A332994 -id:A332994 - cp's OEIS frontend

A333784 a(n) = sigma(n) - A332994(n).

0, 0, 0, 0, 0, 3, 0, 0, 0, 5, 0, 9, 0, 7, 5, 0, 0, 12, 0, 15, 7, 11, 0, 21, 0, 13, 0, 21, 0, 33, 0, 0, 11, 17, 7, 36, 0, 19, 13, 35, 0, 45, 0, 33, 20, 23, 0, 45, 0, 30, 17, 39, 0, 39, 11, 49, 19, 29, 0, 89, 0, 31, 28, 0, 13, 69, 0, 51, 23, 61, 0, 84, 0, 37, 30, 57, 11, 81, 0, 75, 0, 41, 0, 121, 17, 43, 29, 77, 0, 117, 13, 69, 31

Offset: 1

Antti Karttunen, Apr 05 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
Index entries for sequences related to sigma(n)

Cf. A000203, A000961 (positions of zeros), A001065, A052126, A322382, A332994, A333783, A333791.

A332994(n) = if(1==n,n,n + A332994(n/vecmax(factor(n)[,1])));
A333784(n) = (sigma(n) - A332994(n));

a(n) = A000203(n) - A332994(n).
a(n) = A001065(n) - A322382(n).
a(n) = A333783(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).

A332993 a(1) = 1, for n > 1, a(n) = n + a(A032742(n)).

Original entry on oeis.org

1, 3, 4, 7, 6, 10, 8, 15, 13, 16, 12, 22, 14, 22, 21, 31, 18, 31, 20, 36, 29, 34, 24, 46, 31, 40, 40, 50, 30, 51, 32, 63, 45, 52, 43, 67, 38, 58, 53, 76, 42, 71, 44, 78, 66, 70, 48, 94, 57, 81, 69, 92, 54, 94, 67, 106, 77, 88, 60, 111, 62, 94, 92, 127, 79, 111, 68, 120, 93, 113, 72, 139, 74, 112, 106, 134, 89, 131, 80, 156, 121

Offset: 1

Antti Karttunen, Apr 04 2020

nonn

Sum of those divisors of n that can be obtained by repeatedly taking the largest proper divisor (of previous such divisor, starting from n, which is included in the sum), up to and including the terminal 1.

			a(18) = 18 + 18/2 + 9/3 + 3/3 = 18 + 9 + 3 + 1 = 31.

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, A006022, A032742, A064097, A073934, A332994, A333783, A333791, A333794.

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

a(1) = 1; and for n > 1, a(n) = n + a(A032742(n)).
a(n) = n + A006022(n).
a(n) = A332994(n) + A333791(n).
a(n) = A000203(n) - A333783(n).
It seems that for all n >= 1, a(n) <= A073934(n) <= A333794(n).

A333794 a(1) = 1, for n > 1, a(n) = n + a(n-A052126(n)).

Original entry on oeis.org

1, 3, 6, 7, 12, 13, 20, 15, 22, 25, 36, 27, 40, 41, 42, 31, 48, 45, 64, 51, 66, 73, 96, 55, 76, 81, 72, 83, 112, 85, 116, 63, 118, 97, 120, 91, 128, 129, 130, 103, 144, 133, 176, 147, 136, 193, 240, 111, 182, 153, 162, 163, 216, 145, 208, 167, 202, 225, 284, 171, 232, 233, 208, 127, 236, 237, 304, 195, 306, 241, 312, 183, 256, 257

Offset: 1

Antti Karttunen, Apr 05 2020

nonn

Conjecturally, also the largest path sum when iterating from n to 1 with nondeterministic map k -> k - k/p, where p is any prime factor of k.

			For n=119, the graph obtained is this:
              119
             _/\_
            /    \
          102    112
         _/|\_    | \_
       _/  |  \_  |   \_
      /    |    \ |     \
    51     68    96     56
    /|   _/ |   _/|   _/ |
   / | _/   | _/  | _/   |
  /  |/     |/    |/     |
(48) 34    64     48    28
     |\_    |    _/|   _/|
     |  \_  |  _/  | _/  |
     |    \_|_/    |/    |
    17     32     24    14
      \_    |    _/|   _/|
        \_  |  _/  | _/  |
          \_|_/    |/    |
           16      12    7
            |    _/|    _/
            |  _/  |  _/
            |_/    |_/
            8     _6
            |  __/ |
            |_/    |
            4      3
             \     /
              \_ _/
                2
                |
                1.
If we always subtract A052126(n) (i.e., n divided by its largest prime divisor), i.e., iterate with A171462 (starting from 119), we obtain 119-(119/17) = 112 -> 112-(112/7) -> 96-(96/3) -> 64-(64/2) -> 32-(32/2) -> 16-(16/2) -> 8-(8/2) -> 4-(4/2) -> 2-(2/2) -> 1, with sum 119+112+96+64+32+16+8+4+2+1 = 554, thus a(119) = 554. This happens also to be maximal sum of any path in above diagram.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Michael De Vlieger, Graph montage of k -> k - k/p, with prime p|k for 2 <= k <= 211, red line showing path of greatest sum, blue the path of least sum (cf. A333790), and purple where the two paths coincide, with other paths in gray.

Cf. A052126, A073934, A171462, A332994, A333790, A333793.

Array[Total@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # > 1 &] &, 74] (* Michael De Vlieger, Apr 14 2020 *)

A333794(n) = if(1==n,n,n + A333794(n-(n/vecmax(factor(n)[, 1]))));

a(1) = 1; and for n > 1, a(n) = n + a(A171462(n)) = n + a(n-A052126(n)).
a(n) = A073934(n) + A333793(n).
a(n) = n + Max a(n - n/p), for p prime and dividing n. [Conjectured, holds at least up to n=2^24]
For all n >= 1, A333790(n) <= a(n) <= A332904(n).
For all n >= 1, a(n) >= A332993(n). [Apparently, have to check!]

A332904 Sum of distinct integers encountered on all possible paths from n to 1 when iterating with nondeterministic map k -> k - k/p, where p is any of the prime factors of k.

Original entry on oeis.org

1, 3, 6, 7, 12, 16, 23, 15, 25, 30, 41, 36, 49, 57, 66, 31, 48, 63, 82, 66, 105, 99, 122, 76, 91, 115, 90, 125, 154, 156, 187, 63, 222, 114, 240, 139, 176, 196, 217, 138, 179, 251, 294, 215, 264, 284, 331, 156, 300, 213, 258, 247, 300, 220, 345, 261, 334, 348, 407, 336, 397, 429, 395, 127, 492, 512, 579, 246, 650, 546, 617, 291, 364

Offset: 1

Antti Karttunen, Apr 04 2020

nonn

			a(12): we have three alternative paths: {12, 8, 4, 2, 1}, {12, 6, 4, 2, 1} or {12, 6, 3, 2, 1}, with numbers [1, 2, 3, 4, 6, 8, 12] present, therefore a(12) = 1+2+3+4+6+8+12 = 36.
For n=15 we have five alternative paths from 15 to 1: {15, 10, 5, 4, 2, 1}, {15, 10, 8, 4, 2, 1}, {15, 12, 8, 4, 2, 1},  {15, 12, 6, 4, 2, 1},  {15, 12, 6, 3, 2, 1}. These form a lattice illustrated below:
        15
       / \
      /   \
    10     12
    / \   / \
   /   \ /   \
  5     8     6
   \__  |  __/|
      \_|_/   |
        4     3
         \   /
          \ /
           2
           |
           1,
therefore a(15) = 1+2+3+4+5+6+8+10+12+15 = 66.

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

Cf. A064097, A073934, A332809, A332993, A332994, A333000, A333123.

Cf. A333790 (sum of the route with minimal sum), A333794.

Total /@ Nest[Function[{a, n}, Append[a, Union@ Flatten@ Table[Append[a[[n - n/p]], n], {p, FactorInteger[n][[All, 1]]}]]] @@ {#, Length@ # + 1} &, {{1}}, 72] (* Michael De Vlieger, Apr 15 2020 *)

up_to = 20000;
A332904list(up_to) = { my(v=vector(up_to)); v[1] = Set([1]); for(n=2,up_to, my(f=factor(n)[, 1]~, s=Set([n])); for(i=1,#f,s = setunion(s,v[n-(n/f[i])])); v[n] = s); apply(vecsum,v); }
v332904 = A332904list(up_to);
A332904(n) = v332904[n];

For all primes p, a(p) = a(p-1) + p.

For all n >= 1, A333000(n) >= a(n) >= A333794(n) >= A333790(n).

A348991 a(n) = A333791(A276086(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 4, 0, 3, 2, 12, 8, 39, 0, 18, 12, 67, 48, 214, 0, 93, 62, 342, 248, 1089, 0, 468, 312, 1717, 1248, 5464, 0, 5, 4, 20, 16, 65, 2, 30, 24, 109, 90, 346, 12, 155, 124, 554, 460, 1751, 62, 780, 624, 2779, 2310, 8776, 312, 3905, 3124, 13904, 11560, 43901, 0, 40, 32, 153, 128, 492, 16, 219, 178, 788, 664, 2495

Offset: 0

Antti Karttunen, Nov 07 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A276086, A332993, A332994, A333791.

Cf. A060735 (gives the positions of other zeros after the initial a(0)=0).

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
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));
A348991(n) = A333791(A276086(n));

cp's OEIS Frontend

A333784 a(n) = sigma(n) - A332994(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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A332993 a(1) = 1, for n > 1, a(n) = n + a(A032742(n)).

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A333794 a(1) = 1, for n > 1, a(n) = n + a(n-A052126(n)).

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A332904 Sum of distinct integers encountered on all possible paths from n to 1 when iterating with nondeterministic map k -> k - k/p, where p is any of the prime factors of k.

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