A373145 -id:A373145 - cp's OEIS frontend

A373151 Lexicographically earliest infinite sequence such that a(i) = a(j) => A083345(i) = A083345(j) and A373145(i) = A373145(j), for all i, j >= 1.

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

Offset: 1

Antti Karttunen, May 27 2024

nonn

Restricted growth sequence transform of the ordered pair [A083345(n), A373145(n)].
For all i, j >= 1:
  A373150(i) = A373150(j) => a(i) = a(j),
  a(i) = a(j) => A369001(i) = A369001(j),
  a(i) = a(j) => A369004(i) = A369004(j),
  a(i) = a(j) => A373143(i) = A373143(j).

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

Cf. A003415, A002110, A083345, A276085, A373145.

Cf. A369001, A369004, A373143, A373150.

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]));
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A373145(n) = gcd(A003415(n), A276085(n));
Aux373151(n) = [A083345(n), A373145(n)];
v373151 = rgs_transform(vector(up_to, n, Aux373151(n)));
A373151(n) = v373151[n];

A373152 Lexicographically earliest infinite sequence such that a(i) = a(j) => A085731(i) = A085731(j) and A373145(i) = A373145(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 03 2024

nonn

Restricted growth sequence transform of the ordered pair [A085731(n), A373145(n)], i.e., the ordered pair [gcd(n, A003415(n)), gcd(A003415(n), A276085(n))].

For all i, j >= 1: A373150(i) = A373150(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A085731, A276085, A373145.

Cf. also A373150, A373151.

up_to = 100000;
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]));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };
Aux373152(n) = [gcd(n, A003415(n)), gcd(A003415(n), A276085(n))];
v373152 = rgs_transform(vector(up_to, n, Aux373152(n)));
A373152(n) = v373152[n];

A373268 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A085731(i) = A085731(j) and A373145(i) = A373145(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, 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, 21, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 46, 55, 2, 56, 57, 58, 2, 59, 41, 60, 61, 62, 2, 63, 64, 65, 66, 67, 68, 69, 2, 70, 71, 72, 2, 73, 2, 74, 55

Offset: 1

Antti Karttunen, Jun 09 2024

nonn

Restricted growth sequence transform of the triple [A003415(n), A085731(n), A373145(n)].
For all i, j >= 1:
  A373150(i) = A373150(j) => a(i) = a(j),
  a(i) = a(j) => A373151(i) = A373151(j) => A373485(i) = A373485(j),
  a(i) = a(j) => A373152(i) = A373152(j),
  a(i) = a(j) => A373486(i) = A373486(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A003415, A083345, A085731, A276085, A373145.
Cf. A373150, A373151, A373152, A373485, A373486.
Differs from A344025 and A369046 for the first time at n=91, where a(91) = 64, while A344025(91) = A369046(91) = 37.
Differs from A351236 for the first time at n=143, where a(143) = 100, while A351236(143) = 68.

up_to = 100000;
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]));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };
Aux373268(n) = { my(d=A003415(n)); [d, gcd(d,n), gcd(d, A276085(n))]; };
v373268 = rgs_transform(vector(up_to, n, Aux373268(n)));
A373268(n) = v373268[n];

A373380 Lexicographically earliest infinite sequence such that a(i) = a(j) => A373145(i) = A373145(j), A373362(i) = A373362(j), and A373364(i) = A373364(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, 2, 2, 12, 13, 14, 2, 2, 13, 15, 16, 17, 2, 2, 2, 18, 19, 20, 21, 22, 2, 23, 24, 2, 2, 25, 2, 26, 2, 27, 2, 28, 19, 29, 30, 31, 2, 2, 24, 2, 32, 33, 2, 3, 2, 34, 35, 36, 37, 2, 2, 12, 38, 2, 2, 39, 2, 40, 2, 41, 37, 42, 2, 28, 43, 44, 2, 45, 32, 46, 47, 2, 2, 2, 48, 26, 49, 50, 51, 2, 2, 2, 2, 52

Offset: 1

Antti Karttunen, Jun 03 2024

nonn

Restricted growth sequence transform of the triple [A373145(n), A373362(n), A373364(n)], i.e., the triple [gcd(x, y), gcd(x, z), gcd(y, z)], where x=A001414(n), y=A003415(n), z=A276085(n).
For all i, j >= 1:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A373367(i) = A373367(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A001414, A003415, A276085, A305800, A373145, A373362, A373364, A373367.

Cf. also A373150, A373379.

up_to = 100000;
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; };
A001414(n) = ((n=factor(n))[, 1]~*n[, 2]);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };
Aux373380(n) = { my(x=A001414(n), y=A003415(n), z=A276085(n)); [gcd(x, y), gcd(x, z), gcd(y, z)]; };
v373380 = rgs_transform(vector(up_to, n, Aux373380(n)));
A373380(n) = v373380[n];

A276085 Primorial base log-function: fully additive with a(p) = p#/p, where p# = A034386(p).

Original entry on oeis.org

0, 1, 2, 2, 6, 3, 30, 3, 4, 7, 210, 4, 2310, 31, 8, 4, 30030, 5, 510510, 8, 32, 211, 9699690, 5, 12, 2311, 6, 32, 223092870, 9, 6469693230, 5, 212, 30031, 36, 6, 200560490130, 510511, 2312, 9, 7420738134810, 33, 304250263527210, 212, 10, 9699691, 13082761331670030, 6, 60, 13, 30032, 2312, 614889782588491410, 7, 216, 33, 510512, 223092871, 32589158477190044730, 10

Offset: 1

Antti Karttunen, Aug 21 2016

nonn

Completely additive with a(p^e) = e * A002110(A000720(p)-1).
This is a left inverse of A276086 ("primorial base exp-function"), hence the name "primorial base log-function". When the domain is restricted to the terms of A048103, this works also as a right inverse, as A276086(a(A048103(n))) = A048103(n) for all n >= 1. - Antti Karttunen, Apr 24 2022
On average, every third term is a multiple of 4. See A369001. - Antti Karttunen, May 26 2024

A left inverse of A276086.
Cf. A000040, A000720, A002110, A028234, A034386, A048103, A049345, A055396, A067029, A108951, A143293, A276154, A328316, A328624, A328625, A328768, A328832, A346105, A351576, A376398 (partial sums).
Positions of multiples of k in this sequence, for k=2, 3, 4, 5, 8, 27, 3125: A003159, A339746, A369002, A373140, A373138, A377872, A377878.
Cf. A036554 (positions of odd terms), A035263, A096268 (parity of terms).
Cf. A372575 (rgs-transform), A372576 [a(n) mod 360], A373842 [= A003415(a(n))].
Cf. A373145 [= gcd(A003415(n), a(n))], A373361 [= gcd(n, a(n))], A373362 [= gcd(A001414(n), a(n))], A373485 [= gcd(A083345(n), a(n))], A373835 [= gcd(bigomega(n), a(n))], and also A373367 and A373147 [= A003415(n) mod a(n)], A373148 [= a(n) mod A003415(n)].
Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A001414 (with a(p)=p), A059975 (with a(p)=p-1), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#).
Cf. also A276075 for factorial base and A048675, A054841 for base-2 and base-10 analogs.

nn = 60; b = MixedRadix[Reverse@ Prime@ Range@ PrimePi[nn + 1]]; Table[FromDigits[#, b] &@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, nn}] (* Version 10.2, or *)
f[w_List] := Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ w - 1]], Reverse@ w}]; Table[f@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, 60}] (* Michael De Vlieger, Aug 30 2016 *)

A276085(n) = { my(f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= prime(i); i++); s += f[k, 2]*pr); (s); }; \\ Antti Karttunen, Nov 11 2024

from sympy import primorial, primepi, factorint
def a002110(n):
    return 1 if n<1 else primorial(n)
def a(n):
    f=factorint(n)
    return sum(f[i]*a002110(primepi(i) - 1) for i in f)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 22 2017

a(1) = 0; for n > 1, a(n) = a(A028234(n)) + (A067029(n) * A002110(A055396(n)-1)).
a(1) = 0, a(n) = (e1*A002110(i1-1) + ... + ez*A002110(iz-1)) when n = prime(i1)^e1 * ... * prime(iz)^ez.
Other identities.
For all n >= 0:
a(A276086(n)) = n.
a(A000040(1+n)) = A002110(n).
a(A002110(1+n)) = A143293(n).
From Antti Karttunen, Apr 24 & Apr 29 2022: (Start)
a(A283477(n)) = A283985(n).
a(A108951(n)) = A346105(n). [The latter has a similar additive formula as this sequence, but instead of primorials, uses their partial sums]
When applied to sequences where a certain subset of the divisors of n has been multiplicatively encoded with the help of A276086, this yields a corresponding number-theoretical sequence, i.e. completes their computation:
a(A319708(n)) = A001065(n) and a(A353564(n)) = A051953(n).
a(A329350(n)) = A069359(n) and a(A329380(n)) = A323599(n).
In the following group, the sum of the rhs-sequences is n [on each row, as say, A328841(n)+A328842(n)=n], because the pointwise product of the corresponding lhs-sequences is A276086:
a(A053669(n)) = A053589(n) and a(A324895(n)) = A276151(n).
a(A328571(n)) = A328841(n) and a(A328572(n)) = A328842(n).
a(A351231(n)) = A351233(n) and a(A327858(n)) = A351234(n).
a(A351251(n)) = A351253(n) and a(A324198(n)) = A351254(n).
The sum or difference of the rhs-sequences is A108951:
a(A344592(n)) = A346092(n) and a(A346091(n)) = A346093(n).
a(A346106(n)) = A346108(n) and a(A346107(n)) = A346109(n).
Here the two sequences are inverse permutations of each other:
a(A328624(n)) = A328625(n) and a(A328627(n)) = A328626(n).
a(A346102(n)) = A328622(n) and a(A346233(n)) = A328623(n).
a(A346101(n)) = A289234(n). [Self-inverse]
Other correspondences:
a(A324350(x,y)) = A324351(x,y).
a(A003961(A276086(n))) = A276154(n). [The primorial base left shift]
a(A276076(n)) = A351576(n). [Sequence reinterpreting factorial base representation as a primorial base representation]
(End)

Name amended by Antti Karttunen, Apr 24 2022

Name simplified, the old name moved to the comments - Antti Karttunen, Jun 23 2024

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

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

Antti Karttunen, Feb 03 2022

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

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.

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 *)

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));

a(n) = gcd(n, A327860(n)) = gcd(n, A003415(A276086(n))).

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

A373144 Numbers k such that both A003415(k) and A276085(k) are multiples of 3, where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

Original entry on oeis.org

1, 8, 27, 35, 36, 64, 65, 77, 95, 119, 125, 135, 143, 155, 161, 162, 180, 185, 189, 203, 209, 215, 216, 221, 252, 275, 280, 287, 288, 297, 299, 305, 323, 329, 335, 341, 343, 351, 365, 371, 377, 395, 396, 407, 413, 425, 437, 459, 468, 473, 485, 497, 512, 513, 515, 520, 527, 533, 545, 551, 575, 581, 605, 611, 612, 616

Offset: 1

Antti Karttunen, May 26 2024

nonn

This is a multiplicative semigroup; if m and n are in the sequence then so is m*n.

			65 is present as A003415(65) = 18 = 3*6 and A276085(65) = 2316 = 3*772.
77 is present as A003415(77) = 18 = 3*6 and A276085(77) = 240 = 3*80.
5005 (= 65*77) is present as A003415(5005) = A276085(5005) = 2556 = 3*852. (See A369650).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003415, A276085, A373143 (characteristic function).
Intersection of A327863 and A339746.
Positions of multiples of 3 in A373145.
Cf. also A369650.

PARI
```
\\ See A373143.
```

A368998 Numbers k such that A003415(k) and A276085(k) are both even, where A003415 is the arithmetic derivative and A276085 is the primorial base log-function; Numbers k for which A342001(k) is even.

Original entry on oeis.org

1, 4, 9, 12, 15, 16, 20, 21, 25, 28, 33, 35, 36, 39, 44, 48, 49, 51, 52, 55, 57, 60, 64, 65, 68, 69, 76, 77, 80, 81, 84, 85, 87, 91, 92, 93, 95, 100, 108, 111, 112, 115, 116, 119, 121, 123, 124, 129, 132, 133, 135, 140, 141, 143, 144, 145, 148, 155, 156, 159, 161, 164, 169, 172, 176, 177, 180, 183, 185, 187, 188, 189

Offset: 1

Antti Karttunen, Jan 14 2024

nonn

From Antti Karttunen, Jun 03 2024: (Start)
Also numbers k for which A373145(k) is even, or in other words, numbers k such that A003415(k) and A276085(k) are both even.
Because both A003159 and A235992 are multiplicative semigroups, this is also: if m and n are in the sequence then so is m*n.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000290, A003415, A276085, A368995 (subsequence), A368997 (characteristic function), A368999 (complement).
Positions of even terms in A342001 and in A373145.
Intersection of A003159 and A235992.
Disjoint union of A345452 and 2*A358776.

PARI
```
isA368998 = A368997;
```

Alternative definition added as a new primary definition by Antti Karttunen, Jun 04 2024

A373148 a(n) = A276085(n) mod A003415(n), where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

Original entry on oeis.org

0, 0, 2, 0, 3, 0, 3, 4, 0, 0, 4, 0, 4, 0, 4, 0, 5, 0, 8, 2, 3, 0, 5, 2, 1, 6, 0, 0, 9, 0, 5, 2, 11, 0, 6, 0, 1, 8, 9, 0, 33, 0, 20, 10, 16, 0, 6, 4, 13, 12, 16, 0, 7, 8, 33, 2, 7, 0, 10, 0, 1, 34, 6, 12, 30, 0, 8, 2, 37, 0, 7, 0, 1, 14, 32, 6, 41, 0, 10, 8, 31, 0, 34, 6, 16, 8, 73, 0, 11, 0, 44, 2, 36, 12, 7, 0, 61

Offset: 2

Antti Karttunen, May 27 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 2..65537

Cf. A002110, A003415, A276085.

Cf. also A373145, A373147, and also A328382.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A373148(n) = (A276085(n)%A003415(n));

cp's OEIS Frontend

A373151 Lexicographically earliest infinite sequence such that a(i) = a(j) => A083345(i) = A083345(j) and A373145(i) = A373145(j), for all i, j >= 1.

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A373152 Lexicographically earliest infinite sequence such that a(i) = a(j) => A085731(i) = A085731(j) and A373145(i) = A373145(j), for all i, j >= 1.

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A373268 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A085731(i) = A085731(j) and A373145(i) = A373145(j), for all i, j >= 1.

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A373380 Lexicographically earliest infinite sequence such that a(i) = a(j) => A373145(i) = A373145(j), A373362(i) = A373362(j), and A373364(i) = A373364(j), for all i, j >= 1.

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A276085 Primorial base log-function: fully additive with a(p) = p#/p, where p# = A034386(p).

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

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

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A373144 Numbers k such that both A003415(k) and A276085(k) are multiples of 3, where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

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