A333784 -id:A333784 - cp's OEIS frontend

A332994 a(1) = 1, for n > 1, a(n) = n + a(A052126(n)).

1, 3, 4, 7, 6, 9, 8, 15, 13, 13, 12, 19, 14, 17, 19, 31, 18, 27, 20, 27, 25, 25, 24, 39, 31, 29, 40, 35, 30, 39, 32, 63, 37, 37, 41, 55, 38, 41, 43, 55, 42, 51, 44, 51, 58, 49, 48, 79, 57, 63, 55, 59, 54, 81, 61, 71, 61, 61, 60, 79, 62, 65, 76, 127, 71, 75, 68, 75, 73, 83, 72, 111, 74, 77, 94, 83, 85, 87, 80, 111, 121

Offset: 1

Antti Karttunen, Apr 04 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A052126, A322382, A332904, A332993, A333784, A333791, A333794.

A332994(n) = if(1==n,n,n + A332994(n/vecmax(factor(n)[,1])));

a(1) = 1; and for n > 1, a(n) = n + a(A052126(n)).
a(n) = n + A322382(n).
a(n) = A332993(n) - A333791(n).
a(n) = A000203(n) - A333784(n).

A333783 a(n) = sigma(n) - A332993(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 6, 0, 2, 3, 0, 0, 8, 0, 6, 3, 2, 0, 14, 0, 2, 0, 6, 0, 21, 0, 0, 3, 2, 5, 24, 0, 2, 3, 14, 0, 25, 0, 6, 12, 2, 0, 30, 0, 12, 3, 6, 0, 26, 5, 14, 3, 2, 0, 57, 0, 2, 12, 0, 5, 33, 0, 6, 3, 31, 0, 56, 0, 2, 18, 6, 7, 37, 0, 30, 0, 2, 0, 69, 5, 2, 3, 14, 0, 78, 7, 6, 3, 2, 5, 62, 0, 16, 12

Offset: 1

Antti Karttunen, Apr 05 2020

Sum of all other divisors of n, except those divisors that can be obtained by repeatedly taking the largest proper divisor (of previous such divisor, starting from n), up to and including the terminal 1.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A000961 (positions of zeros), A001065, A006022, A032742, A332993, A333784, A333791.

A332993(n) = if(1==n,n,n + A332993(n/vecmin(factor(n)[,1])));
A333783(n) = (sigma(n) - A332993(n));

a(n) = A000203(n) - A332993(n).
a(n) = A001065(n) - A006022(n).
a(n) = A333784(n) - A333791(n).

A333791 Difference of sums of two subsets of divisors of n, those obtained by repeatedly dividing with the smallest remaining prime factor (A332993) and those obtained by repeatedly dividing with the largest remaining prime factor (A332994).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 3, 0, 5, 2, 0, 0, 4, 0, 9, 4, 9, 0, 7, 0, 11, 0, 15, 0, 12, 0, 0, 8, 15, 2, 12, 0, 17, 10, 21, 0, 20, 0, 27, 8, 21, 0, 15, 0, 18, 14, 33, 0, 13, 6, 35, 16, 27, 0, 32, 0, 29, 16, 0, 8, 36, 0, 45, 20, 30, 0, 28, 0, 35, 12, 51, 4, 44, 0, 45, 0, 39, 0, 52, 12, 41, 26, 63, 0, 39, 6, 63, 28, 45, 14, 31

Offset: 1

Antti Karttunen, Apr 05 2020

nonn

			For n = 12 = 2*2*3, we obtain the A332993(12) = 22 as 12 + 12/2 + 6/2 + 3/3 = 12+6+3+1, and A332994(12) = 19 as 12 + 12/3 + 4/2 + 2/2 = 12+4+2+1, thus a(12) = 22 - 19 = 3.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000961 (positions of zeros), A006022, A032742, A052126, A322382, A332993, A332994, A333783, A333784.

A332993(n) = if(1==n,n,n + A332993(n/vecmin(factor(n)[,1])));
A332994(n) = if(1==n,n,n + A332994(n/vecmax(factor(n)[,1])));
A333791(n) = (A332993(n)-A332994(n));

a(n) = A332993(n) - A332994(n).
a(n) = A333784(n) - A333783(n).
a(n) = A006022(n) - A322382(n).
a(p^k) = 0, for all primes p and exponents k >= 0.

A348988 Numerator of A332994(n) / sigma(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 13, 1, 19, 1, 17, 19, 1, 1, 9, 1, 9, 25, 25, 1, 13, 1, 29, 1, 5, 1, 13, 1, 1, 37, 37, 41, 55, 1, 41, 43, 11, 1, 17, 1, 17, 29, 49, 1, 79, 1, 21, 55, 59, 1, 27, 61, 71, 61, 61, 1, 79, 1, 65, 19, 1, 71, 25, 1, 25, 73, 83, 1, 37, 1, 77, 47, 83, 85, 29, 1, 37, 1, 85, 1, 103, 91, 89, 91, 103

Offset: 1

Antti Karttunen, Nov 06 2021

Ratio A332994(n) / sigma(n) tells how large proportion of the divisor sum we obtain if we sum just those divisors of n that can be obtained by repeatedly dividing a single instance of the largest prime divisor out of previous such divisor (when starting from n, which is included in the sum), up to and including the terminal 1. Pair a(n) / A348989(n) shows the ratio in the lowest terms: 1/1, 1/1, 1/1, 1/1, 1/1, 3/4, 1/1, 1/1, 1/1, 13/18, 1/1, 19/28, 1/1, 17/24, 19/24, 1/1, 1/1, 9/13, 1/1, 9/14, 25/32, 25/36, 1/1, 13/20, 1/1, 29/42, 1/1, 5/8, 1/1, 13/24, 1/1, etc. The ratio is 1 for all powers of primes (A000961).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A000961, A332994, A333784, A348987, A348989 (denominators).

Cf. also A348978.

f[n_] := n/FactorInteger[n][[-1, 1]]; g[1] = 1; g[n_] := g[n] = n + g[f[n]]; a[n_] := Numerator[g[n]/DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

A332994(n) = if(1==n,n,n + A332994(n/vecmax(factor(n)[,1])));
A348988(n) = { my(u=A332994(n)); (u/gcd(sigma(n), u)); };

a(n) = A332994(n) / A348987(n) = A332994(n) / gcd(A000203(n), A332994(n)).

A348989 Denominator of A332994(n) / sigma(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 1, 18, 1, 28, 1, 24, 24, 1, 1, 13, 1, 14, 32, 36, 1, 20, 1, 42, 1, 8, 1, 24, 1, 1, 48, 54, 48, 91, 1, 60, 56, 18, 1, 32, 1, 28, 39, 72, 1, 124, 1, 31, 72, 98, 1, 40, 72, 120, 80, 90, 1, 168, 1, 96, 26, 1, 84, 48, 1, 42, 96, 144, 1, 65, 1, 114, 62, 140, 96, 56, 1, 62, 1, 126, 1, 224, 108, 132, 120

Offset: 1

Antti Karttunen, Nov 06 2021

See comments in A348988.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A332994, A333784, A348987, A348988 (numerators).

Cf. also A348978, A348979.

f[n_] := n/FactorInteger[n][[-1, 1]]; g[1] = 1; g[n_] := g[n] = n + g[f[n]]; a[n_] := Denominator[g[n]/DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

A332994(n) = if(1==n,n,n + A332994(n/vecmax(factor(n)[,1])));
A348989(n) = { my(s=sigma(n)); (s/gcd(s, A332994(n))); };

a(n) = A000203(n) / A348987(n) = A000203(n) / gcd(A000203(n), A332994(n)).

A348987 a(n) = gcd(sigma(n), A332994(n)).

Original entry on oeis.org

1, 3, 4, 7, 6, 3, 8, 15, 13, 1, 12, 1, 14, 1, 1, 31, 18, 3, 20, 3, 1, 1, 24, 3, 31, 1, 40, 7, 30, 3, 32, 63, 1, 1, 1, 1, 38, 1, 1, 5, 42, 3, 44, 3, 2, 1, 48, 1, 57, 3, 1, 1, 54, 3, 1, 1, 1, 1, 60, 1, 62, 1, 4, 127, 1, 3, 68, 3, 1, 1, 72, 3, 74, 1, 2, 1, 1, 3, 80, 3, 121, 1, 84, 1, 1, 1, 1, 1, 90, 117, 1, 3, 1, 1, 1, 3

Offset: 1

Antti Karttunen, Nov 06 2021

nonn

Index entries for sequences related to sigma(n)

Cf. A000203, A332994, A333784, A348988, A348989.

Cf. also A348977.

f[n_] := n/FactorInteger[n][[-1, 1]]; g[1] = 1; g[n_] := g[n] = n + g[f[n]]; a[n_] := GCD[g[n], DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

A332994(n) = if(1==n,n,n + A332994(n/vecmax(factor(n)[,1])));
A348987(n) = gcd(sigma(n), A332994(n));

a(n) = gcd(A000203(n), A332994(n)).
a(n) = gcd(A000203(n), A333784(n)) = gcd(A332994(n), A333784(n)).
a(n) = A332994(n) / A348988(n) = A000203(n) / A348989(n).

cp's OEIS Frontend

A332994 a(1) = 1, for n > 1, a(n) = n + a(A052126(n)).

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A333783 a(n) = sigma(n) - A332993(n).

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A333791 Difference of sums of two subsets of divisors of n, those obtained by repeatedly dividing with the smallest remaining prime factor (A332993) and those obtained by repeatedly dividing with the largest remaining prime factor (A332994).

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A348988 Numerator of A332994(n) / sigma(n).

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A348989 Denominator of A332994(n) / sigma(n).

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A348987 a(n) = gcd(sigma(n), A332994(n)).

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