A345000 -id:A345000 - cp's OEIS frontend

A354347 Dirichlet inverse of A345000, where A345000(n) = gcd(A003415(n), A003415(A276086(n))), with A003415 the arithmetic derivative, and A276086 the primorial base exp-function.

1, -1, -1, -1, -1, 1, -1, -1, 0, 1, -1, 1, -1, 1, 1, -9, -1, -2, -1, 1, -3, 1, -1, 1, -4, -3, 0, 1, -1, -1, -1, 21, 1, 1, -1, -6, -1, 1, 1, 3, -1, 7, -1, -1, 0, -3, -1, 23, 0, 4, -3, 7, -1, 2, 1, 3, 1, 1, -1, -1, -1, 1, 8, 15, -1, -1, -1, 1, 1, 3, -1, 14, -1, 1, -46, -7, -1, 7, -1, 5, 0, 1, -1, 3, 1, -3, 1, -131

Offset: 1

Antti Karttunen, Jun 07 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..30030
Index entries for sequences related to primorial base

Cf. A003415, A276086, A345000.
Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms), A354815 (positions of 0's), A354816 (of -1's).
Cf. also A346242, A354348, A354823, A354824.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
memoA354347 = Map();
A354347(n) = if(1==n,1,my(v); if(mapisdefined(memoA354347,n,&v), v, v = -sumdiv(n,d,if(dA345000(n/d)*A354347(d),0)); mapput(memoA354347,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA345000(n/d) * a(d).

For all n >= 1, A000035(a(n)) = A353627(n).

A347959 Dirichlet convolution of A342001 with A345000.

Original entry on oeis.org

0, 1, 1, 3, 1, 7, 1, 7, 3, 9, 1, 18, 1, 11, 10, 15, 1, 16, 1, 24, 12, 15, 1, 40, 3, 17, 6, 30, 1, 54, 1, 39, 16, 21, 14, 41, 1, 23, 18, 54, 1, 72, 1, 42, 25, 27, 1, 92, 3, 24, 22, 52, 1, 30, 18, 68, 24, 33, 1, 132, 1, 35, 35, 73, 20, 96, 1, 60, 28, 92, 1, 99, 1, 41, 27, 66, 20, 114, 1, 120, 10, 45, 1, 176, 24, 47

Offset: 1

Antti Karttunen, Sep 21 2021

Antti Karttunen, Table of n, a(n) for n = 1..11550
Index entries for sequences related to primorial base

Cf. A003415, A327860, A342001, A345000.

Cf. also A347234, A347954, A347958.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A345000(n) = gcd(A003415(n), A003415(A276086(n)));
A347959(n) = sumdiv(n,d,A342001(n/d)*A345000(d));

a(n) = Sum_{d|n} A342001(n/d) * A345000(d).

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

Antti Karttunen, Feb 03 2022

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).

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A003415, A276086, A327858, A345000, A351086.

Cf. also A305800, A351080.

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];

A351235 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j), A327858(i) = A327858(j) and A345000(i) = A345000(j) for all i, j >= 1.

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, 7, 2, 15, 16, 17, 18, 8, 2, 19, 2, 20, 21, 7, 22, 23, 2, 24, 10, 25, 2, 19, 2, 26, 27, 28, 2, 29, 30, 31, 14, 32, 2, 33, 10, 25, 10, 7, 2, 34, 2, 9, 27, 35, 36, 19, 2, 13, 10, 19, 2, 37, 2, 9, 38, 39, 36, 19, 2, 40, 41, 7, 2, 34, 10, 17, 10, 42, 2

Offset: 1

Antti Karttunen, Feb 06 2022

Restricted growth sequence transform of the triplet [A046523(n), A327858(n), A345000(n)].
For all i, j >= 1:
  A305800(i) = A305800(j) => a(i) = a(j) => A351085(i) = A351085(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A046523, A276086, A327858, A345000, A351086.

Cf. also A101296, A305800, A329620, A351085, A351236.

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
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); };
A327858(n) = gcd(A003415(n),A276086(n));
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
Aux351235(n) = [A046523(n), A327858(n), A345000(n)];
v351235 = rgs_transform(vector(up_to,n,Aux351235(n)));
A351235(n) = v351235[n];

A347958 Inverse Möbius transform of A345000.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 8, 3, 4, 2, 8, 2, 4, 4, 24, 2, 8, 2, 8, 8, 4, 2, 16, 7, 8, 4, 8, 2, 8, 2, 34, 4, 4, 6, 24, 2, 4, 4, 14, 2, 12, 2, 10, 6, 8, 2, 34, 3, 14, 8, 14, 2, 12, 4, 14, 4, 4, 2, 16, 2, 4, 10, 46, 6, 8, 2, 8, 4, 10, 2, 44, 2, 4, 64, 16, 6, 12, 2, 46, 5, 4, 2, 20, 4, 8, 4, 154, 2, 16, 4, 26, 4, 52, 6, 46

Offset: 1

Antti Karttunen, Sep 21 2021

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Cf. A003415, A276086, A327860, A345000, A347959.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A345000(n) = gcd(A003415(n), A003415(A276086(n)));
A347958(n) = sumdiv(n,d,A345000(d));

a(n) = Sum_{d|n} A345000(d).

A355830 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A345000(i) = A345000(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 4, 2, 7, 2, 4, 4, 8, 2, 9, 2, 7, 10, 4, 2, 11, 12, 10, 13, 7, 2, 14, 2, 15, 4, 4, 16, 17, 2, 4, 4, 18, 2, 14, 2, 19, 20, 10, 2, 21, 6, 22, 10, 19, 2, 23, 4, 18, 4, 4, 2, 24, 2, 4, 20, 25, 16, 14, 2, 7, 4, 14, 2, 26, 2, 4, 27, 28, 16, 14, 2, 29, 30, 4, 2, 24, 4, 10, 4, 31, 2, 32, 4, 33, 4, 34, 16, 35, 2, 36, 20, 37, 2, 38, 2, 18, 14

Offset: 1

Antti Karttunen, Jul 20 2022

nonn

Restricted growth sequence transform of the ordered pair [A046523(n), A345000(n)].

For all i, j: A351235(i) = A351235(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to primorial base

Cf. A003415, A046523, A276086, A345000.

Cf. also A351235, A355000, A355831, A355832, A355836.

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) = 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); };
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
Aux355830(n) = [A046523(n), A345000(n)];
v355830 = rgs_transform(vector(up_to,n,Aux355830(n)));
A355830(n) = v355830[n];

A369047 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A345000(i) = A345000(j), for all i, j >= 1.

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, 14, 17, 18, 19, 2, 20, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 22, 34, 35, 36, 2, 37, 27, 38, 39, 20, 2, 38, 2, 40, 41, 42, 43, 44, 2, 45, 46, 47, 2, 48, 2, 31, 49, 21, 43, 50, 2, 51, 52, 53, 2, 54, 39, 34, 55, 56, 2, 57

Offset: 1

Antti Karttunen, Jan 21 2024

nonn

Restricted growth sequence transform of the ordered pair [A003415(n), A345000(n)], or equally, of the pair [A003415(n), A369038(n)], or equally, of the pair [A345000(n), A369038(n)].
For all i, j >= 1:
  A351236(i) = A351236(j) => a(i) = a(j),
  a(i) = a(j) => A369034(i) = A369034(j),
  a(i) = a(j) => A369036(i) = A369036(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A345000, A369034, A369036, A369038.

Cf. also A351236, A369051.

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) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
Aux369047(n) = [A003415(n), A345000(n)];
v369047 = rgs_transform(vector(up_to, n, Aux369047(n)));
A369047(n) = v369047[n];

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

Antti Karttunen, Sep 30 2019

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

			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.

Antti Karttunen, Table of n, a(n) for n = 0..2310
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..30030
Index entries for sequences related to primorial base

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).

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 *)

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));

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

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)

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

A166486 Periodic sequence [0,1,1,1] of length 4; Characteristic function of numbers that are not multiples of 4.

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0

Offset: 0

Jaume Oliver Lafont, Oct 15 2009

			G.f. = x + x^2 + x^3 + x^5 + x^6 + x^7 + x^9 + x^10 + x^11 + x^13 + x^14 + ...

Antti Karttunen, Table of n, a(n) for n = 0..65537
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Characteristic function of A042968, whose complement A008586 gives the positions of zeros (after its initial term).
Absolute values of A046978, A075553, A131729, A358839, and for n >= 1, also of A112299 and of A257196.
Sequence A152822 shifted by two terms.
Parity of A327860, A331733, A342002, A351083 and A345000.
Row 3 of A225145, Column 2 of A229940 (after the initial term).
First differences of A057353. Sum of A359370 and A359372.
Cf. A000035, A011655, A011558, A097325, A109720, A168181, A168182, A168184, A145568, A168185 (characteristic functions for numbers that are not multiples of k = 2, 3 and 5..12).
Cf. A016628, A164985, A165132.
Cf. A010873, A033436, A069733 (inverse Möbius transform), A121262 (one's complement), A190621 [= n*a(n)], A355689 (Dirichlet inverse).

[Ceiling(n/4)-Floor(n/4) : n in [0..50]]; // Wesley Ivan Hurt, Jun 20 2014

seq(1/2*((n^3+n) mod 4), n=0..50); # Gary Detlefs, Mar 20 2010

PadRight[{},120,{0,1,1,1}] (* Harvey P. Dale, Jul 04 2013 *)
Table[Ceiling[n/4] - Floor[n/4], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 20 2014 *)
a[ n_] := Sign[ Mod[n, 4]]; (* Michael Somos, May 05 2015 *)

PARI
```
{a(n) = !!(n%4)};
```

def A166486(n): return (0,1,1,1)[n&3] # Chai Wah Wu, Jan 03 2023

G.f.: (x + x^2 + x^3) / (1 - x^4) = x * (1 + x + x^2) / ((1 - x) * (1 + x) * (1 + x^2)) = x * (1 - x^3) / ((1 - x) * (1 - x^4)).
a(n) = (3 - i^n - (-i)^n - (-1)^n) / 4, where i=sqrt(-1).
Sum_{k>0} a(k)/(k*3^k) = log(5)/4.
From Reinhard Zumkeller, Nov 30 2009: (Start)
Multiplicative with a(p^e) = (if p=2 then 0^(e-1) else 1), p prime and e>0.
a(n) = 1-A121262(n).
a(A042968(n))=1; a(A008586(n))=0.
A033436(n) = Sum{k=0..n} a(k)*(n-k). (End)
a(n) = 1/2*((n^3+n) mod 4). - Gary Detlefs, Mar 20 2010
a(n) = (Fibonacci(n)*Fibonacci(3n) mod 3)/2. - Gary Detlefs Dec 21 2010
Euler transform of length 4 sequence [ 1, 0, -1, 1]. - Michael Somos, Feb 12 2011
Dirichlet g.f. (1-1/4^s)*zeta(s). - R. J. Mathar, Feb 19 2011
a(n) = Fibonacci(n)^2 mod 3. - Gary Detlefs, May 16 2011
a(n) = -1/4*cos(Pi*n)-1/2*cos(1/2*Pi*n)+3/4. - Leonid Bedratyuk, May 13 2012
For the general case: the characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013
a(n) = ceiling(n/4) - floor(n/4). - Wesley Ivan Hurt, Jun 20 2014
a(n) = a(-n) for all n in Z. - Michael Somos, May 05 2015
For n >= 1, a(n) = A053866(A225546(n)) = A000035(A331733(n)). - Antti Karttunen, Jul 07 2020
a(n) = signum(n mod 4). - Alois P. Heinz, May 12 2021
From Antti Karttunen, Dec 28 2022: (Start)
a(n) = [A010873(n) > 0], where [ ] is the Iverson bracket.
a(n) = abs(A046978(n)) = abs(A075553(n)) = abs(A131729(n)) = abs(A358839(n)).
For all n >= 1, a(n) = abs(A112299(n)) = abs(A257196(n))
a(n) = A152822(2+n).
a(n) = A359370(n) + A359372(n). (End)
E.g.f.: (cosh(x) - cos(x))/2 + sinh(x). - Stefano Spezia, Aug 04 2025

Secondary definition (from Reinhard Zumkeller's Nov 30 2009 comment) added to the name by Antti Karttunen, Dec 20 2022

A327858 Greatest common divisor of the arithmetic derivative and the primorial base exp-function: a(n) = gcd(A003415(n), A276086(n)).

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 5, 1, 3, 6, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 10, 1, 1, 1, 10, 15, 3, 1, 1, 1, 1, 1, 14, 1, 6, 5, 1, 21, 2, 1, 1, 1, 1, 3, 3, 25, 1, 7, 14, 15, 10, 7, 1, 1, 2, 1, 2, 1, 1, 1, 1, 3, 3, 3, 18, 1, 1, 3, 2, 1, 1, 1, 1, 3, 5, 5, 18, 1, 1, 1, 6, 1, 1, 1, 2, 15, 2, 35, 1, 1, 2, 3, 2, 49, 6, 1, 1, 7, 15, 35, 1, 7, 1, 1, 1

Offset: 0

Antti Karttunen, Sep 30 2019

Sequence contains only terms of A048103.

Proof that A046337 gives the positions of even terms: see Charlie Neder's Feb 25 2019 comment in A235992 and recall that A276086 is never a multiple of 4, as it is a permutation of A048103, and furthermore it toggles the parity. See also comment in A327860. - Antti Karttunen, May 01 2022

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Cf. A003415, A048103, A235992, A276086, A327859, A328382, A351234, A354348, A356299, A358669, A359423, A359589 (Dirichlet inverse of a(n)-1), A369971, A373849.
Cf. A046337 (positions of even terms), A356311 (positions of 1's), A356310 (their characteristic function).
Cf. also A085731, A324198, A328572 [= gcd(A276086(n), A327860(n))], A345000, A373145, A373843.

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12], f, g}, f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]]; g[n_] := Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[n, b]; Array[GCD[f@ #, g@ #] &, 105]] (* Michael De Vlieger, Sep 30 2019 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
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));

a(n) = gcd(A003415(n), A276086(n)).
a(p) = 1 for all primes p.
a(n) = A276086(A351234(n)). - Antti Karttunen, May 01 2022
From Antti Karttunen, Dec 05 2022: (Start)
For n >= 2, a(n) = gcd(A003415(n), A328382(n)).
(End)
For n >= 2, a(n) = A358669(n) / A359423(n). For n >= 1, A356299(n) | a(n). - Antti Karttunen, Jan 09 2023
a(n) = gcd(A003415(n), A373849(n)) = gcd(A276086(n), A369971(n)) = A373843(A276086(n)). - Antti Karttunen, Jun 21 & 23 2024

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

cp's OEIS Frontend

A354347 Dirichlet inverse of A345000, where A345000(n) = gcd(A003415(n), A003415(A276086(n))), with A003415 the arithmetic derivative, and A276086 the primorial base exp-function.

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A347959 Dirichlet convolution of A342001 with A345000.

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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.

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A351235 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j), A327858(i) = A327858(j) and A345000(i) = A345000(j) for all i, j >= 1.

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A347958 Inverse Möbius transform of A345000.

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A355830 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A345000(i) = A345000(j) for all i, j >= 1.

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A369047 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A345000(i) = A345000(j), for all i, j >= 1.

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

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A166486 Periodic sequence [0,1,1,1] of length 4; Characteristic function of numbers that are not multiples of 4.

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