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 1-4 of 4 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

A351083 a(n) = gcd(n, A003415(A276086(n))), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 7, 8, 1, 1, 1, 2, 1, 1, 5, 16, 1, 3, 1, 10, 1, 1, 1, 4, 25, 1, 1, 2, 1, 1, 1, 2, 1, 17, 5, 12, 1, 1, 13, 2, 1, 1, 1, 4, 5, 1, 1, 2, 1, 25, 1, 4, 1, 3, 5, 2, 1, 1, 1, 2, 1, 1, 7, 4, 1, 1, 1, 2, 1, 7, 1, 24, 1, 1, 5, 2, 7, 1, 1, 80, 1, 1, 1, 14, 5, 1, 1, 8, 1, 3, 91, 4, 1, 1, 1, 2, 1, 49, 1, 4
Offset: 0

Views

Author

Antti Karttunen, Feb 03 2022

Keywords

Links

Crossrefs

Cf. A003415, A276086, A324198, A327860, A328572, A351080, A351084, A351087 (fixed points), A354823 (Dirichlet inverse), A373145, A373599 (indices of multiples of 3 in this sequence).
Cf. A166486 (a(n) mod 2, parity of terms, see comment in A327860).
Cf. also A345000.

Programs

  • Mathematica
    Array[Block[{i, m, n = #, p}, m = i = 1; While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; GCD[#, If[m < 2, 0, m Total[#2/#1 & @@@ FactorInteger[m]]]]] &, 101, 0] (* Michael De Vlieger, Feb 04 2022 *)
  • 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); };
A351083(n) = gcd(n, A327860(n));

Formula

a(n) = gcd(n, A327860(n)) = gcd(n, A003415(A276086(n))).
a(n) = A373145(A276086(n)). - Antti Karttunen, Jun 18 2024

A351084 a(n) = gcd(n, A328572(n)), where A328572 converts the primorial base expansion of n into its prime product form, but with 1 subtracted from all nonzero digits.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 25, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 5, 1, 7, 1, 1, 5, 1, 1, 1, 7, 5, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 49, 1, 1, 1, 1, 1, 1, 35
Offset: 0

Views

Author

Antti Karttunen, Feb 03 2022

Keywords

Links

Crossrefs

Cf. A003415, A276086, A324198, A327860, A328572, A351080, A351083.

Programs

  • PARI
    A328572(n) = { my(m=1, p=2); while(n, if(n%p, m *= p^((n%p)-1)); n = n\p; p = nextprime(1+p)); (m); };
A351084(n) = gcd(n, A328572(n));
  • PARI
    A351084(n) = { my(m=1, p=2, orgn=n); while(n, if(n%p, m *= (p^min((n%p)-1, valuation(orgn, p)))); n = n\p; p = nextprime(1+p)); (m); };

Formula

a(n) = gcd(n, A328572(n)) = gcd(A324198(n), A351083(n)).
a(n) = gcd(n, A085731(A276086(n))) = gcd(n, A276086(n), A327860(n)).

A351085 Lexicographically earliest infinite sequence such that a(i) = a(j) => A327858(i) = A327858(j) and A345000(i) = A345000(j) for all i, j >= 0.

Original entry on oeis.org

1, 2, 3, 3, 4, 3, 5, 3, 6, 7, 3, 3, 4, 3, 8, 2, 9, 3, 10, 3, 11, 12, 3, 3, 13, 12, 14, 8, 4, 3, 3, 3, 15, 16, 3, 17, 18, 3, 19, 2, 4, 3, 3, 3, 6, 8, 20, 3, 21, 16, 14, 12, 22, 3, 10, 2, 4, 2, 3, 3, 4, 3, 8, 8, 23, 24, 3, 3, 11, 2, 3, 3, 25, 3, 8, 26, 27, 24, 3, 3, 9, 7, 3, 3, 4, 2, 14, 2, 28, 3, 10, 2, 29, 2, 30
Offset: 0

Views

Author

Antti Karttunen, Feb 03 2022

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A327858(n), A345000(n)].
For all i, j >= 1: A305800(i) = A305800(j) => a(i) = a(j).
For all i, j >= 0: a(i) = a(j) => A351086(i) = A351086(j).

Links

Crossrefs

Cf. A003415, A276086, A327858, A345000, A351086.
Cf. also A305800, A351080.

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
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327858(n) = gcd(A003415(n),A276086(n));
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
Aux351085(n) = [A327858(n), A345000(n)];
v351085 = rgs_transform(vector(1+up_to,n,Aux351085(n-1)));
A351085(n) = v351085[1+n];
Showing 1-4 of 4 results.

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