A297112 -id:A297112 - cp's OEIS frontend

A324190 Number of distinct values A297167 obtains over the divisors > 1 of n; a(1) = 0.

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 4, 2, 2, 1, 4, 2, 2, 3, 4, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 5, 1, 3, 1, 4, 3, 2, 1, 5, 2, 3, 2, 4, 1, 4, 2, 6, 2, 2, 1, 4, 1, 2, 4, 6, 2, 3, 1, 4, 2, 3, 1, 5, 1, 2, 3, 4, 2, 3, 1, 6, 4, 2, 1, 5, 2, 2, 2, 6, 1, 4, 2, 4, 2, 2, 2, 6, 1, 3, 4, 5, 1, 3, 1, 6, 3

Offset: 1

Antti Karttunen, Feb 19 2019

nonn

Number of distinct values of the sum {excess of d} + {the index of the largest prime factor of d} (that is, A046660(d) + A061395(d)) that occurs over all divisors d > 1 of n.

Number of distinct values A297112 obtains over the divisors > 1 of n; a(1) = 0.

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

Cf. A000005, A001221, A001222, A046660, A061395, A324120, A324179, A324191, A324192, A324202, A324203.

Cf. also A156552, A297112.

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A324190(n) = #Set(apply(A297167, select(d -> d>1,divisors(n))));

a(n) = A001221(A324202(n)).
a(n) >= A324120(n).
a(n) >= A001222(n) >= A001221(n). [See A324179 and A324192 for differences]
a(n) <= A000005(n)-1. [See A324191 for differences]
For all primes p, a(p^k) = k.

A324120 Binary weight of SumXOR variant of A297168: a(n) = A000120(A324180(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 19 2019

nonn

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

Cf. A000043, A324179, A324180, A324181, A324190, A324191, A324192, A324201.

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A297112(n) = if(1==n, 0, 2^A297167(n));
A324180(n) = { my(v=0); fordiv(n, d, if(dA297112(d)))); (v); };
A324120(n) = hammingweight(A324180(n));

a(n) = A000120(A324180(n)).
a(n) <= A324190(n).
a(p^k) = k-1 for all primes p and exponents k >= 1.

A364557 Möbius transform of A005941.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 8, 4, 4, 4, 16, 4, 32, 8, 4, 8, 64, 4, 128, 8, 8, 16, 256, 8, 8, 32, 8, 16, 512, 4, 1024, 16, 16, 64, 8, 8, 2048, 128, 32, 16, 4096, 8, 8192, 32, 8, 256, 16384, 16, 16, 8, 64, 64, 32768, 8, 16, 32, 128, 512, 65536, 8, 131072, 1024, 16, 32, 32, 16, 262144, 128, 256, 8, 524288, 16, 1048576, 2048

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

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

Cf. A005941, A008683, A297112, A297113, A297167, A364558.

A364557(n) = if(1==n, 1, 2^(primepi(vecmax(factor(n)[, 1]))+(bigomega(n)-omega(n))-1));

A005941(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]); (1+res) }; \\ (After David A. Corneth's program for A156552)
A364557(n) = sumdiv(n,d,moebius(n/d)*A005941(d));

from sympy import factorint, primepi
def A364557(n): return 1<1 else 1 # Chai Wah Wu, Jul 29 2023

a(n) = Sum_{d|n} A008683(n/d) * A005941(d).

a(1) = 1; for n > 1, a(n) = A297112(n) = 2^(A297113(n)-1) = 2^A297167(n).

A297171 Möbius transform of A243071.

Original entry on oeis.org

0, 1, 3, 1, 7, 2, 15, 2, 2, 6, 31, 5, 63, 14, 3, 4, 127, 2, 255, 13, 11, 30, 511, 10, 4, 62, 4, 29, 1023, 4, 2047, 8, 27, 126, 5, 4, 4095, 254, 59, 26, 8191, 12, 16383, 61, 10, 510, 32767, 20, 8, 4, 123, 125, 65535, 4, 21, 58, 251, 1022, 131071, 7, 262143, 2046, 26, 16, 53, 28, 524287, 253, 507, 6, 1048575, 8

Offset: 1

Antti Karttunen, Dec 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A008683, A064989, A243071, A297161 (rgs-transform of this sequence).

Cf. also A297112, A297156, A297172.

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)};
A243071(n) = if(n<=2, n-1, if(!(n%2), 2*A243071(n/2), 1+(2*A243071(A064989(n)))));
A297171(n) = sumdiv(n,d,moebius(n/d)*A243071(d));

a(n) = Sum_{d|n} A008683(n/d)*A243071(d).

A324181 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A324180(n) for n > 1 and f(1) = -1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 5, 6, 2, 4, 2, 7, 8, 9, 2, 6, 2, 4, 10, 11, 2, 4, 12, 13, 8, 4, 2, 6, 2, 14, 15, 16, 17, 9, 2, 18, 19, 14, 2, 7, 2, 4, 8, 20, 2, 4, 21, 7, 22, 4, 2, 7, 23, 24, 25, 26, 2, 14, 2, 27, 8, 28, 29, 11, 2, 4, 30, 7, 2, 4, 2, 31, 10, 4, 32, 13, 2, 24, 33, 34, 2, 24, 35, 36, 37, 38, 2, 7, 39, 4, 40, 41, 42, 4, 2, 11, 8, 43, 2, 16, 2, 44, 10

Offset: 1

Antti Karttunen, Feb 19 2019

nonn

For all i, j: a(i) = a(j) => A324120(i) = A324120(j).

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

Cf. A000040 (positions of 2's), A156552, A297112, A324120, A324180.

Cf. also A300827, A323914, A324203.

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; };
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A297112(n) = if(1==n, 0, 2^A297167(n));
A324180(n) = { my(v=0); fordiv(n, d, if(dA297112(d)))); (v); };
Aux324181(n) = if((1==n),-n,A324180(n));
v324181 = rgs_transform(vector(up_to, n, Aux324181(n)));
A324181(n) = v324181[n];

A322994 Möbius transform of A322993.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 7, 1, 4, 1, 15, 3, 8, 1, 6, 1, 8, 7, 31, 1, 8, 2, 63, 4, 16, 1, 5, 1, 16, 15, 127, 3, 8, 1, 255, 31, 16, 1, 9, 1, 32, 4, 511, 1, 16, 2, 14, 63, 64, 1, 12, 7, 32, 127, 1023, 1, 8, 1, 2047, 8, 32, 15, 17, 1, 128, 255, 13, 1, 16, 1, 4095, 6, 256, 3, 33, 1, 32, 8, 8191, 1, 16, 31, 16383, 511, 64, 1, 12, 7, 512, 1023

Offset: 1

Antti Karttunen, Jan 02 2019

nonn

Möbius transform of A000265(A156552(n)).

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000265, A008683, A156552, A297112, A322993.

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));

a(n) = Sum_{d|n} A008683(n/d)*A322993(d).

A297156 Möbius transform of A243354.

Original entry on oeis.org

0, 1, 3, 1, 7, 2, 15, 3, 1, 6, 31, 6, 63, 14, 2, 5, 127, 2, 255, 14, 10, 30, 511, 10, 1, 62, 7, 30, 1023, 4, 2047, 11, 26, 126, 2, 2, 4095, 254, 58, 26, 8191, 12, 16383, 62, 14, 510, 32767, 22, 1, 2, 122, 126, 65535, 6, 18, 58, 250, 1022, 131071, 4, 262143, 2046, 30, 21, 50, 28, 524287, 254, 506, 4, 1048575, 6

Offset: 1

Antti Karttunen, Dec 28 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A006068, A156552, A243354, A297157 (rgs-transform of this sequence).

Cf. also A297112, A297171, A297172.

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))));
A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ Essentially Joerg Arndt's Jul 19 2012 code.
A243354(n) = A006068(A156552(n));
A297156(n) = sumdiv(n,d,moebius(n/d)*A243354(d));

A297172 Möbius transform of A253564.

Original entry on oeis.org

0, 1, 3, 1, 7, 1, 15, 2, 3, 3, 31, 3, 63, 7, 3, 4, 127, 2, 255, 7, 9, 15, 511, 6, 7, 31, 6, 15, 1023, 3, 2047, 8, 21, 63, 7, 4, 4095, 127, 45, 14, 8191, 7, 16383, 31, 9, 255, 32767, 12, 15, 4, 93, 63, 65535, 4, 21, 30, 189, 511, 131071, 5, 262143, 1023, 21, 16, 49, 15, 524287, 127, 381, 5, 1048575, 8, 2097151, 2047, 6

Offset: 1

Antti Karttunen, Dec 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A008683, A064989, A122111, A156552, A253564, A297162 (rgs-transform of this sequence).

Cf. also A297112, A297156, A297171.

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)));
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A253564(n) = A156552(A122111(n));
A297172(n) = sumdiv(n,d,moebius(n/d)*A253564(d));

a(n) = Sum_{d|n} A008683(n/d)*A253564(d).

A318891 Filter sequence combining the prime signature of n (A046523) with the largest prime factor of n (A006530).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 16 2018

nonn

Restricted growth sequence transform of A286356.

For all i, j: a(i) = a(j) => A297112(i) = A297112(j). (Also, equivalently, A297113 or A297167 in place of A297112.)

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

Cf. A006530, A046523, A061395, A286356, A318890.

Cf. also A297112, A297113, A297167.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A318891aux(n) = [A046523(n), A061395(n)];
v318891 = rgs_transform(vector(up_to,n,A318891aux(n)));
A318891(n) = v318891[n];

A324179 Number of distinct values A297167 obtains over divisors > 1 of n, minus number of prime factors of n counted with multiplicity: a(n) = A324190(n) - A001222(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0

Offset: 1

Antti Karttunen, Feb 19 2019

nonn

a(n) is zero for all prime powers (A000961), but also for many other numbers.

			Divisors of 56 larger than 1 are [2, 4, 7, 8, 14, 28, 56]. When A297167 is applied to each, one obtains values: [0, 1, 3, 2, 3, 4, 5], of which 6 values are distinct (as one of them, 3, occurs twice). On the other hand, 56 = 2 * 2 * 2 * 7 has four prime factors in total, thus a(56) = 6 - 4 = 2.

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

Cf. A000961, A001222, A061395, A297112, A297167, A324120, A324190, A324191, A324192.

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A324190(n) = #Set(apply(A297167, select(d -> d>1,divisors(n))));
A324179(n) = (A324190(n)-bigomega(n));

a(n) = A324190(n) - A001222(n).

a(n) <= A324192(n).

cp's OEIS Frontend

A324190 Number of distinct values A297167 obtains over the divisors > 1 of n; a(1) = 0.

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A324120 Binary weight of SumXOR variant of A297168: a(n) = A000120(A324180(n)).

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A364557 Möbius transform of A005941.

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A297171 Möbius transform of A243071.

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A324181 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A324180(n) for n > 1 and f(1) = -1.

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A322994 Möbius transform of A322993.

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A297156 Möbius transform of A243354.

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A297172 Möbius transform of A253564.

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A318891 Filter sequence combining the prime signature of n (A046523) with the largest prime factor of n (A006530).

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