A320111 -id:A320111 - cp's OEIS frontend

A320110 Restricted growth sequence transform of function f: f(1) = 0, f(n) = A046523(A252463(n)) for n > 1.

1, 2, 3, 3, 3, 3, 3, 4, 4, 3, 3, 5, 3, 3, 5, 6, 3, 4, 3, 5, 5, 3, 3, 7, 4, 3, 6, 5, 3, 5, 3, 8, 5, 3, 5, 7, 3, 3, 5, 7, 3, 5, 3, 5, 7, 3, 3, 9, 4, 4, 5, 5, 3, 6, 5, 7, 5, 3, 3, 10, 3, 3, 7, 11, 5, 5, 3, 5, 5, 5, 3, 12, 3, 3, 7, 5, 5, 5, 3, 9, 8, 3, 3, 10, 5, 3, 5, 7, 3, 7, 5, 5, 5, 3, 5, 13, 3, 4, 7, 7, 3, 5, 3, 7, 10

Offset: 1

Antti Karttunen, Nov 22 2018

nonn

For all i, j: a(i) = a(j) => A320111(i) = A320111(j).

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

Cf. A046523, A252463, A320111.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
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)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A320110aux(n) = if(1==n,0,A046523(A252463(n)));
v320110 = rgs_transform(vector(up_to,n,A320110aux(n)));
A320110(n) = v320110[n];

A323080 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(1)=0, and f(n) = A252463(n) for n > 1.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 6, 6, 5, 7, 8, 9, 7, 8, 10, 11, 12, 13, 14, 14, 9, 15, 16, 12, 11, 10, 17, 18, 19, 20, 21, 17, 13, 19, 22, 23, 15, 24, 25, 26, 27, 28, 24, 16, 18, 29, 30, 31, 31, 32, 32, 33, 34, 27, 35, 36, 20, 37, 38, 39, 23, 25, 40, 41, 41, 42, 36, 43, 44, 45, 46, 47, 26, 22, 43, 44, 48, 49, 50, 21, 28, 51, 52, 48, 29, 53, 54, 55, 56, 57, 53, 58, 33, 59, 60, 61, 62, 35, 63, 64, 59, 65, 66, 38

Offset: 1

Antti Karttunen, Jan 03 2019

For all i, j:
  a(i) = a(j) => A252463(i) = A252463(j) => A285727(i) = A285727(j),
  a(i) = a(j) => A320110(i) = A320110(j) => A320111(i) = A320111(j).

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

Cf. A252463, A285727, A320107, A320110, A320111, A322805, A323081.

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; };
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)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A323080aux(n) = if(1==n,0,A252463(n));
v323080 = rgs_transform(vector(up_to,n,A323080aux(n)));
A323080(n) = v323080[n];

A338117 Number of partitions of n into two parts (s,t) such that (t-s) | n, where s < t.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 3, 1, 2, 1, 3, 3, 1, 1, 5, 2, 1, 3, 3, 1, 3, 1, 4, 3, 1, 3, 5, 1, 1, 3, 5, 1, 3, 1, 3, 5, 1, 1, 7, 2, 2, 3, 3, 1, 3, 3, 5, 3, 1, 1, 7, 1, 1, 5, 5, 3, 3, 1, 3, 3, 3, 1, 8, 1, 1, 5, 3, 3, 3, 1, 7, 4, 1, 1, 7, 3, 1, 3, 5, 1, 5, 3, 3, 3

Offset: 1

Wesley Ivan Hurt, Oct 10 2020

Apparently a(n) = A320111(n) - 1. - Hugo Pfoertner, Oct 30 2020

The above observation is true, which can be seen from the formula A320111(2n) = A000005(n), A320111(2n+1) = A000005(2n+1). For odd numbers, the difference (t-s) may range over all the divisors of n except the n itself, and for even numbers the difference (t-s) [which is always even] may range only over the even divisors of n, except the n itself. Note that A000005(2n) = A000005(n) + A001227(n). - Antti Karttunen, Dec 12 2021

			a(8) = 2; The partitions of 8 into two parts (s,t) such that s < t are (7,1), (6,2), (5,3) and (4,4). Only the partitions (6,2) and (5,3) have (6-2) | 8 and (5-3) | 8, so a(8) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..20160
Index entries for sequences related to partitions

Cf. A000005, A001227, A320111.

Cf. also A337101, A338021.

Table[Sum[(1 - Ceiling[n/(n - 2 i)] + Floor[n/(n - 2 i)]), {i, Floor[(n - 1)/2]}], {n, 100}]

for(n=1,85,my(j=0);forpart(x=n,if(#x==2,if(x[2]!=x[1]&&!(n%(x[2]-x[1])),j++)));print1(j,", ")) \\ Hugo Pfoertner, Oct 30 2020

A338117(n) = sum(s=1,(n-1)\2,!(n%(n-(2*s)))); \\ Antti Karttunen, Dec 12 2021

a(n) = Sum_{i=1..floor((n-1)/2)} (1 - ceiling(n/(n-2*i)) + floor(n/(n-2*i))).

A366362 Triangle read by rows: T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^3 - x^2 - y^2 - y.

Original entry on oeis.org

1, 0, 4, 5, 0, 4, 0, 8, 0, 8, 21, 0, 0, 0, 4, 0, 20, 0, 0, 0, 16, 40, 0, 0, 0, 0, 0, 9, 0, 32, 0, 16, 0, 0, 0, 16, 45, 0, 24, 0, 0, 0, 0, 0, 12, 0, 84, 0, 0, 0, 0, 0, 0, 0, 16, 111, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 40, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32

Offset: 1

Mats Granvik, Oct 08 2023

Row n appears to have sum n^2. T(prime(m),1) = A366346(m). The number of nonzero terms in row n appears to be A320111(n).

			{
{1}, = 1^2
{0, 4}, = 2^2
{5, 0, 4}, = 3^2
{0, 8, 0, 8}, = 4^2
{21, 0, 0, 0, 4}, = 5^2
{0, 20, 0, 0, 0, 16}, = 6^2
{40, 0, 0, 0, 0, 0, 9}, = 7^2
{0, 32, 0, 16, 0, 0, 0, 16}, = 8^2
{45, 0, 24, 0, 0, 0, 0, 0, 12}, = 9^2
{0, 84, 0, 0, 0, 0, 0, 0, 0, 16}, = 10^2
{111, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10}, = 11^2
{0, 40, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32} = 12^2
}

Cf. A127649, A366346, A000290, A060457, A002070, A036689, A001248, A272196.

f = x^3 - x^2 - y^2 - y; nn = 12; Flatten[Table[Table[Sum[Sum[If[GCD[f, n] == k, 1, 0], {x, 1, n}], {y, 1, n}], {k, 1, n}], {n, 1, nn}]]

T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^3 - x^2 - y^2 - y.

Conjecture: T(n,n) = A060457(n).

A318889 a(n) = A001065(n) - A001065(A252463(n)).

Original entry on oeis.org

0, 1, 0, 2, 0, 5, 0, 4, 1, 7, 0, 10, 0, 9, 3, 8, 0, 17, 0, 14, 3, 13, 0, 20, 2, 15, 6, 18, 0, 33, 0, 16, 5, 19, 4, 34, 0, 21, 3, 28, 0, 43, 0, 26, 17, 25, 0, 40, 2, 37, 5, 30, 0, 53, 6, 36, 3, 31, 0, 66, 0, 33, 19, 32, 4, 63, 0, 38, 5, 61, 0, 68, 0, 39, 28, 42, 6, 73, 0, 56, 25, 43, 0, 86, 6, 45, 7, 52, 0, 111, 4, 50, 3, 49, 4, 80, 0

Offset: 1

Antti Karttunen, Nov 22 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A001065, A252463.

Cf. also A319699, A319700, A319703, A319989, A320107, A320111.

A001065(n) = (sigma(n)-n);
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)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A318889(n) = (A001065(n) - A001065(A252463(n)));

a(n) = A001065(n) - A319699(n) = A001065(n) - A001065(A252463(n)).

A366563 Number of nonzero terms in row n of A366561(n,k).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 3, 3, 4, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 4, 4, 2, 6, 3, 4, 4, 4, 2, 8, 2, 5, 4, 4, 4, 6, 2, 4, 4, 6, 2, 8, 2, 4, 6, 4, 2, 8, 3, 6, 4, 4, 2, 8, 4, 6, 4, 4, 2, 8, 2, 4, 6, 6, 4, 8, 2, 4, 4, 8, 2, 9, 2, 4

Offset: 1

Mats Granvik, Oct 13 2023

nonn

Cf. A366561, A366562, A000005, A320111.

nn = 74; f = x^2 - y^2; g[n_] := DivisorSum[n, MoebiusMu[#] # &]; Table[Sum[Abs[Sign[Sum[Sum[If[GCD[f, n] == k, 1, 0], {x, 1, n}], {y, 1, n}]]], {k, 1, n}], {n, 1, nn}]

a(n) = Sum_{k=1..n} abs(sign(A366561(n,k))).

A366799 Number of divisors d of n that are not of the form 4k+2, as permuted by the Doudna sequence.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 4, 4, 3, 3, 4, 4, 2, 2, 4, 4, 4, 4, 6, 6, 3, 3, 6, 6, 4, 4, 5, 5, 2, 2, 4, 4, 4, 4, 6, 6, 4, 4, 8, 8, 6, 6, 8, 8, 3, 3, 6, 6, 6, 6, 9, 9, 4, 4, 8, 8, 5, 5, 6, 6, 2, 2, 4, 4, 4, 4, 6, 6, 4, 4, 8, 8, 6, 6, 8, 8, 4, 4, 8, 8, 8, 8, 12, 12, 6, 6, 12, 12, 8, 8, 10, 10, 3, 3, 6, 6, 6, 6, 9, 9

Offset: 0

Antti Karttunen, Oct 27 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A005940, A320111, A366800 (rgs-transform).

Cf. also A366797.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A320111(n) = sumdiv(n,d,(2!=(d%4)));
A366799(n) = A320111(A005940(1+n));

a(n) = A320111(A005940(1+n)).

A366800 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366799(i) = A366799(j) for all i, j >= 0, where A366799 is the number of divisors d of n that are not of the form 4k+2, as permuted by the Doudna sequence.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 4, 4, 3, 3, 4, 4, 2, 2, 4, 4, 4, 4, 5, 5, 3, 3, 5, 5, 4, 4, 6, 6, 2, 2, 4, 4, 4, 4, 5, 5, 4, 4, 7, 7, 5, 5, 7, 7, 3, 3, 5, 5, 5, 5, 8, 8, 4, 4, 7, 7, 6, 6, 5, 5, 2, 2, 4, 4, 4, 4, 5, 5, 4, 4, 7, 7, 5, 5, 7, 7, 4, 4, 7, 7, 7, 7, 9, 9, 5, 5, 9, 9, 7, 7, 10, 10, 3, 3, 5, 5, 5, 5, 8, 8

Offset: 0

Antti Karttunen, Oct 27 2023

nonn

Restricted growth sequence transform of A366799.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A005940, A320111, A366799.

Cf. also A366798.

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; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A320111(n) = sumdiv(n,d,(2!=(d%4)));
A366799(n) = A320111(A005940(1+n));
v366800 = rgs_transform(vector(1+up_to,n,A366799(n-1)));
A366800(n) = v366800[1+n];

cp's OEIS Frontend

A320110 Restricted growth sequence transform of function f: f(1) = 0, f(n) = A046523(A252463(n)) for n > 1.

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A323080 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(1)=0, and f(n) = A252463(n) for n > 1.

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A338117 Number of partitions of n into two parts (s,t) such that (t-s) | n, where s < t.

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A366362 Triangle read by rows: T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^3 - x^2 - y^2 - y.

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A318889 a(n) = A001065(n) - A001065(A252463(n)).

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A366563 Number of nonzero terms in row n of A366561(n,k).

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A366799 Number of divisors d of n that are not of the form 4k+2, as permuted by the Doudna sequence.

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A366800 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366799(i) = A366799(j) for all i, j >= 0, where A366799 is the number of divisors d of n that are not of the form 4k+2, as permuted by the Doudna sequence.

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