A322865 -id:A322865 - cp's OEIS frontend

A334107 a(n) = A329697(A122111(n)).

0, 0, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 2, 0, 2, 0, 1, 2, 1, 0, 2, 3, 1, 2, 1, 0, 2, 0, 2, 2, 1, 3, 3, 0, 1, 2, 2, 0, 2, 0, 1, 2, 1, 0, 2, 4, 3, 2, 1, 0, 3, 3, 2, 2, 1, 0, 3, 0, 1, 2, 2, 3, 2, 0, 1, 2, 3, 0, 3, 0, 1, 3, 1, 4, 2, 0, 2, 4, 1, 0, 3, 3, 1, 2, 2, 0, 3, 4, 1, 2, 1, 3, 2, 0, 4, 2, 4, 0, 2, 0, 2, 3

Offset: 1

Antti Karttunen, Apr 29 2020

nonn

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

Cf. A001248, A008578 (positions of zeros), A064989, A105560, A122111, A322865, A329697, A334108, A334109, A334201.

Map[Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] - 1 &, Array[Times @@ Table[Prime[LengthWhile[#1, # >= j &] /. 0 -> 1], {j, #2}] & @@ {#, Max[#]} &@ PrimePi@ Flatten[ConstantArray[#1, {#2}] & @@@ FactorInteger@ #] &, 105] ] (* Michael De Vlieger, May 14 2020, after Robert G. Wilson v at A329697 *)

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)));
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A334107(n) = A329697(A122111(n));

a(n) = A329697(A122111(n)) = A329697(A322865(n)).
a(n) = A329697(A105560(n)) + a(A064989(n)).
For n >= 1, a(A001248(n)) = n, and these seem to be also the first occurrences of each n.

A322867 a(n) = A000120(A122111(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 3, 1, 4, 1, 2, 2, 2, 1, 3, 4, 2, 3, 2, 1, 4, 1, 3, 2, 2, 4, 3, 1, 2, 2, 3, 1, 4, 1, 2, 3, 2, 1, 3, 3, 4, 2, 2, 1, 3, 4, 3, 2, 2, 1, 3, 1, 2, 3, 3, 4, 4, 1, 2, 2, 4, 1, 2, 1, 2, 4, 2, 3, 4, 1, 3, 3, 2, 1, 3, 4, 2, 2, 3, 1, 3, 3, 2, 2, 2, 4, 3, 1, 4, 3, 6, 1, 4, 1, 3, 4

Offset: 1

Antti Karttunen, Dec 30 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization

Cf. A000120, A001222, A122111, A322863, A322865, A322866.

Array[DigitCount[#, 2, 1] &@ If[# < 3, 1, Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@Length@k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ FactorInteger@ #]] &, 105] (* Michael De Vlieger, Dec 31 2018, after JungHwan Min at A122111 *)

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)));
A322867(n) = hammingweight(A122111(n));

a(n) = A000120(A122111(n)) = A000120(A322865(n)) = A001222(A322863(n)).

A323903 a(n) = A002487(A122111(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 4, 2, 1, 3, 1, 2, 4, 3, 1, 4, 1, 3, 4, 2, 1, 3, 8, 2, 7, 3, 1, 4, 1, 5, 4, 2, 8, 8, 1, 2, 4, 3, 1, 4, 1, 3, 7, 2, 1, 5, 14, 12, 4, 3, 1, 9, 8, 3, 4, 2, 1, 8, 1, 2, 7, 5, 8, 4, 1, 3, 4, 12, 1, 6, 1, 2, 18, 3, 14, 4, 1, 5, 9, 2, 1, 8, 8, 2, 4, 3, 1, 9, 14, 3, 4, 2, 8, 5, 1, 16, 7, 6, 1, 4, 1, 3, 18

Offset: 1

Antti Karttunen, Feb 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to Stern's sequences

Cf. A002487, A122111, A322865.

Cf. also A323168, A323174, A323902.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
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)));
A323903(n) = A002487(A122111(n));

a(n) = A002487(A122111(n)) = A002487(A322865(n)).

a(p) = 1 for all primes p.

A334108 a(n) = A331410(A122111(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 2, 1, 0, 2, 0, 1, 2, 1, 0, 3, 0, 2, 2, 1, 0, 1, 3, 1, 4, 2, 0, 3, 0, 2, 2, 1, 3, 2, 0, 1, 2, 1, 0, 3, 0, 2, 4, 1, 0, 2, 4, 4, 2, 2, 0, 3, 3, 1, 2, 1, 0, 2, 0, 1, 4, 2, 3, 3, 0, 2, 2, 4, 0, 3, 0, 1, 5, 2, 4, 3, 0, 2, 2, 1, 0, 2, 3, 1, 2, 1, 0, 3, 4, 2, 2, 1, 3, 2, 0, 5, 4, 3, 0, 3, 0, 1, 5

Offset: 1

Antti Karttunen, Apr 29 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A008578 (positions of zeros), A064989, A105560, A122111, A322865, A331410, A334107.

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)));
A331410(n) = if(!bitand(n,n-1),0,1+A331410(n+(n/vecmax(factor(n)[, 1]))));
A334108(n) = A331410(A122111(n));

a(n) = A331410(A122111(n)) = A331410(A322865(n)).

a(n) = A331410(A105560(n)) + a(A064989(n)).

A331731 Odd part of A331595(n), where A331595(n) = gcd(A122111(n), A241909(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 5, 3, 3, 1, 5, 1, 3, 9, 7, 1, 15, 1, 5, 9, 3, 1, 7, 3, 3, 5, 5, 1, 15, 1, 11, 9, 3, 9, 7, 1, 3, 9, 7, 1, 15, 1, 5, 25, 3, 1, 11, 3, 45, 9, 5, 1, 7, 27, 7, 9, 3, 1, 7, 1, 3, 25, 13, 27, 15, 1, 5, 9, 45, 1, 11, 1, 3, 15, 5, 9, 15, 1, 11, 7, 3, 1, 7, 27, 3, 9, 7, 1, 7, 27, 5, 9, 3, 27, 13, 1, 135, 25, 7, 1, 15, 1, 7, 75

Offset: 1

Antti Karttunen, Jan 25 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A122111, A241909, A331595, A331600.

Cf. also A322865, A331732.

PARI
```
A331731(n) = A000265(A331595(n));
```

a(n) = A000265(A331595(n)).

A322866 Lexicographically earliest such sequence a that a(i) = a(j) => A046523(A322863(i)) = A046523(A322863(j)) for all i, j.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 30 2018

nonn

Restricted growth sequence transform of A046523(A322863(n)).
Equally, restricted growth sequence transform of A278222(A322865(n)).
For all i, j: a(i) = a(j) => A322867(i) = A322867(j).

Cf. A005940, A046523, A122111, A278222, A286622, A322863, A322865, A322867.

up_to = 8192;
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 };
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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A322863(n) = if(!n,1,A005940(1+A122111(n)));
v322866 = rgs_transform(vector(up_to,n,A046523(A322863(n))));
A322866(n) = v322866[n];

A331286 Odd part of number of divisors of primorial inflation of n: a(n) = A000265(A329605(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 1, 9, 3, 1, 1, 1, 3, 9, 5, 1, 3, 1, 1, 9, 3, 1, 5, 27, 3, 1, 1, 1, 3, 1, 3, 9, 3, 27, 15, 1, 3, 9, 5, 1, 3, 1, 1, 1, 3, 1, 3, 81, 9, 9, 1, 1, 5, 27, 5, 9, 3, 1, 15, 1, 3, 1, 7, 27, 3, 1, 1, 9, 9, 1, 9, 1, 3, 3, 1, 81, 3, 1, 3, 25, 3, 1, 15, 27, 3, 9, 5, 1, 5, 81, 1, 9, 3, 27, 7, 1, 27, 1, 45, 1, 3, 1, 5, 3

Offset: 1

Antti Karttunen, Jan 14 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000265, A108951, A212181, A329605.

Cf. also A322865.

A331286(n) = if(1==n,1,my(f=factor(n),e=1,m=1); forstep(i=#f~,1,-1, e += f[i,2]; m *= (e>>valuation(e,2))^(primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1])))); (m));

a(n) = A000265(A329605(n)).

a(n) = A212181(A108951(n)).

A331732 Odd part of A241909(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 9, 1, 5, 3, 27, 1, 25, 1, 81, 9, 7, 1, 15, 1, 125, 27, 243, 1, 49, 3, 729, 5, 625, 1, 75, 1, 11, 81, 2187, 9, 35, 1, 6561, 243, 343, 1, 375, 1, 3125, 25, 19683, 1, 121, 3, 45, 729, 15625, 1, 21, 27, 2401, 2187, 59049, 1, 245, 1, 177147, 125, 13, 81, 1875, 1, 78125, 6561, 225, 1, 77, 1, 531441, 15, 390625, 9, 9375, 1, 1331

Offset: 1

Antti Karttunen, Jan 25 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A241909, A331601.

Cf. also A322865, A331731.

A000265(n) = (n/2^valuation(n, 2));
A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m));
A331732(n) = A000265(A241909(n));

a(n) = A000265(A241909(n)).

cp's OEIS Frontend

A334107 a(n) = A329697(A122111(n)).

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A322867 a(n) = A000120(A122111(n)).

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A323903 a(n) = A002487(A122111(n)).

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A334108 a(n) = A331410(A122111(n)).

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A331731 Odd part of A331595(n), where A331595(n) = gcd(A122111(n), A241909(n)).

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A322866 Lexicographically earliest such sequence a that a(i) = a(j) => A046523(A322863(i)) = A046523(A322863(j)) for all i, j.

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A331286 Odd part of number of divisors of primorial inflation of n: a(n) = A000265(A329605(n)).

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A331732 Odd part of A241909(n).

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