A324182 -id:A324182 - cp's OEIS frontend

A366804 Lexicographically earliest infinite sequence such that a(i) = a(j) => A324182(i) = A324182(j) for all i, j >= 0.

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

Offset: 0

Antti Karttunen, Oct 24 2023

Restricted growth sequence transform of A324182.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A083254, A163511, A324182.

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; };
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));
A083254(n) = (2*eulerphi(n)-n);
A324182(n) = A083254(A163511(n));
v366804 = rgs_transform(vector(1+up_to,n,A324182(n-1)));
A366804(n) = v366804[1+n];

A324185 Deficiency of n permuted by A163511: a(n) = A033879(A163511(n)) = 2*A163511(n) - sigma(A163511(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 5, 0, 4, 1, 14, -3, 19, -4, 6, 2, 6, 1, 41, -12, 94, -19, 26, 7, 41, -12, 12, -12, 22, -2, 10, 4, 10, 1, 122, -39, 469, -64, 126, 32, 286, -51, 47, -72, 148, -17, 66, 25, 109, -28, 30, -54, 102, -48, 18, -4, 58, -10, 22, -12, 38, 0, 18, 8, 12, 1, 365, -120, 2344, -199, 626, 157, 2001, -168, 222, -372, 1030, -92, 458, 172, 1198

Offset: 0

Antti Karttunen, Feb 17 2019

If there are no odd perfect numbers, then all n for which a(n) is 0 are given by sequence A324200.

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

Cf. A000203, A033879, A054429, A163511, A324055, A324182 (compare the scatter plot), A324184, A324187, A324188, A324189, A324199, A324200.

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));
A033879(n) = (2*n-sigma(n));
A324185(n) = A033879(A163511(n));

A324184(n) = if(!n,1,my(p=2,mp=p*p,m=1); while(n>1, if(n%2, p=nextprime(1+p); mp = p*p, if((2==n)||!(n%4),mp *= p,m *= (mp-1)/(p-1))); n >>= 1); (m*(mp-1)/(p-1)));
A324185(n) = (2*A163511(n)) - A324184(n);

a(n) = A033879(A163511(n)) = 2*A163511(n) - A324184(n) = 2*A163511(n) - A000203(A163511(n)).

For n > 0, a(n) = A324055(A054429(n)).

A324052 a(n) = A083254(A005940(1+n)).

Original entry on oeis.org

1, 0, 1, 0, 3, -2, 3, 0, 5, -2, 1, -4, 15, -6, 9, 0, 9, -2, 3, -4, 13, -14, 3, -8, 35, -10, 5, -12, 75, -18, 27, 0, 11, -2, 7, -4, 25, -18, 9, -8, 43, -22, -9, -28, 65, -42, 9, -16, 99, -14, 21, -20, 91, -70, 15, -24, 245, -50, 25, -36, 375, -54, 81, 0, 15, -2, 9, -4, 31, -26, 21, -8, 53, -30, -5, -36, 125, -54, 27, -16, 97, -34, 9

Offset: 0

Antti Karttunen, Feb 18 2019

sign

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A005940, A083254, A054429, A290077, A324055, A324182.

Cf. also A324103.

A324052(n) = { my(m1=1,m2=2,p=2); while(n, if(!(n%2), p=nextprime(1+p), m1 *= p; m2 *= (p-(1==(n%4)))); n>>=1); (m2-m1); };

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A083254(n) = (2*eulerphi(n)-n);
A324052(n) = A083254(A005940(1+n));

a(n) = A083254(A005940(1+n)).
a(n) = 2*A290077(n) - A005940(1+n).
For n >= 1, a(n) = A324182(A054429(n)).
For n >= 1, a((2^n)-1) = 0.

A324103 a(1) = 0; for n > 1, a(n) = A083254(A156552(n)).

Original entry on oeis.org

0, 1, 0, 1, 0, 3, 0, 5, -2, 3, 0, 9, 0, 15, -2, 1, 0, 11, 0, 17, -6, 7, 0, 21, -4, 31, -2, 13, 0, 3, 0, 29, -2, 39, -4, 9, 0, 255, -26, 9, 0, 35, 0, 65, -2, 135, 0, 45, -8, 15, -34, 129, 0, 27, -12, 69, -90, 575, 0, 41, 0, 679, -2, 9, -4, 19, 0, 173, -2, 39, 0, 25, 0, 3583, -2, 301, -8, 83, 0, 77, -14, 2727, 0, 5, -52, 8703, -378, 9, 0, 3

Offset: 1

Antti Karttunen, Feb 18 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..4473

Cf. A000010, A083254, A156552, A323244, A324104, A324105.

Cf. also A324052, A324182.

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)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A083254(n) = (2*eulerphi(n)-n);
A324103(n) = if(1==n,0,A083254(A156552(n)));

a(1) = 0; for n > 1, a(n) = A083254(A156552(n)).

a(n) = 2*A324104(n) - A156552(n).

cp's OEIS Frontend

A366804 Lexicographically earliest infinite sequence such that a(i) = a(j) => A324182(i) = A324182(j) for all i, j >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A324185 Deficiency of n permuted by A163511: a(n) = A033879(A163511(n)) = 2*A163511(n) - sigma(A163511(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A324052 a(n) = A083254(A005940(1+n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A324103 a(1) = 0; for n > 1, a(n) = A083254(A156552(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula