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 1-6 of 6 results.

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Feb 19 2019

Keywords

Comments

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.

Links

Crossrefs

Cf. A000005, A001221, A001222, A046660, A061395, A324120, A324179, A324191, A324192, A324202, A324203.
Cf. also A156552, A297112.

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));
A324190(n) = #Set(apply(A297167, select(d -> d>1,divisors(n))));

Formula

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

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.

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];

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).

A324537 a(n) = A003557(k), where k = Product_{d|n, d>2} prime(A297167(d)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 4, 1, 5, 3, 1, 1, 6, 1, 3, 5, 7, 1, 12, 1, 11, 1, 5, 1, 54, 1, 1, 7, 13, 5, 36, 1, 17, 11, 9, 1, 250, 1, 7, 9, 19, 1, 60, 1, 15, 13, 11, 1, 30, 7, 5, 17, 23, 1, 1620, 1, 29, 5, 1, 11, 686, 1, 13, 19, 375, 1, 540, 1, 31, 15, 17, 7, 2662, 1, 45, 1, 37, 1, 3500, 13, 41, 23, 7, 1, 2430, 11, 19, 29, 43, 17, 420, 1, 35, 7, 75, 1
Offset: 1

Views

Author

Antti Karttunen, Mar 07 2019

Keywords

Links

Crossrefs

Cf. A000961 (positions of ones), A003557, A297167, A300827, A324191, A324193, A324202, A324538.

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));
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A324537(n) = { my(m=1); fordiv(n, d, if(d>2, m *= prime(A297167(d)))); A003557(m); };

Formula

A001222(a(n)) = A324191(n) - 1.

A324538 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A324537(n) for all other numbers, except f(1) = 0.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Mar 07 2019

Keywords

Comments

For all i, j:
a(i) = a(j) => A069513(i) = A069513(j),
a(i) = a(j) => A324191(i) = A324191(j).

Links

Crossrefs

Cf. A000961 (positions of terms <= 2), A069513, A297167, A324191, A324537.
Cf. also A300827, A324181, A324203, A324196, A324197.

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));
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A324537(n) = { my(m=1); fordiv(n, d, if(d>2, m *= prime(A297167(d)))); A003557(m); };
Aux324538(n) = if(1==n,0,A324537(n));
v324538 = rgs_transform(vector(up_to,n,Aux324538(n)));
A324538(n) = v324538[n];
Showing 1-6 of 6 results.

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