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.

A327860 Arithmetic derivative of the primorial base exp-function: a(n) = A003415(A276086(n)).

Original entry on oeis.org

0, 1, 1, 5, 6, 21, 1, 7, 8, 31, 39, 123, 10, 45, 55, 185, 240, 705, 75, 275, 350, 1075, 1425, 3975, 500, 1625, 2125, 6125, 8250, 22125, 1, 9, 10, 41, 51, 165, 12, 59, 71, 247, 318, 951, 95, 365, 460, 1445, 1905, 5385, 650, 2175, 2825, 8275, 11100, 30075, 4125, 12625, 16750, 46625, 63375, 166125, 14, 77, 91, 329, 420
Offset: 0

Views

Author

Antti Karttunen, Sep 30 2019

Keywords

Comments

Are there any other fixed points after 0, 1, 7, 8 and 2556? (A328110, see also A351087 and A351088).
Out of the 30030 initial terms, 19220 are multiples of 5. (See A327865).
Proof that a(n) is even if and only if n is a multiple of 4: Consider Charlie Neder's Feb 25 2019 comment in A235992. As A276086 is never a multiple of 4, and as it toggles the parity, we only need to know when A001222(A276086(n)) = A276150(n) is even. The condition for that is given in the latter sequence by David A. Corneth's Feb 27 2019 comment. From this it also follows that A166486 gives similarly the parity of terms of A342002, A351083 and A345000. See also comment in A327858. - Antti Karttunen, May 01 2022

Examples

			2556 has primorial base expansion [1,1,1,1,0,0] as 1*A002110(5) + 1*A002110(4) + 1*A002110(3) + 1*A002110(2) = 2310 + 210 + 30 + 6 = 2556. That in turn is converted by A276086 to 13^1 * 11^1 * 7^1 * 5^1 = 5005, whose arithmetic derivative is 5' * 1001 + 1001' * 5 = 1*1001 + 311*5 = 2556, thus 2556 is one of the rare fixed points (A328110) of this sequence.

Links

Crossrefs

Cf. A002110 (positions of 1's), A003415, A048103, A276086, A327858, A327859, A327865, A328110 (fixed points), A328233 (positions of primes), A328242 (positions of squarefree terms), A328388, A328392, A328571, A328572, A329031, A329032, A329041, A342002.
Cf. A345000, A351074, A351075, A351076, A351077, A351080, A351083, A351084, A351087 (numbers k such that a(k) is a multiple of k), A351088.
Coincides with A329029 on positions given by A276156.
Cf. A166486 (a(n) mod 2), A353630 (a(n) mod 4).
Cf. A267263, A276150, A324650, A324653, A324655 for omega, bigomega, phi, sigma and tau applied to A276086(n).
Cf. also A351950 (analogous sequence).

Programs

  • Mathematica
    Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12]}, Array[Function[k, If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]] ] &@ Abs[Times @@ Power @@@ # &@ Transpose@{Prime@ Range@ Length@ k, Reverse@ k}]]@ IntegerDigits[#, b] &, 65, 0]] (* Michael De Vlieger, Mar 12 2021 *)
  • 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
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A327860(n) = A003415(A276086(n));
  • PARI
    A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); }; \\ (Standalone version) - Antti Karttunen, Nov 07 2019

Formula

a(n) = A003415(A276086(n)).
a(A002110(n)) = 1 for all n >= 0.
From Antti Karttunen, Nov 03 2019: (Start)
Whenever A329041(x,y) = 1, a(x + y) = A003415(A276086(x)*A276086(y)) = a(x)*A276086(y) + a(y)*A276086(x). For example, we have:
a(n) = a(A328841(n)+A328842(n)) = A329031(n)*A328572(n) + A329032(n)*A328571(n).
A051903(a(n)) = A328391(n).
A328114(a(n)) = A328392(n).
(End)
From Antti Karttunen, May 01 2022: (Start)
a(n) = A328572(n) * A342002(n).
For all n >= 0, A000035(a(n)) = A166486(n). [See comments]
(End)

Extensions

Verbal description added to the definition by Antti Karttunen, May 01 2022

A328391 Maximal exponent in the prime factorization of A327860(n): a(n) = A051903(A327860(n)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 15 2019

Keywords

Links

Crossrefs

Cf. A002110, A003415, A051903, A276086, A327860, A327969, A328114, A328388, A328389, A328390, A328392.

Programs

  • PARI
    A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
A328391(n) = A051903(A327860(n));

Formula

a(n) = A051903(A327860(n)) = A051903(A003415(A276086(n))).
a(A002110(n)) = 0 for all n >= 0.
For all n >= 1, a(n) >= A328114(n)-1. [Because arithmetic derivative will decrease the maximal prime exponent (A051903) of its argument by at most one]

A328395 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A276087(i)) = A046523(A276087(j)) for all i, j.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 5, 5, 6, 1, 7, 4, 8, 9, 10, 11, 7, 5, 12, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 2, 5, 9, 5, 6, 8, 6, 23, 24, 1, 25, 4, 26, 27, 28, 11, 29, 8, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 9, 9, 42, 12, 24, 12, 7, 43, 44, 45, 46, 47, 48, 27, 49, 33, 50, 35, 51, 52, 53, 54, 55, 56, 57, 58, 59, 20, 60, 61, 30, 27, 62, 63, 64, 65, 66, 15, 67, 68, 69, 8
Offset: 0

Views

Author

Antti Karttunen, Oct 15 2019

Keywords

Comments

Restricted growth sequence transform of function f(n) = A278226(A276086(n)) = A046523(A276086(A276086(n))).
For all i, j:
a(i) = a(j) => A328397(i) = A328397(j) => A328389(i) = A328389(j).

Links

Crossrefs

Cf. A046523, A276086, A276087, A328389.
Cf. A143293 (positions of 1's after the initial one).
Cf. also A278226, A286626, A328388, A328396, A328397.

Programs

  • PARI
    up_to = 32589;
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
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A276087(n) = A276086(A276086(n));
v328395 = rgs_transform(vector(1+up_to, n, A046523(A276087(n-1))));
A328395(n) = v328395[1+n];

A328396 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = A046523(A276086(A003415(A276086(n)))).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 5, 5, 6, 7, 3, 8, 7, 9, 5, 10, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 2, 22, 3, 8, 23, 24, 25, 26, 11, 6, 27, 28, 29, 24, 11, 30, 31, 32, 27, 33, 10, 34, 35, 36, 37, 38, 39, 19, 40, 41, 3, 42, 43, 44, 25, 45, 46, 11, 9, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 4, 69, 70, 71, 72, 73, 74, 32
Offset: 0

Views

Author

Antti Karttunen, Oct 15 2019

Keywords

Comments

Restricted growth sequence transform of function f(n) = A278226(A327860(n)) = A046523(A276086(A003415(A276086(n)))).
For all i, j: a(i) = a(j) => A328392(i) = A328392(j).

Links

Crossrefs

Cf. A003415, A046523, A276086, A327860, A328392.
Cf. also A278226, A286626, A328388, A328395.

Programs

  • PARI
    up_to = 30030;
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) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
Aux328396(n) = A046523(A276086(A003415(A276086(n))));
v328396 = rgs_transform(vector(1+up_to, n, Aux328396(n-1)));
A328396(n) = v328396[1+n];

A328397 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328400(A276087(i)) = A328400(A276087(j)) for all i, j.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 3, 3, 1, 3, 2, 4, 5, 3, 6, 3, 3, 7, 3, 8, 9, 8, 10, 11, 12, 13, 14, 15, 16, 1, 3, 5, 3, 3, 4, 3, 3, 17, 1, 17, 2, 18, 19, 17, 6, 20, 4, 7, 21, 17, 22, 23, 24, 25, 26, 27, 28, 23, 21, 5, 5, 4, 7, 17, 7, 3, 3, 17, 29, 30, 31, 18, 19, 32, 22, 33, 24, 34, 35, 36, 37, 38, 39, 15, 40, 41, 14, 42, 43, 7, 19, 44, 45, 19, 46, 21, 8, 17, 47, 48, 4, 49
Offset: 0

Views

Author

Antti Karttunen, Oct 15 2019

Keywords

Comments

Restricted growth sequence transform of function f(n) = A328400(A276087(n)).
For all i, j:
A328395(i) = A328395(j) => a(i) = a(j) => A328389(i) = A328389(j).

Links

Crossrefs

Cf. A276086, A276087, A328389, A328400.
Cf. also A278226, A286626, A328388, A328395, A328396.

Programs

  • PARI
    up_to = 32589;
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; };
A007947(n) = factorback(factorint(n)[, 1]);
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A181821(n) = { my(f=factor(n),p=0,m=1); forstep(i=#f~,1,-1,while(f[i,2], f[i,2]--; m *= (p=nextprime(p+1))^primepi(f[i,1]))); (m); };
A328400(n) = A181821(A007947(A181819(n)));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A276087(n) = A276086(A276086(n));
v328397 = rgs_transform(vector(1+up_to, n, A328400(A276087(n-1))));
A328397(n) = v328397[1+n];

A342022 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342002(i) = A342002(j) for all i, j >= 0.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 2, 5, 6, 7, 8, 9, 4, 10, 11, 12, 13, 14, 15, 11, 16, 17, 18, 19, 20, 8, 21, 22, 23, 24, 2, 10, 25, 9, 21, 26, 27, 24, 28, 29, 30, 31, 18, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 4, 11, 8, 14, 49, 50, 21, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 7, 62, 63, 64, 65, 66
Offset: 0

Views

Author

Antti Karttunen, Mar 12 2021

Keywords

Comments

Restricted growth sequence transform of A342002.
For all i, j >= 1:
a(i) = a(j) => A342017(i) = A342017(j) => A342019(i) = A342019(j).

Links

Crossrefs

Cf. A003415, A276086, A327860, A328572, A342002, A342017, A342019.
Cf. also A286626, A328388.

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; };
A342002(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= p^(e>0); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
v342022 = rgs_transform(vector(1+up_to,n,A342002(n-1)));
A342022(n) = v342022[1+n];
Showing 1-6 of 6 results.

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