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 1-7 of 7 results.

A316556 Number of distinct LCMs of nonempty submultisets of the integer partition with Heinz number n.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Jul 06 2018

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Number of distinct values obtained when A290103 is applied to all divisors of n larger than one. - Antti Karttunen, Sep 25 2018

Examples

			462 is the Heinz number of (5,4,2,1) which has possible LCMs of nonempty submultisets {1,2,4,5,10,20} so a(462) = 6.

Links

Crossrefs

Cf. A056239, A074761, A108917, A122768, A275972, A290103, A296150, A301957, A316313, A316314, A316430, A316555, A316557.
Cf. also A304793, A305611, A319685, A319695 for other similarly constructed sequences.

Programs

  • Mathematica
    Table[Length[Union[LCM@@@Rest[Subsets[If[n==1,{},Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]]]]],{n,100}]
  • PARI
    A290103(n) = lcm(apply(p->primepi(p),factor(n)[,1]));
A316556(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s=A290103(d)), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 25 2018

Extensions

More terms from Antti Karttunen, Sep 25 2018

A319695 Number of distinct values obtained when Euler phi (A000010) is applied to proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 3, 3, 1, 3, 1, 3, 3, 2, 1, 3, 2, 2, 3, 3, 1, 4, 1, 4, 3, 2, 3, 4, 1, 2, 3, 4, 1, 4, 1, 3, 5, 2, 1, 4, 2, 3, 3, 3, 1, 4, 3, 5, 3, 2, 1, 4, 1, 2, 4, 5, 3, 4, 1, 3, 3, 4, 1, 6, 1, 2, 5, 3, 3, 4, 1, 5, 4, 2, 1, 5, 3, 2, 3, 5, 1, 6, 3, 3, 3, 2, 3, 5, 1, 3, 5, 5, 1, 4, 1, 5, 7
Offset: 1

Views

Author

Antti Karttunen, Oct 02 2018

Keywords

Examples

			For n = 6, it has three proper divisors: 1, 2, 3, and applying A000010 to these gives 1, 1 and 2, with just two distinct values, thus a(6) = 2.

Links

Crossrefs

Cf. A000010, A319696.
Cf. also A304793, A305611, A316555, A316556, A319685 for similarly constructed sequences.

Programs

  • PARI
    A319695(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d

A316555 Number of distinct GCDs of nonempty submultisets of the integer partition with Heinz number n.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Jul 06 2018

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Number of distinct values obtained when A289508 is applied to all divisors of n larger than one. - Antti Karttunen, Sep 28 2018

Examples

			455 is the Heinz number of (6,4,3) which has possible GCDs of nonempty submultisets {1,2,3,4,6} so a(455) = 5.

Links

Crossrefs

Cf. A056239, A108917, A122768, A275972, A289508, A289509, A296150, A316313, A316314, A316430, A316556, A316557.
Cf. also A304793, A305611, A319685, A319695 for other similarly constructed sequences.

Programs

  • Mathematica
    Table[Length[Union[GCD@@@Rest[Subsets[If[n==1,{},Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]]]]],{n,100}]
  • PARI
    A289508(n) = gcd(apply(p->primepi(p),factor(n)[,1]));
A316555(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s=A289508(d)), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 28 2018

Extensions

More terms from Antti Karttunen, Sep 28 2018

A319686 Number of distinct values obtained when arithmetic derivative (A003415) is applied to the divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 3, 5, 2, 5, 2, 5, 3, 3, 2, 7, 3, 3, 4, 5, 2, 6, 2, 6, 3, 3, 3, 8, 2, 3, 3, 7, 2, 6, 2, 5, 5, 3, 2, 9, 3, 5, 3, 5, 2, 7, 3, 7, 3, 3, 2, 10, 2, 3, 5, 7, 3, 6, 2, 5, 3, 6, 2, 11, 2, 3, 5, 5, 3, 6, 2, 9, 5, 3, 2, 10, 3, 3, 3, 7, 2, 10, 3, 5, 3, 3, 3, 11, 2, 5, 5, 8, 2, 6, 2, 7, 6, 3, 2, 11, 2, 6, 3, 8, 2, 6, 3, 5, 5, 3, 3, 14
Offset: 1

Views

Author

Antti Karttunen, Oct 02 2018

Keywords

Links

Crossrefs

One more than A319685.
Cf. A003415.
Cf. also A184395, A319696.

Programs

  • Mathematica
    d[0] = d[1] = 0; d[n_] := d[n] = n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := CountDistinct[d /@ Divisors[n]]; Array[a, 100] (* Amiram Eldar, Apr 17 2024 *)
  • 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
A319686(n) = { my(m=Map(),s,k=0); fordiv(n,d,if(!mapisdefined(m,s=A003415(d)), mapput(m,s,s); k++)); (k); };
  • PARI
    a(n) = my(d = divisors(n)); for(i = 1, #d, d[i] = A003415(d[i])); #Set(d) \\ uses A003415 listed at Antti's programs. David A. Corneth, Oct 02 2018

Formula

a(n) = 1+A319685(n).

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.

A319356 a(n) = Product_{d|n, dA003415(d)), where A003415(d) gives arithmetic derivative of d.

Original entry on oeis.org

1, 2, 2, 6, 2, 18, 2, 66, 6, 18, 2, 2574, 2, 18, 18, 2706, 2, 3978, 2, 3762, 18, 18, 2, 6226506, 6, 18, 102, 5742, 2, 306774, 2, 370722, 18, 18, 18, 203956038, 2, 18, 18, 14961474, 2, 631098, 2, 8514, 7038, 18, 2, 168047170434, 6, 10602, 18, 10494, 2, 33626034, 18, 32252814, 18, 18, 2, 2529917014482, 2, 18, 9486, 155332518, 18, 1418742, 2, 14058, 18, 1219914, 2
Offset: 1

Views

Author

Antti Karttunen, Oct 02 2018

Keywords

Links

Crossrefs

Cf. A003415, A319357 (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
A319356(n) = { my(m=1); fordiv(n, d, if(dA003415(d)))); (m); };

Formula

a(n) = Product_{d|n, dA000040(1+A003415(d)).
A001221(a(n)) = A319685(n).
A056239(A064989(a(n))) = A319683(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 1-7 of 7 results.

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