A322034 -id:A322034 - cp's OEIS frontend

A322036 a(n) = A322035(n) - A322034(n).

1, 1, 2, 1, 4, 1, 6, 1, 5, 2, 10, 1, 12, 3, 3, 1, 16, 5, 18, 1, 13, 5, 22, 1, 19, 6, 14, 3, 28, 3, 30, 1, 7, 8, 27, 5, 36, 9, 25, 1, 40, 13, 42, 5, 8, 11, 46, 1, 41, 19, 11, 3, 52, 7, 43, 3, 37, 14, 58, 3, 60, 15, 34, 1, 51, 7, 66, 4, 15, 27, 70

Offset: 1

N. J. A. Sloane and David James Sycamore, Nov 28 2018

Let s be the fraction defined in A322034 and A322035. Then for n >= 2, 1-s is a(n)/A322035(n).

Note that a(n) >= 1, see A322034.

			Let s be the fraction defined in A322034 and A322035. The fractions 1-s for n >= 2 are 1/2, 2/3, 1/4, 4/5, 1/3, 6/7, 1/8, 5/9, 2/5, 10/11, 1/6, 12/13, 3/7, 3/5, 1/16, 16/17, 5/18, 18/19, 1/5, 13/21, 5/11, 22/23, 1/12, 19/25, 6/13, 14/27, ...

Seiichi Manyama, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Cf. A006022, A027746, A322034, A322035.

# This generates the terms starting at n=2:
P:=proc(n) local FM: FM:=ifactors(n)[2]: seq(seq(FM[j][1], k=1..FM[j][2]), j=1..nops(FM)) end: # A027746
f0:=[]; f1:=[]; f2:=[];
for n from 2 to 120 do
a:=0; b:=1; t1:=[P(n)];
for i from 1 to nops(t1) do b:=b/t1[i]; a:=a+b; od;
f0:=[op(f0),a]; f1:=[op(f1), numer(a)]; f2:=[op(f2),denom(a)]; od:
f0;    # s
f1;    # A322034
f2;    # A322035
f2-f1; # A322036

f[x_] := Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[x]]; {1}~Join~Table[Denominator[#] - Numerator[#] &@ Total@ Table[1/Times @@ #[[;; i]], {i, Length[#]}] &@ f[n], {n, 2, 120}] (* Michael De Vlieger, Jun 20 2025 *)

A006022 Sprague-Grundy (or Nim) values for the game of Maundy cake on an n X 1 sheet.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 7, 4, 6, 1, 10, 1, 8, 6, 15, 1, 13, 1, 16, 8, 12, 1, 22, 6, 14, 13, 22, 1, 21, 1, 31, 12, 18, 8, 31, 1, 20, 14, 36, 1, 29, 1, 34, 21, 24, 1, 46, 8, 31, 18, 40, 1, 40, 12, 50, 20, 30, 1, 51, 1, 32, 29, 63, 14, 45, 1, 52, 24, 43, 1, 67, 1, 38, 31, 58, 12, 53, 1

Offset: 1

N. J. A. Sloane

nonn

There are three equivalent formulas for a(n). Suppose n >= 2, and let p1 <= p2 <= ... <= pk be the prime factors of n, with repetition.
Theorem 1: a(1) = 0. For n >= 2, a(n) = n*s(n), where
s(n) = 1/p1 + 1/(p1*p2) + 1/(p1*p2*p3) + ... + 1/(p1*p2*...*pk).
This is implicit in Berlekamp, Conway and Guy, Winning Ways, 2 vols., 1982, pp. 28, 53.
Note that s(n) = A322034(n) / A322035(n).
David James Sycamore observed on Nov 24 2018 that Theorem 1 implies a(n) < n for all n (see comments in A322034), and also leads to a simple recurrence for a(n):
Theorem 2: a(1) = 0. For n >= 2, a(n) = p*a(n/p) + 1, where p is the largest prime factor of n.
Proof. (Th. 1 implies Th. 2) If n is a prime, Theorem 1 gives a(n) = 1 = n*a(1)+1. For a nonprime n, let n = m*p where p is the largest prime factor of n and m >= 2. From Theorem 1, a(m) = m*s(m), a(n) = q*m*(s(m) + 1/n) = q*a(m) + 1.
(Th. 2 implies Th. 1) The reverse implication is equally easy.
Theorem 2 is equivalent to the following more complicated recurrence:
Theorem 3: a(1) = 0. For n >= 2, a(n) = max_{p|n, p prime} (p*a(n/p)+1).

			For n=24, s(24) = 1/2 + 1/4 + 1/8 + 1/24 = 11/12, so a(24) = 24*11/12 = 22.

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 28, 53.
E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Second Edition, Vol. 1, A K Peters, 2001, pp. 27, 51.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jonathan Blanchette and Robert Laganière, A Curious Link Between Prime Numbers, the Maundy Cake Problem and Parallel Sorting, arXiv:1910.11749 [cs.DS], 2019.

Cf. A020639, A032742, A001348, A002182, A008864.
Cf. also A027746, A306264, A306363, A322034, A322035, A322036, A322382, A328898, A333783, A332993, A333791.
Row sums of A328969.

a006022 1 = 0
a006022 n = (+ 1) $ sum $ takeWhile (> 1) $
          iterate (\x -> x `div` a020639 x) (a032742 n)
-- Reinhard Zumkeller, Jun 03 2012

P:=proc(n) local FM: FM:=ifactors(n)[2]: seq(seq(FM[j][1], k=1..FM[j][2]), j=1..nops(FM)) end: # A027746
s:=proc(n) local i,t,b; global P;t:=0; b:=1; for i in [P(n)] do b:=b*i; t:=t+1/b; od; t; end; # A322034/A322035
A006022 := n -> if n = 1 then 0 else n*s(n); fi;
# N. J. A. Sloane, Nov 28 2018

Nest[Function[{a, n}, Append[a, Max@ Map[# a[[n/#]] + 1 &, Rest@ Divisors@ n]]] @@ {#, Length@ # + 1} &, {0, 1}, 77] (* Michael De Vlieger, Nov 23 2018 *)

lista(nn) = {my(v = vector(nn)); for (n=1, nn, if (n>1, my(m = 0); fordiv (n, d, if (d>1, m = max(m, d*v[n/d]+1))); v[n] = m;); print1(v[n], ", "););} \\ Michel Marcus, Nov 25 2018

a(n) = n * Sum_{k=1..N} (1/(p1^m1*p2^m2*...*pk^mk)) * (pk^mk-1)/(pk-1) for n>=2, where pk is the k-th distinct prime factor of n, N is the number of distinct prime factors of n, and mk is the multiplicity of pk occurring in n. To prove this, expand the factors in Theorem 1 and use the geometrical series identity. - Jonathan Blanchette, Nov 01 2019
From Antti Karttunen, Apr 12 2020: (Start)
a(n) = A322382(n) + A333791(n).
a(n) = A332993(n) - n = A001065(n) - A333783(n). (End)
a(n) = Sum_{k=1..bigomega(n)} F^k(n), where F^k(n) is the k-th iterate of F(n) = A032742(n). - Ridouane Oudra, Jan 26 2024

Edited and extended by Christian G. Bower, Oct 18 2002

Entry revised by N. J. A. Sloane, Nov 28 2018

A322035 Let p1 <= p2 <= ... <= pk be the prime factors of n, with repetition; let s = 1/p1 + 1/(p1p2) + 1/(p1p2p3) + ... + 1/(p1p2...pk); a(n) = denominator of s. a(1)=1 by convention.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 6, 13, 7, 5, 16, 17, 18, 19, 5, 21, 11, 23, 12, 25, 13, 27, 14, 29, 10, 31, 32, 11, 17, 35, 36, 37, 19, 39, 10, 41, 42, 43, 22, 15, 23, 47, 24, 49, 50, 17, 13, 53, 27, 55, 28, 57, 29, 59, 20, 61, 31, 63, 64, 65, 22

Offset: 1

N. J. A. Sloane and David James Sycamore, Nov 28 2018

			If n=12 we get the prime factors 2,2,3, and s = 1/2 + 1/4 + 1/12 = 5/6. So a(12) = 6.
The fractions s for n >= 2 are 1/2, 1/3, 3/4, 1/5, 2/3, 1/7, 7/8, 4/9, 3/5, 1/11, 5/6, 1/13, 4/7, 2/5, 15/16, 1/17, 13/18, 1/19, 4/5, 8/21, ...

Seiichi Manyama, Table of n, a(n) for n = 1..16384

Cf. A006022, A027746, A322034, A322036.

A017665/A017666 = sum of reciprocals of all divisors of n.

# This generates the terms starting at n=2:
P:=proc(n) local FM: FM:=ifactors(n)[2]: seq(seq(FM[j][1], k=1..FM[j][2]), j=1..nops(FM)) end: # A027746
f0:=[]; f1:=[]; f2:=[];
for n from 2 to 120 do
a:=0; b:=1; t1:=[P(n)];
for i from 1 to nops(t1) do b:=b/t1[i]; a:=a+b; od;
f0:=[op(f0),a]; f1:=[op(f1), numer(a)]; f2:=[op(f2),denom(a)]; od:
f0;    # s
f1;    # A322034
f2;    # A322035
f2-f1; # A322036

f[x_] := Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[x]]; Table[Denominator@ Total@ Table[1/Times @@ #[[;; i]], {i, Length[#]}] &@ f[n], {n, 120}] (* Michael De Vlieger, Jun 20 2025 *)

A322382 a(n) = p*a(n/p) + 1, where p is the smallest prime divisor of n; a(1)=0.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 7, 4, 3, 1, 7, 1, 3, 4, 15, 1, 9, 1, 7, 4, 3, 1, 15, 6, 3, 13, 7, 1, 9, 1, 31, 4, 3, 6, 19, 1, 3, 4, 15, 1, 9, 1, 7, 13, 3, 1, 31, 8, 13, 4, 7, 1, 27, 6, 15, 4, 3, 1, 19, 1, 3, 13, 63, 6, 9, 1, 7, 4, 13, 1, 39, 1, 3, 19, 7, 8, 9, 1, 31, 40, 3, 1, 19, 6

Offset: 1

David James Sycamore, Dec 05 2018

nonn

An equivalent definition of this sequence is: a(1) = 0; and for n > 1, a(n) = n*w(n), where if p1, ..., pk are the prime divisors of n (with repetition) and p1 <= p2 <= ... <= pk, then w(n) = 1/pk + 1/(pk-1*pk) + ... + 1/(p2*p3*...*pk) + ... + 1/(p1*p2*...*pk). Since 2 is smallest prime w(n) <= 1/2 + 1/(2^2) + ... + 1/(2^k), a partial sum of a series which -->1 as n-->oo. Therefore w(n) < 1 and n-a(n) is a sequence of positive numbers (1,1,2,1,4,3,6,...). For n=p^k, p a prime and n >= 1, a(n) = a(p^k) = p^(n-1) + p^(n-2) + ... + p^2 + p + 1 = (p^k-1)/(p-1); e.g., a(2^k) = 2^k - 1.

			For any prime p, a(p) = p*a(p/p)+1 = p*a(1)+1 = 1, because a(1) = 0.
For n = 6, the least prime divisor is 2, so a(6) = 2*a(6/2)+1 = 2*a(3)+1 = 3.
Using the equivalent definition we get w(6) = 1/3 + 1/6 = 1/2, so a(6) = 6*w(6) = 6*1/2 = 3. For n=32, a(32) = a(2^5) = 2^5 - 1 = 32 - 1 = 31.

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A003557, A006022, A020639, A032742, A322034, A322035, A052126.

a[n_] := #*a[n/#] + 1 &[FactorInteger[n][[1, 1]] ]; a[1] = 0; Array[a, 1000] (* Michael De Vlieger, Jun 20 2025 *)

a(n) = if (n==1, 0, my(p = vecmin(factor(n)[,1])); p*a(n/p)+1); \\ Michel Marcus, Jan 25 2019

From Antti Karttunen, Feb 28 2019 , Mar 04 2019: (Start)
a(1) = 0; for n > 1, a(n) = 1 + A020639(n)*a(A032742(n)).
If n is of the form p^k, k >= 1, then a(n) = A000203(A003557(n)). [Based on author's comments above] (End)
a(n) = Sum_{k=1..bigomega(n)} F^k(n), where F^k(n) is the k-th iterate of F(n) = A052126(n). - Ridouane Oudra, Aug 17 2024

More terms from Michel Marcus, Jan 25 2019

cp's OEIS Frontend

A322036 a(n) = A322035(n) - A322034(n).

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A006022 Sprague-Grundy (or Nim) values for the game of Maundy cake on an n X 1 sheet.

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A322035 Let p1 <= p2 <= ... <= pk be the prime factors of n, with repetition; let s = 1/p1 + 1/(p1*p2) + 1/(p1*p2*p3) + ... + 1/(p1*p2*...*pk); a(n) = denominator of s. a(1)=1 by convention.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A322382 a(n) = p*a(n/p) + 1, where p is the smallest prime divisor of n; a(1)=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A322035 Let p1 <= p2 <= ... <= pk be the prime factors of n, with repetition; let s = 1/p1 + 1/(p1p2) + 1/(p1p2p3) + ... + 1/(p1p2...pk); a(n) = denominator of s. a(1)=1 by convention.