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 1-5 of 5 results.

A300827 Lexicographically earliest sequence such that a(i) = a(j) => A324193(i) = A324193(j) for all i, j >= 1.

Original entry on oeis.org

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

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Author

Antti Karttunen, Mar 13 2018

Keywords

Comments

Apart from primes, the sequence contains duplicate values at points p*q and p^3, where p*q are the product of two successive primes, with p < q (sequences A006094, A030078). Question: are there any other cases where a(x) = a(y), with x < y ?
The reason why this is not equal to A297169: Even though A297112 contains only powers of two after the initial zero, as A297112(n) = 2^A033265(A156552(d)) for n > 1, and A297168(n) is computed as Sum_{d|n, dA297112(d), still a single 1-bit in binary expansion of A297168(n) might be formed as a sum of several terms of A297112(d), i.e., could be born of carries.
From Antti Karttunen, Feb 28 2019: (Start)
A297168(n) = Sum_{d|n, dA297112(d) will not produce any carries (in base-2) if and only if n is a power of prime. Only in that case the number of summands (A000005(n)-1) is equal to the number of prime factors counted with multiplicity, A001222(n) = A000120(A156552(n)). (A notable subset of such numbers is A324201, numbers that are mapped to even perfect numbers by A156552). Precisely because there are so few points with duplicate values (apart from primes), this sequence is not particularly good for filtering other sequences, because the number of false positives is high. Any of the related sequences like A324203, A324196, A324197 or A324181 might work better in that respect. In any case, the following implications hold (see formula section of A324193 for the latter): (End)
For all i, j:
a(i) = a(j) => A297168(i) = A297168(j). (The same holds for A297169).
a(i) = a(j) => A324181(i) = A324181(j) => A324120(i) = A324120(j).

Examples

			For n = 15, with proper divisors 3 and 5, we have f(n) = prime(1+A297167(3)) * prime(1+A297167(5)) = prime(2)*prime(3) = 3*5 = 15.
For n = 27, with proper divisors 3 and 9, we have f(n) = prime(1+A297167(3)) * prime(1+A297167(9)) = prime(2)*prime(3) = 3*5 = 15.
Because f(15) = f(27), the restricted growth sequence transform allots the same number (in this case 9) for both, so a(15) = a(27) = 9.

Links

Crossrefs

Cf. A000120, A000961, A006094, A030078, A046660, A061395, A101296, A156552, A297112, A297167, A297168, A297169, A324193, A324120, A324201.
Cf. also A324181, A323914, A324196, A324197, A324203.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1]))); \\ After M. F. Hasler's code for A006530.
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) -1));
Aux300827(n) = { my(m=1); if(n<=2, n-1, fordiv(n,d,if((d>1)&(dA297167(d)))); (m)); };
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux300827(n))),"b300827.txt");

Formula

Restricted growth sequence transform of sequence f, defined as f(1) = 0, f(2) = 1, and for n > 2, f(n) = Product_{d|n, 1A297167(d)).
a(p) = 2 for all primes p.
a(A006094(n)) = a(A030078(n)), for all n >= 1.

Extensions

Name changed by Antti Karttunen, Feb 21 2019

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.

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

Views

Author

Antti Karttunen, Feb 19 2019

Keywords

Comments

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

Links

Crossrefs

Cf. A000040 (positions of 2's), A156552, A297112, A324120, A324180.
Cf. also A300827, A323914, A324203.

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));
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];

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).

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 1-5 of 5 results.

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