A250469 -id:A250469 - cp's OEIS frontend

A249810 a(1) = 0, a(n) = A078898(A003961(n)).

0, 1, 1, 2, 1, 3, 1, 5, 2, 4, 1, 8, 1, 6, 3, 14, 1, 13, 1, 11, 4, 7, 1, 23, 2, 9, 9, 17, 1, 18, 1, 41, 5, 10, 3, 38, 1, 12, 6, 32, 1, 28, 1, 20, 12, 15, 1, 68, 2, 25, 7, 26, 1, 63, 4, 50, 8, 16, 1, 53, 1, 19, 19, 122, 5, 33, 1, 29, 10, 39, 1, 113, 1, 21, 17, 35, 3, 43, 1, 95, 42, 22, 1, 83, 6, 24, 11, 59, 1, 88, 4, 44, 13, 27, 7, 203

Offset: 1

Antti Karttunen, Dec 08 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..3071

Cf. A003961, A078898, A246277, A249817, A249818, A249820, A249821, A249822, A251721, A251722, A250469, A250470.

(define (A249810 n) (if (= 1 n) 0 (A078898 (A003961 n))))

a(1) = 0, a(n) = A078898(A003961(n)).

a(1) = 0, a(n) = A078898(n) + A249820(n).

A346477 Dirichlet inverse of A346476.

Original entry on oeis.org

1, -1, -1, 2, -3, 5, -3, 2, 8, 13, -9, -2, -9, 17, 11, 8, -15, -8, -15, -12, 19, 37, -17, 18, 8, 41, -4, -12, -27, -33, -25, 20, 37, 61, 25, 56, -33, 65, 35, 38, -39, -45, -39, -42, -36, 77, -41, 32, 32, -20, 53, -42, -47, 96, 35, 58, 61, 109, -57, 132, -55, 109, -48, 56, 43, -121, -63, -72, 71, -109, -69, 56

Offset: 1

Antti Karttunen, Jul 29 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves

Cf. A000040, A000720, A062234, A250469, A252748, A346476, A346478.

Cf. also A323910, A346248, A346479.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA346476(n) = (n+n-A250469(n));
v346477 = DirInverseCorrect(vector(up_to,n,A346476(n)));
A346477(n) = v346477[n];

a(1) = 1; and for n > 2, a(n) = -Sum_{d|n, dA346476(n/d).
a(n) = A346478(n) - A346476(n).
a(p) = A252748(p) = A346248(p) = -A346476(p) = -A062234(A000720(p)), for any prime p.

A346478 Sum of A346476 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 2, 0, -3, 1, 6, 0, -11, 0, 6, 6, -5, 0, -23, 0, -29, 6, 18, 0, -3, 9, 18, -15, -37, 0, -60, 0, -9, 18, 30, 18, 23, 0, 30, 18, 1, 0, -84, 0, -83, -61, 34, 0, -13, 9, -67, 30, -91, 0, 45, 54, 5, 30, 54, 0, 75, 0, 50, -77, -5, 54, -184, 0, -137, 34, -176, 0, -13, 0, 66, -55, -145, 54, -188, 0, -37, 49

Offset: 1

Antti Karttunen, Jul 30 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves

Cf. A000040, A000720, A062234, A250469, A252748, A346476, A346477.

Cf. also A323911, A346250, A346480.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA346476(n) = (n+n-A250469(n));
v346477 = DirInverseCorrect(vector(up_to,n,A346476(n)));
A346477(n) = v346477[n];
A346478(n) = (A346476(n)+A346477(n));

a(n) = A346476(n) + A346477(n).

a(1) = 2; and for n > 2, a(n) = -Sum_{d|n, 1A346476(n/d) * A346477(d).

A302043 a(n) = n - A302042(n); an analog of A060681 based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 6, 4, 6, 5, 10, 6, 12, 7, 10, 8, 16, 9, 18, 10, 12, 11, 22, 12, 20, 13, 20, 14, 28, 15, 30, 16, 18, 17, 28, 18, 36, 19, 28, 20, 40, 21, 42, 22, 24, 23, 46, 24, 42, 25, 26, 26, 52, 27, 30, 28, 30, 29, 58, 30, 60, 31, 50, 32, 54, 33, 66, 34, 36, 35, 70, 36, 72, 37, 58, 38, 66, 39, 78, 40, 42, 41, 82, 42, 50, 43, 52, 44, 88, 45, 42

Offset: 1

Antti Karttunen, Mar 31 2018

nonn

An analog of A060681 based on the sieve of Eratosthenes (A083221).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences generated by sieves

Cf. A060681, A302042.

A302042(n) = if(1==n,n,my(k=0); while((n%2), n = A268674(n); k++); n = n/2; while(k>0, n = A250469(n); k--); (n));
A302043(n) = (n - A302042(n));

a(n) = n - A302042(n).

A302046 A filter sequence analogous to A101296 for nonstandard factorization based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 6, 5, 4, 2, 8, 3, 4, 4, 6, 2, 9, 2, 10, 6, 4, 4, 11, 2, 4, 4, 8, 2, 8, 2, 6, 7, 4, 2, 12, 3, 6, 6, 6, 2, 9, 5, 8, 6, 4, 2, 13, 2, 4, 4, 14, 4, 13, 2, 6, 8, 9, 2, 15, 2, 4, 4, 6, 4, 9, 2, 12, 6, 4, 2, 15, 6, 4, 9, 8, 2, 12, 5, 6, 10, 4, 4, 16, 2, 6, 4, 11, 2, 13, 2, 8, 11

Offset: 1

Antti Karttunen, Mar 31 2018

nonn

Restricted growth sequence transform of A278524.
See A302042 for the description of the nonstandard factorization employed here.
For all i, j:
  a(i) = a(j) => A253557(i) = A253557(j).
  a(i) = a(j) => A302041(i) = A302041(j).
  a(i) = a(j) => A302050(i) = A302050(j).
  a(i) = a(j) => A302051(i) = A302051(j) => A302052(i) = A302052(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences generated by sieves

Cf. A101296, A250246, A278524, A302042, A302044, A302045.

up_to = 32769;
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };
A078898(n) = { if(n<=1,n, my(spf=A020639(n),k=1,m=n/spf); while(m>1,if(A020639(m)>=spf,k++); m--); (k)); };
A001511(n) = 1+valuation(n,2);
A302045(n) = A001511(A078898(n));
A302044(n) = if(1==n,n,my(k=0); while((n%2), n = A268674(n); k++); n = (n/2^valuation(n, 2)); while(k>0, n = A250469(n); k--); (n));
A302041(n) = if(1==n, 0,1+A302041(A302044(n)));
Aux302046(n) = if(1==n,n, my(k=A302041(n), v = vector(k),i=1); while(n>1,v[i] = A302045(n); n = A302044(n); i++); vecsort(v));
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux302046(n))),"b302046.txt");

A280701 a(n) = n - A280704(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 1, 8, 1, 10, 1, 12, 1, 14, 11, 16, 1, 18, 1, 20, 1, 22, 1, 24, 1, 14, 19, 28, 1, 30, 1, 16, 1, 34, 29, 36, 1, 20, 27, 40, 1, 42, 1, 22, 1, 46, 1, 48, 49, 26, 51, 52, 1, 54, 51, 28, 1, 58, 1, 60, 1, 32, 57, 64, 65, 66, 1, 34, 1, 70, 1, 72, 1, 38, 51, 76, 1, 78, 1, 40, 1, 82, 1, 84, 1, 44, 59, 88, 1, 90, 1, 46

Offset: 1

Antti Karttunen, Mar 08 2017

nonn

Questions: Are all terms nonnegative? Where do ones occur?

Antti Karttunen, Table of n, a(n) for n = 1..13098

Cf. A003961, A250469, A280702, A280704.

f[n_] := f[n] = Which[n == 1, 1, PrimeQ@ n, NextPrime@ n, True, Times @@ Replace[FactorInteger[n], {p_, e_} :> f[p]^e, 1]]; g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[ Function[t, First@ #2 - t/GCD[t, f@ First@ #2]][Lookup[s, g[First@ #2] + 1][[#1]] - Boole[First@ #2 == 1]] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 08 2017, Version 10 *)

Scheme
```
(define (A280701 n) (- n (A280704 n)))
```

a(n) = n - A280704(n) = n - (A250469(n)/gcd(A003961(n),A250469(n))).

cp's OEIS Frontend

A249810 a(1) = 0, a(n) = A078898(A003961(n)).

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A346477 Dirichlet inverse of A346476.

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A346478 Sum of A346476 and its Dirichlet inverse.

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A302043 a(n) = n - A302042(n); an analog of A060681 based on the sieve of Eratosthenes (A083221).

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A302046 A filter sequence analogous to A101296 for nonstandard factorization based on the sieve of Eratosthenes (A083221).

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A280701 a(n) = n - A280704(n).

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