A323167 -id:A323167 - cp's OEIS frontend

A323174 Deficiency computed for conjugated prime factorization: a(n) = A033879(A122111(n)).

1, 1, 1, 2, 1, 0, 1, 4, 5, -4, 1, 2, 1, -12, -3, 6, 1, 6, 1, -2, -19, -28, 1, 4, 14, -60, 19, -10, 1, -12, 1, 10, -51, -124, -12, 10, 1, -252, -115, 0, 1, -48, 1, -26, 7, -508, 1, 8, 41, 12, -243, -58, 1, 22, -64, -8, -499, -1020, 1, -12, 1, -2044, -17, 12, -168, -120, 1, -122, -1011, -54, 1, 18, 1, -4092, 26, -250, -39, -264, 1, 4

Offset: 1

Antti Karttunen, Jan 10 2019

sign

Zeros occur at A122111(A000396(k)), k >= 1: 6, 40, 11264, 18253611008, ...

Cf. A033879, A122111, A323167, A323168.

Cf. also A323244.

A122111[n_] := Product[Prime[Sum[If[jA122111[n]}, 2k - DivisorSigma[1, k]];
Array[a, 80] (* Jean-François Alcover, Sep 23 2020 *)

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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A323174(n) = { my(k=A122111(n)); ((2*k)-sigma(k)); }

a(n) = A033879(A122111(n)).

a(n) = 2*A122111(n) - A323173(n).

A323248 a(n) = A323247(n) - A323243(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 3, -2, 3, 0, 7, 0, 14, 0, 2, 0, 9, 0, 15, -5, 16, 0, 18, -6, 44, 1, 19, 0, 7, 0, 25, 12, 80, -4, 10, 0, 254, -14, 18, 0, 33, 0, 63, 5, 224, 0, 41, -14, 16, 6, 127, 0, 24, -21, 66, -14, 746, 0, 38, 0, 1360, 13, 16, 8, 39, 0, 211, 252, 37, 0, 33, 0, 3836, 7, 403, -12, 103, 0, 73, -16, 5456, 0, 22, -74, 12248, -350, 26, 0, 8

Offset: 1

Antti Karttunen, Jan 10 2019

sign

Cf. A005187, A156552, A323243, A323246, A323247.

Cf. also A323167.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
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))));
A323247(n) = A005187(A156552(n));
A323248(n) = (A323247(n) - A323243(n));

a(n) = A323247(n) - A323243(n).
a(n) = A323244(n) - A001222(n).
For n > 1, a(n) = A294898(A156552(n)).

A323168 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = [A322867(n), A323174(n)] for n > 1, and f(1) = 0.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 3, 2, 8, 9, 10, 2, 11, 2, 12, 13, 14, 2, 15, 16, 17, 18, 19, 2, 20, 2, 21, 22, 23, 20, 21, 2, 24, 25, 26, 2, 27, 2, 28, 29, 30, 2, 31, 32, 33, 34, 35, 2, 36, 37, 38, 39, 40, 2, 41, 2, 42, 43, 44, 45, 46, 2, 47, 48, 49, 2, 50, 2, 51, 52, 53, 54, 55, 2, 15, 32, 56, 2, 57, 58, 59, 60, 61, 2, 62, 63, 64, 65, 66, 67, 21, 2, 68, 69, 70, 2, 71, 2, 57

Offset: 1

Antti Karttunen, Jan 10 2019

nonn

Restricted growth sequence transform of function f, with f(1) = 0 and f(n) = [A322867(n), A323174(n)] for n > 1.
Equally, restricted growth sequence transform of function f, with f(1) = 0 and f(n) = A318310(A122111(n)) for n > 1.
For all i, j:
  a(i) = a(j) => A322867(i) = A322867(j),
  a(i) = a(j) => A323167(i) = A323167(j),
  a(i) = a(j) => A323174(i) = A323174(j).

Cf. A000203, A005187, A122111, A294898, A318310, A322867, A323167, A323173, A323174.

Cf. also A323240.

up_to = 4096;
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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(n));
A318310aux(n) = [hammingweight(n), A294898(n)];
A323168aux(n) = if(1==n,0,A318310aux(A122111(n)));
v323168 = rgs_transform(vector(up_to, n, A323168aux(n)));
A323168(n) = v323168[n];

cp's OEIS Frontend

A323174 Deficiency computed for conjugated prime factorization: a(n) = A033879(A122111(n)).

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A323248 a(n) = A323247(n) - A323243(n).

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A323168 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = [A322867(n), A323174(n)] for n > 1, and f(1) = 0.

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