A347240(k) >= A006530(k), as for any odd perfect number x, A347240(x) = -1 by its escape clause.
If k is included as a term, then 2*k is also present.
Not all odd squares of primes are present. For example, 67^2 and 79^2 are not included. See also A091490, which seems to be a subsequence of those exceptions.
Conjecture: There are no primes in this sequence. Checked up to the 2^20-th prime, 16290047.

			For n = 55 = 5*11, on the first iteration we get A000593(55) = 72 = 2^3 * 3^2, but both 2 and 3 are less than 11; therefore we iterate a second time to get A000593(72) = 13, which is the first value whose largest prime factor is larger than that of 55 (13 > 11), thus 55 is included in the sequence.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000593, A006530, A091490, A161942, A336353, A336361, A347240, A347241, A347243 (complement), A347244 (characteristic function).

Positions of nonzero terms in A347245.

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A000265(n) = (n >> valuation(n, 2));
A000593(n) = sigma(A000265(n));
A347244(n) = { my(gpf=A006530(n)); while(n>1, n = A000593(n); if(A006530(n)>=gpf,return(1))); (0); };
isA347242(n) = A347244(n);

A347374 Lexicographically earliest infinite sequence such that a(i) = a(j) => A331410(i) = A331410(j) and A000593(i) = A000593(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 17, 5, 18, 10, 19, 3, 20, 11, 21, 6, 22, 12, 23, 2, 24, 13, 25, 7, 26, 14, 25, 4, 27, 15, 28, 8, 29, 16, 30, 1, 31, 17, 32, 9, 33, 17, 34, 5, 35, 18, 36, 10, 33, 19, 37, 3, 38, 20, 39, 11, 40, 21, 41, 6, 42, 22, 43, 12

Offset: 1

Antti Karttunen, Aug 29 2021

nonn

Restricted growth sequence transform of the ordered pair [A000593(n), A331410(n)].

For all i, j: A003602(i) = A003602(j) => a(i) = a(j) => A347249(i) = A347249(j).

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

Cf. A000593, A003602, A331410, A347249.

Cf. also A335880, A336390, A336391, A336394 for similar constructions.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A000593(n) = sigma(n>>valuation(n, 2));
A331410(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A331410(f[k,1]+1)))); };
Aux347374(n) = [A331410(n), A000593(n)];
v347374 = rgs_transform(vector(up_to, n, Aux347374(n)));
A347374(n) = v347374[n];

cp's OEIS Frontend

A336361 Number of iterations of A000593 (sum of divisors of odd part of n) needed to reach a power of 2, or -1 if never reached.

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A288417 a(n) = Sum_{d|n} A000593(n/d).

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A324186 Sum of odd divisors permuted by A163511: a(n) = A000593(A163511(n)).

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A322819 a(n) = A000593(A122111(n)).

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A347240 a(n) is the largest prime factor (A006530) of all terms encountered when iterating the map x -> A000593(x), when starting from x = n, but excluding the n itself. If n is a power of 2, then a(n) = 1. If 1 is never reached, then a(n) = -1.

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A288418 a(n) = Sum_{d|n} d^2*A000593(n/d).

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A288419 a(n) = Sum_{d|n} d^3*A000593(n/d).

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A288420 a(n) = Sum_{d|n} d^4*A000593(n/d).

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