A322993 -id:A322993 - cp's OEIS frontend

A342651 a(n) = A329697(A156552(n)).

0, 0, 1, 0, 1, 0, 2, 1, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 3, 2, 3, 0, 3, 1, 3, 2, 3, 0, 3, 0, 3, 1, 5, 1, 3, 0, 1, 3, 3, 0, 3, 0, 4, 2, 6, 0, 4, 1, 2, 3, 4, 0, 3, 2, 4, 5, 4, 0, 4, 0, 7, 3, 4, 1, 4, 0, 5, 1, 2, 0, 3, 0, 4, 2, 4, 1, 5, 0, 4, 2, 8, 0, 3, 3, 7, 6, 4, 0, 3, 2, 6, 4, 9, 3, 4, 0, 4, 3, 2, 0, 5, 0, 5, 3

Offset: 2

Antti Karttunen, Mar 18 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 2..6381 (based on Hans Havermann's factorization of A156552)
Index entries for sequences computed from indices in prime factorization

Cf. A156552, A329697, A322993, A342652, A350067.

Cf. A000040 (positions of 0's), A350069 (of 1's).

A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res};
A329697(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A329697(f[k,1]-1)))); };
A342651(n) = A329697(A156552(n));

\\ Version using the factorization file available at https://oeis.org/A156552/a156552.txt
v156552sigs = readvec("a156552.txt");
A329697(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A329697(f[k,1]-1)))); };
A342651(n) = if(isprime(n),0,my(prsig=v156552sigs[n],ps=prsig[1],es=prsig[2]); sum(i=2-(ps[1]%2),#ps,es[i]*(1+A329697(ps[i]-1)))); \\ Antti Karttunen, Jan 29 2022

a(n) = A329697(A156552(n)) = A329697(A322993(n)).
a(n) = A329697(A342666(n)) + A342656(n).
a(p) = 0 for all primes p.
a(A003961(n)) = a(n).

A350065 Lexicographically earliest infinite sequence such that a(i) = a(j) => A350063(i) = A350063(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 29 2022

nonn

Restricted growth sequence transform of A350063.
For all i, j >= 1: A305897(i) = A305897(j) => a(i) = a(j) => A324117(i) = A324117(j).
For all i, j >= 2: a(i) = a(j) => A342656(i) = A342656(j).

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A046523, A156552, A322993, A350063, A350064, A350067, A350068.

Cf. also A305897, A324117, A342656.

up_to = 3000;
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; };
A000265(n) = (n>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A350063(n) = if(1==n,0,A046523(A000265(A156552(n))));
v350065 = rgs_transform(vector(up_to, n, A350063(n)));
A350065(n) = v350065[n];

\\ Version using the factorization file available at https://oeis.org/A156552/a156552.txt
v156552sigs = readvec("a156552.txt");
up_to = #v156552sigs;
A350063(n) = if(n<=2,n-1,my(es=v156552sigs[n][2]); if(n%2, es = vector(#es-1,i,es[1+i])); my(f=vecsort(es, , 4), p=0); prod(i=1, #f, (p=nextprime(p+1))^f[i]));
v350065 = rgs_transform(vector(up_to, n, A350063(n)));
A350065(n) = v350065[n]; \\ Antti Karttunen, Jan 29 2022

A350067 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A342666(n), A350063(n)] for n > 1, with f(1) = 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 5, 2, 6, 2, 3, 3, 7, 2, 4, 2, 8, 5, 9, 2, 10, 3, 11, 4, 11, 2, 11, 2, 12, 3, 13, 3, 14, 2, 3, 9, 11, 2, 8, 2, 15, 6, 16, 2, 17, 3, 5, 11, 18, 2, 19, 5, 20, 13, 21, 2, 22, 2, 23, 8, 24, 3, 25, 2, 26, 3, 6, 2, 9, 2, 27, 4, 28, 3, 26, 2, 29, 7, 30, 2, 31, 9, 32, 16, 33, 2, 31, 5, 34, 21, 35

Offset: 1

Antti Karttunen, Jan 29 2022

nonn

Restricted growth sequence transform of the ordered pair [A342666(n), A350063(n)], when assuming that A342666(1) = 0.
Restricted growth sequence transform of the function f(1) = 0, f(n) = A336470(A156552(n)) for n > 1.
For all i, j >= 1: A305897(i) = A305897(j) => a(i) = a(j) => A350065(i) = A350065(j).
For all i, j >= 2: a(i) = a(j) => A342651(i) = A342651(j).

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences computed from indices in prime factorization

Cf. A156552, A322993, A336466, A342666, A350063, A336470.

Cf. also A305897, A350065, A342651.

up_to = 3003;
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; };
A000265(n) = (n>>valuation(n,2));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A336466(n) = { my(f=factor(n)); prod(k=1, #f~, A000265(f[k, 1]-1)^f[k, 2]); };
Aux350067(n) = if(1==n,1,my(u=A000265(A156552(n))); [A046523(u),A336466(u)]);
v350067 = rgs_transform(vector(up_to, n, Aux350067(n)));
A350067(n) = v350067[n];

A322995 a(1) = 0; for n > 1, a(n) = A000265(A289271(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 9, 5, 1, 1, 17, 1, 3, 9, 33, 1, 17, 1, 65, 1, 5, 1, 7, 1, 1, 33, 129, 3, 9, 1, 257, 65, 5, 1, 11, 1, 17, 9, 513, 1, 129, 1, 1025, 257, 33, 1, 2049, 17, 3, 513, 4097, 1, 7, 1, 8193, 5, 1, 33, 19, 1, 65, 1025, 13, 1, 3, 1, 16385, 2049, 129, 9, 35, 1, 65, 1, 32769, 1, 11, 129, 65537, 4097, 5, 1, 21, 17, 257

Offset: 1

Antti Karttunen, Jan 02 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000120, A001221, A000265, A289271.

Cf. also A322993.

A000265(n) = (n/2^valuation(n, 2));
A289271(n) = { my(v=0,i=0,x=1); for(d=2,oo,if(n==1, return(v)); if(1==gcd(x,d)&&1==omega(d), if(!(n%d)&&1==gcd(d,n/d), v += 2^i; n /= d; x *= d); i++)); }; \\ After Rémy Sigrist's program for A289271.
A322995(n) = if(1==n,0,A000265(A289271(n)));

a(1) = 0; for n > 1, a(n) = A000265(A289271(n)).

For all n >= 1, A000120(a(n)) = A001221(n).

A323914 Lexicographically earliest sequence such that a(i) = a(j) => A322994(i) = A322994(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 6, 2, 5, 2, 7, 4, 8, 2, 9, 2, 8, 6, 10, 2, 8, 3, 11, 5, 12, 2, 13, 2, 12, 7, 14, 4, 8, 2, 15, 10, 12, 2, 16, 2, 17, 5, 18, 2, 12, 3, 19, 11, 20, 2, 21, 6, 17, 14, 22, 2, 8, 2, 23, 8, 17, 7, 24, 2, 25, 15, 26, 2, 12, 2, 27, 9, 28, 4, 29, 2, 17, 8, 30, 2, 12, 10, 31, 18, 20, 2, 21, 6, 32, 22, 33, 11, 17, 2, 34, 12, 12, 2, 35, 2, 25, 13

Offset: 1

Antti Karttunen, Feb 12 2019

nonn

Restricted growth sequence transform of A322994.

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

Cf. A156552, A297112, A322993, A322994.

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; };
A000265(n) = (n/2^valuation(n, 2));
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))));
A322993(n) = if(1==n,0,A000265(A156552(n)));
A322994(n) = sumdiv(n,d,moebius(n/d)*A322993(d));
v323914 = rgs_transform(vector(up_to,n,A322994(n)));
A323914(n) = v323914[n];

A324910 Multiplicative with a(p^e) = (2^e)-1.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 7, 3, 1, 1, 3, 1, 1, 1, 15, 1, 3, 1, 3, 1, 1, 1, 7, 3, 1, 7, 3, 1, 1, 1, 31, 1, 1, 1, 9, 1, 1, 1, 7, 1, 1, 1, 3, 3, 1, 1, 15, 3, 3, 1, 3, 1, 7, 1, 7, 1, 1, 1, 3, 1, 1, 3, 63, 1, 1, 1, 3, 1, 1, 1, 21, 1, 1, 3, 3, 1, 1, 1, 15, 15, 1, 1, 3, 1, 1, 1, 7, 1, 3, 1, 3, 1, 1, 1, 31, 1, 3, 3, 9, 1, 1, 1, 7, 1

Offset: 1

Antti Karttunen, Apr 14 2019

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

Cf. A000225, A156552, A246674, A322993, A324911.

Array[Times @@ (2^(FactorInteger[#][[All, -1]]) - 1) &, 105] (* Michael De Vlieger, Apr 14 2019 *)

A324910(n) = factorback(apply(e -> -1+(2^e), factor(n)[,2]~));

Multiplicative with a(p^e) = A000225(e).
Multiplicative with a(p^e) = A322993(p^e).
a(n) = A246674(A156552(n)).

A342657 The difference between floor(log_2(.)) of and the number of prime factors in A156552(n) (when counted with multiplicity).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 1, 1, 0, 2, 0, 3, 1, 3, 0, 3, 0, 4, 1, 3, 0, 2, 0, 3, 3, 5, 1, 1, 0, 7, 3, 3, 0, 4, 0, 5, 2, 5, 0, 4, 0, 2, 4, 6, 0, 3, 1, 5, 5, 7, 0, 4, 0, 9, 3, 2, 3, 4, 0, 6, 7, 4, 0, 3, 0, 10, 2, 7, 1, 5, 0, 5, 1, 11, 0, 3, 3, 11, 5, 3, 0, 2, 1, 8, 7, 11, 4, 4, 0, 3, 3, 3, 0, 5, 0, 7, 2

Offset: 2

Antti Karttunen, Mar 18 2021

nonn

Cf. A000430 (positions of 0's), A000523, A001222, A003961, A061395, A070939, A156552, A246277, A322993, A325134, A339893, A342655, A342656.

Cf. also A339895.

A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res};
A342657(n) = { my(u=A156552(n)); (#binary(u)-bigomega(u))-1; };

a(n) = A339893(A156552(n)) = A339893(A322993(n)) = A000523(A156552(n)) - A001222(A156552(n)).
a(n) = (A252464(n)-A342655(n))-1 = (A325134(n)-A342655(n)) - 2.
a(p) = a(p^2) = 0 for all primes p. (Second part added Jul 27 2023)
a(A003961(n)) = a(2*A246277(n)) = a(n).

A342666 a(n) = A336466(A156552(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 9, 1, 5, 1, 11, 1, 3, 3, 3, 1, 3, 1, 15, 1, 21, 1, 1, 1, 1, 5, 3, 1, 9, 1, 33, 5, 9, 1, 23, 1, 1, 3, 65, 1, 7, 1, 35, 21, 5, 1, 21, 1, 341, 9, 3, 1, 11, 1, 27, 1, 5, 1, 5, 1, 15, 3, 51, 1, 27, 1, 39, 1, 1365, 1, 1, 5, 49, 9, 1, 1, 1, 1, 117, 5, 825, 3, 9, 1, 9, 3, 1, 1, 7, 1

Offset: 2

Antti Karttunen, Mar 18 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 2..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A003961, A156552, A322993, A336466.

Cf. also A324104, A342651, A342656, A350067.

A000265(n) = (n>>valuation(n,2));
A336466(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]-1))^f[k,2])); };
A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res};
A342666(n) = A336466(A156552(n));

\\ Version using the factorization file available at https://oeis.org/A156552/a156552.txt
v156552sigs = readvec("a156552.txt");
A000265(n) = (n>>valuation(n,2));
A342666(n) = if(isprime(n),1,my(prsig=v156552sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,A000265(ps[i]-1)^es[i])); \\ Antti Karttunen, Jan 29 2022

a(n) = A336466(A156552(n)) = A336466(A322993(n)).
a(p) = 1 for all primes p.
a(A003961(n)) = a(n).

A350068 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A350063(i) = A350063(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 6, 2, 7, 2, 4, 4, 8, 2, 7, 2, 7, 6, 9, 2, 10, 3, 9, 5, 11, 2, 12, 2, 13, 4, 9, 4, 14, 2, 4, 9, 15, 2, 16, 2, 7, 7, 17, 2, 18, 3, 19, 9, 7, 2, 10, 6, 10, 9, 20, 2, 21, 2, 9, 7, 22, 4, 12, 2, 11, 4, 16, 2, 23, 2, 9, 7, 11, 4, 12, 2, 18, 8, 9, 2, 24, 9, 25, 17, 26, 2, 24, 6, 11, 20, 27, 9

Offset: 1

Antti Karttunen, Jan 29 2022

nonn

Restricted growth sequence transform of the ordered pair [A046523(n), A350063(n)].

For all i, j >= 1: A305897(i) = A305897(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A046523, A156552, A322993, A350063.
Cf. A000040 (positions of 2's), A001248 (of 3's).
Cf. also A101296, A300226, A305897, A350065.

up_to = 3003;
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; };
A000265(n) = (n>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A350063(n) = if(1==n,0,A046523(A000265(A156552(n))));
Aux350068(n) = [A046523(n),A350063(n)];
v350068 = rgs_transform(vector(up_to, n, Aux350068(n)));
A350068(n) = v350068[n];

A331602 a(1) = 0; for n > 1, a(n) = A007947(A156552(n)).

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 2, 7, 6, 3, 2, 11, 2, 17, 10, 15, 2, 13, 2, 19, 6, 33, 2, 23, 6, 65, 14, 35, 2, 21, 2, 31, 34, 129, 10, 3, 2, 257, 66, 39, 2, 37, 2, 67, 22, 57, 2, 47, 6, 5, 130, 131, 2, 29, 6, 71, 258, 205, 2, 43, 2, 2049, 38, 21, 34, 69, 2, 259, 514, 41, 2, 55, 2, 4097, 26, 515, 10, 133, 2, 79, 30, 8193, 2, 15, 66, 16385, 114, 15, 2, 15, 6, 1027, 410, 10923

Offset: 1

Antti Karttunen, Jan 22 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences computed from indices in prime factorization

Cf. A000035, A007947, A156552.

Cf. also A322993.

Array[Times @@ FactorInteger[#][[All, 1]] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &, 94] (* Michael De Vlieger, Jan 24 2020 *)

A007947(n) = factorback(factorint(n)[, 1]);
A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A331602(n) = A007947(A156552(n));

a(1) = 0; and for n > 1, a(n) = A007947(A156552(n)).

A000035(a(n)) = 1 - A000035(n). [Flips the parity]

cp's OEIS Frontend

A342651 a(n) = A329697(A156552(n)).

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A350065 Lexicographically earliest infinite sequence such that a(i) = a(j) => A350063(i) = A350063(j), for all i, j >= 1.

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A350067 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A342666(n), A350063(n)] for n > 1, with f(1) = 1.

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A322995 a(1) = 0; for n > 1, a(n) = A000265(A289271(n)).

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A323914 Lexicographically earliest sequence such that a(i) = a(j) => A322994(i) = A322994(j), for all i, j >= 1.

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A324910 Multiplicative with a(p^e) = (2^e)-1.

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Mathematica

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A342657 The difference between floor(log_2(.)) of and the number of prime factors in A156552(n) (when counted with multiplicity).

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A342666 a(n) = A336466(A156552(n)).

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A350068 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A350063(i) = A350063(j), for all i, j >= 1.

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