cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A324191 Number of divisors of n minus the number of distinct values that A297167 obtains over the divisors > 1 of n: a(n) = A000005(n) - A324190(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 5, 1, 2, 2, 3, 1, 5, 1, 2, 3, 2, 1, 5, 1, 3, 2, 2, 1, 4, 2, 2, 2, 2, 1, 8, 1, 2, 2, 1, 2, 5, 1, 2, 2, 5, 1, 7, 1, 2, 3, 2, 2, 5, 1, 4, 1, 2, 1, 7, 2, 2, 2, 2, 1, 8, 2, 2, 2, 2, 2, 6, 1, 3, 2, 4, 1, 5, 1, 2, 5
Offset: 1

Views

Author

Antti Karttunen, Feb 19 2019

Keywords

Comments

a(p^k) = 1 for all primes p and all exponents k >= 0, because with prime powers there are k divisors larger than 1 and A297167 obtains a distinct value for each one of them.

Links

Crossrefs

Cf. A000005, A001221, A001222, A046660, A061395, A297167, A324120, A324179, A324120, A324190, A324192.
Cf. A000961 (positions of ones).

Programs

Formula

a(n) = A000005(n) - A324190(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

Views

Author

Antti Karttunen, Feb 19 2019

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

Formula

a(n) = A324190(n) - A001222(n).
a(n) <= A324192(n).

A324192 Number of distinct values A297167 obtains over divisors > 1 of n, minus number of distinct prime factors of n: a(n) = A324190(n) - A001221(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 2, 0, 0, 0, 2, 1, 0, 2, 2, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 2, 1, 0, 0, 3, 1, 1, 0, 2, 0, 2, 0, 4, 0, 0, 0, 1, 0, 0, 2, 5, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 1, 2, 0, 0, 0, 4, 3, 0, 0, 2, 0, 0, 0, 4, 0, 1, 0, 2, 0, 0, 0, 4, 0, 1, 2, 3, 0, 0, 0, 4, 0
Offset: 1

Views

Author

Antti Karttunen, Feb 19 2019

Keywords

Links

Crossrefs

Cf. A001222, A324179.

Programs

Formula

a(n) = A324190(n) - A001221(n).
a(n) >= A324179(n).

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

Views

Author

Antti Karttunen, Feb 19 2019

Keywords

Links

Crossrefs

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

Programs

Formula

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

A324202 a(n) = A046523(A332461(n)), where A332461(n) = Product_{d|n, d>1} prime(1+A297167(d)).

Original entry on oeis.org

1, 2, 2, 6, 2, 12, 2, 30, 6, 12, 2, 120, 2, 12, 12, 210, 2, 180, 2, 420, 12, 12, 2, 2520, 6, 12, 30, 420, 2, 720, 2, 2310, 12, 12, 12, 7560, 2, 12, 12, 9240, 2, 720, 2, 420, 120, 12, 2, 138600, 6, 180, 12, 420, 2, 6300, 12, 60060, 12, 12, 2, 151200, 2, 12, 420, 30030, 12, 720, 2, 420, 12, 720, 2, 831600, 2, 12, 180, 420, 12, 720, 2, 360360
Offset: 1

Views

Author

Antti Karttunen, Feb 19 2019

Keywords

Links

Crossrefs

Cf. A032741, A061395, A297113, A297167, A324190, A324203 (rgs-transform), A324537, A332461.

Programs

  • PARI
    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])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A324202(n) = A046523(factorback(apply(x -> prime(1+x),apply(A297167, select(d -> d>1,divisors(n))))));

Formula

a(n) = A046523(A332461(n)).
A001221(a(n)) = A324190(n).
A001222(a(n)) = A032741(n).

A324203 Lexicographically earliest sequence such that a(i) = a(j) => A324202(i) = A324202(j) for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Feb 19 2019

Keywords

Comments

Restricted growth sequence transform of A324202.
For all i, j:
a(i) = a(j) => A324190(i) = A324190(j),
a(i) = a(j) => A324191(i) = A324191(j).

Links

Crossrefs

Cf. A300827, A324181, A324190, A324191, A324202.

Programs

  • PARI
    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])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A324202(n) = A046523(factorback(apply(x -> prime(1+x),apply(A297167, select(d -> d>1,divisors(n))))));
v324203 = rgs_transform(vector(up_to, n, A324202(n)));
A324203(n) = v324203[n];

A324196 Lexicographically earliest sequence such that a(i) = a(j) => A324195(i) = A324195(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 6, 7, 8, 9, 10, 7, 11, 12, 8, 13, 14, 7, 15, 13, 16, 17, 18, 13, 19, 20, 21, 22, 23, 7, 24, 25, 26, 27, 19, 13, 28, 29, 30, 25, 31, 32, 33, 34, 21, 35, 36, 25, 37, 38, 39, 40, 41, 13, 42, 43, 44, 45, 46, 13, 47, 48, 49, 43, 50, 51, 52, 53, 54, 38, 55, 25, 56, 57, 21, 58, 37, 59, 60, 43, 49, 61, 62, 25, 63, 64, 65, 66, 67, 13, 68, 69, 70, 71, 72, 43
Offset: 1

Views

Author

Antti Karttunen, Feb 20 2019

Keywords

Comments

Restricted growth sequence transform of A324195.
For all i, j: a(i) = a(j) => A324197(i) = A324197(j) => A324190(i) = A324190(j).

Links

Crossrefs

Cf. A297112, A297167, A297168, A061395, A324190, A324195, A324197.
Cf. also A324181.

Programs

  • PARI
    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));
A324195(n) = { my(v=0); fordiv(n, d, v = bitor(v,A297112(d))); (v); };
v324196 = rgs_transform(vector(up_to, n, A324195(n)));
A324196(n) = v324196[n];

A324197 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A324195(n) for all other numbers except f(2) = -1 and f(n) = -2 when n is an odd prime.

Original entry on oeis.org

1, 2, 3, 4, 3, 4, 3, 5, 6, 7, 3, 5, 3, 8, 6, 9, 3, 5, 3, 9, 10, 11, 3, 9, 12, 13, 14, 15, 3, 5, 3, 16, 17, 18, 12, 9, 3, 19, 20, 16, 3, 21, 3, 22, 14, 23, 3, 16, 24, 25, 26, 27, 3, 9, 28, 29, 30, 31, 3, 9, 3, 32, 33, 29, 34, 35, 3, 36, 37, 25, 3, 16, 3, 38, 14, 39, 24, 40, 3, 29, 33, 41, 3, 16, 42, 43, 44, 45, 3, 9, 46, 47, 48, 49, 50, 29, 3, 51, 52, 16, 3, 53, 3, 54, 14
Offset: 1

Views

Author

Antti Karttunen, Feb 20 2019

Keywords

Comments

For all i, j: a(i) = a(j) => A324190(i) = A324190(j).

Links

Crossrefs

Cf. A297112, A297167, A297168, A061395, A324190, A324195, A324196.

Programs

  • PARI
    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));
A324195(n) = { my(v=0); fordiv(n, d, v = bitor(v,A297112(d))); (v); };
Aux324197(n) = if(isprime(n),-(n%2)-1,A324195(n));
v324197 = rgs_transform(vector(up_to, n, Aux324197(n)));
A324197(n) = v324197[n];

A324195 Cumulative bitwise-OR of A297112(d), where d ranges over the divisors d of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 3, 8, 7, 6, 5, 16, 7, 32, 9, 6, 15, 64, 7, 128, 15, 10, 17, 256, 15, 12, 33, 14, 27, 512, 7, 1024, 31, 18, 65, 12, 15, 2048, 129, 34, 31, 4096, 11, 8192, 51, 14, 257, 16384, 31, 24, 13, 66, 99, 32768, 15, 20, 63, 130, 513, 65536, 15, 131072, 1025, 30, 63, 36, 19, 262144, 195, 258, 13, 524288, 31, 1048576, 2049, 14, 387, 24, 35, 2097152, 63, 30
Offset: 1

Views

Author

Antti Karttunen, Feb 20 2019

Keywords

Comments

A324180 differs from this one in that it uses XOR instead of OR, and uses only the proper divisors of n.

Links

Crossrefs

Cf. A000120, A003986, A297112, A297167, A297168, A061395, A324190, A324196, A324197.
Cf. also A324180.

Programs

  • PARI
    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));
A324195(n) = { my(v=0); fordiv(n, d, v = bitor(v,A297112(d))); (v); };

Formula

A000120(a(n)) = A324190(n).
Showing 1-9 of 9 results.

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