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 1-3 of 3 results.

A327931 Lexicographically earliest infinite sequence such that for all i, j, a(i) = a(j) => A327930(i) = A327930(j).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 28, 40, 41, 42, 2, 43, 2, 44, 45, 46, 47, 48, 2, 49, 50, 51, 2, 52, 2, 53, 54, 55, 47, 56, 2, 57, 58, 59, 2, 60, 41, 61, 62, 63, 2, 64, 37, 65, 66, 67, 68, 69, 2, 70, 71, 72, 2, 73, 2, 74, 75
Offset: 1

Views

Author

Antti Karttunen, Sep 30 2019

Keywords

Comments

Restricted growth sequence transform of A327930, or equally, of the ordered pair [A003415(n), A319356(n)].
It seems that the sequence takes duplicated values only on primes (A000040) and some subset of squarefree semiprimes (A006881). If this holds, then also the last implication given below is valid.
For all i, j:
a(i) = a(j) => A000005(i) = A000005(j),
a(i) = a(j) => A319684(i) = A319684(j),
a(i) = a(j) => A319685(i) = A319685(j),
a(i) = a(j) => A101296(i) = A101296(j). [Conjectural, see notes above and in A319357]

Examples

			Divisors of 39 are [1, 3, 13, 39], while the divisors of 55 are [1, 5, 11, 55]. Taking their arithmetic derivatives (A003415) yields in both cases [0, 1, 1, 16], thus a(39) = a(55) (= 28, as allotted by restricted growth sequence transform).

Links

Crossrefs

Cf. A000005, A000040 (positions of 2's), A003415, A006881, A101296, A319356, A319357, A319684, A319685, A327930.
Differs from A300249 for the first time at n=105, where a(105)=75, while A300249(105)=56.
Differs from A300235 for the first time at n=153, where a(153)=110, while A300235(153)=106.
Differs from A305895 for the first time at n=3283, where a(3283)=2502, while A305895(3283)=1845.

Programs

  • PARI
    up_to = 8192;
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; };
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
v003415 = vector(up_to,n,A003415(n));
A327930(n) = { my(m=1); fordiv(n,d,if((d>1), m *= prime(v003415[d]))); (m); };
v327931 = rgs_transform(vector(up_to, n, A327930(n)));
A327931(n) = v327931[n];

Formula

a(p) = 2 for all primes p.

A319357 Filter sequence combining A003415(d) from all proper divisors d of n, where A003415(d) = arithmetic derivative of d.

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, 11, 12, 2, 13, 2, 14, 4, 4, 4, 15, 2, 4, 4, 16, 2, 17, 2, 18, 19, 4, 2, 20, 3, 21, 4, 22, 2, 23, 4, 24, 4, 4, 2, 25, 2, 4, 26, 27, 4, 28, 2, 29, 4, 30, 2, 31, 2, 4, 32, 33, 4, 34, 2, 35, 36, 4, 2, 37, 4, 4, 4, 38, 2, 39, 4, 40, 4, 4, 4, 41, 2, 42, 43, 44, 2, 45, 2, 46, 47
Offset: 1

Views

Author

Antti Karttunen, Oct 02 2018

Keywords

Comments

Restricted growth sequence transform of A319356.
The only duplicates in range 1..65537 with a(n) > 4 are the following six pairs: a(1445) = a(2783), a(4205) = a(11849), a(5819) = a(8381), a(6727) = a(15523), a(8405) = a(31211) and a(28577) = a(44573). All these have prime signature p^2 * q^1. If all the other duplicates respect the prime signature as well, then also the last implication given below is valid.
For all i, j:
a(i) = a(j) => A000005(i) = A000005(j),
a(i) = a(j) => A319683(i) = A319683(j),
a(i) = a(j) => A319686(i) = A319686(j),
a(i) = a(j) => A101296(i) = A101296(j). [Conjectural, see notes above]

Examples

			Proper divisors of 1445 are [1, 5, 17, 85, 289], while the proper divisors of 2783 are [1, 11, 23, 121, 253]. 1 contributes 0 and primes contribute 1, so only the last two matter in each set. We have A003415(85) = 22 = A003415(121) and A003415(289) = 34 = A003415(253), thus the value of arithmetic derivative coincides for all proper divisors, thus a(1445) = a(2783).

Links

Crossrefs

Cf. A003415, A319356.
Cf. A000041 (positions of 2's), A001248 (positions of 3's), A006881 (positions of 4's),
Cf. also A300245, A300249, A319353, A319693.

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; };
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A319356(n) = { my(m=1); fordiv(n, d, if(dA003415(d)))); (m); };
v319357 = rgs_transform(vector(up_to,n,A319356(n)));
A319357(n) = v319357[n];

A327930 Product_{d|n, d>1} prime(A003415(d)), where A003415(x) gives the arithmetic derivative of x.

Original entry on oeis.org

1, 2, 2, 14, 2, 44, 2, 518, 26, 68, 2, 16324, 2, 92, 76, 67858, 2, 41756, 2, 42364, 116, 164, 2, 116569684, 58, 188, 2678, 84364, 2, 3609848, 2, 27753922, 172, 268, 148, 4353104756, 2, 292, 212, 528236716, 2, 10506584, 2, 256004, 164996, 388, 2, 9360895334252, 86, 388484, 284, 346108, 2, 1802063692, 212, 1495183172, 316
Offset: 1

Views

Author

Antti Karttunen, Sep 30 2019

Keywords

Links

Crossrefs

Cf. A000040, A001221, A001222, A003415, A007814, A056239, A064989, A319356, A319684, A319685, A327931 (rgs-transform).

Programs

  • PARI
    A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A327930(n) = { my(m=1); fordiv(n,d,if((d>1), m *= prime(A003415(d)))); (m); };

Formula

a(n) = Product_{d|n, d>1} A000040(A003415(d)).
For all n >= 2, a(n) = prime(A003415(n)) * A064989(A319356(n)).
A001221(a(n)) = A319685(n).
A001222(a(n)) = A032741(n).
A007814(a(n)) = A001221(n).
A056239(a(n)) = A319684(n).
Showing 1-3 of 3 results.

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