A372576 -id:A372576 - cp's OEIS frontend

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

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

A339746 Positive integers of the form 2^i3^jk, gcd(k,6)=1, and i == j (mod 3).

Original entry on oeis.org

1, 5, 6, 7, 8, 11, 13, 17, 19, 23, 25, 27, 29, 30, 31, 35, 36, 37, 40, 41, 42, 43, 47, 48, 49, 53, 55, 56, 59, 61, 64, 65, 66, 67, 71, 73, 77, 78, 79, 83, 85, 88, 89, 91, 95, 97, 101, 102, 103, 104, 107, 109, 113, 114, 115, 119, 121, 125, 127, 131, 133, 135

Offset: 1

Griffin N. Macris, Dec 15 2020

nonn

From Peter Munn, Mar 16 2021: (Start)
The positive integers in the multiplicative subgroup of the positive rationals generated by 8, 6, and A215848 (primes greater than 3).
This subgroup, denoted H, has two cosets: 2H = (1/3)H and 3H = (1/2)H. It follows that the sequence is one part of a 3-part partition of the positive integers with the property that each part's terms are half the even terms of one of the other parts and also one third of the multiples of 3 in the remaining part.
(End)
Positions of multiples of 3 in A276085 (and in A276075). Because A276085 is completely additive, this is closed under multiplication: if m and n are in the sequence then so is m*n. - Antti Karttunen, May 27 2024
The coset sequences mentioned in Peter Munn's comment above are A373261 and A373262. - Antti Karttunen, Jun 04 2024

Robert Israel, Table of n, a(n) for n = 1..10000

Sequences of positive integers in a multiplicative subgroup of positive rationals generated by a set S and A215848: S={}: A007310, S={6}: A064615, S={3,4}: A003159, S={2,9}: A007417, S={4,6}: A036668, S={3,8}: A191257, S={4,9}: A339690, S={6,8}: this sequence.
Positions of 0's in A373153, positions of multiples of 3 in A276085 and in A372576.
Cf. A372573 (characteristic function), A373261, A373262.
Subsequences: A064615, A373144, A373373, A373484, A373837, A374042, A374044, A374120, A377872.
Sequences giving positions of multiples of k in A276085, for k=2, 3, 4, 5, 8, 27, 3125: A003159, this sequence, A369002, A373140, A373138, A377872, A377878.
Cf. also A332820, A373992, A383288.

N:= 1000: # for terms <= N
R:= {}:
for k1 from 0 to floor(N/6) do
  for k0 in [1,5] do
    k:= k0 + 6*k1;
    for j from 0 while 3^j*k <= N do
      for i from (j mod 3) by 3 do
        x:= 2^i * 3^j * k;
        if x > N then break fi;
        R:= R union {x}
od od od od:
sort(convert(R,list)); # Robert Israel, Apr 08 2021

Select[Range[130], Mod[IntegerExponent[#, 2] - IntegerExponent[#, 3], 3] == 0 &]

isA339746 = A372573; \\ Antti Karttunen, Jun 04 2024

from sympy import factorint
def ok(n):
  f = factorint(n, limit=4)
  i, j = 0 if 2 not in f else f[2], 0 if 3 not in f else f[3]
  return (i-j)%3 == 0
def aupto(limit): return [m for m in range(1, limit+1) if ok(m)]
print(aupto(200)) # Michael S. Branicky, Mar 26 2021

from itertools import count
def A339746(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in range(x.bit_length()+1):
            i2 = 1<x:
                    break
                m = x//k
                c -= (m-1)//6+(m-5)//6+2
        return c
    return bisection(f,n,n) # Chai Wah Wu, Feb 12 2025

Formula

a(n) ~ (91/43)*n.

A374124 a(n) = A113177(n) mod 360, where A113177 is fully additive with a(p) = Fibonacci(p).

Original entry on oeis.org

0, 1, 2, 2, 5, 3, 13, 3, 4, 6, 89, 4, 233, 14, 7, 4, 157, 5, 221, 7, 15, 90, 217, 5, 10, 234, 6, 15, 149, 8, 229, 5, 91, 158, 18, 6, 17, 222, 235, 8, 301, 16, 77, 91, 9, 218, 73, 6, 26, 11, 159, 235, 293, 7, 94, 16, 223, 150, 161, 9, 161, 230, 17, 6, 238, 92, 293, 159, 219, 19, 289, 7, 73, 18, 12, 223, 102, 236, 301

Offset: 1

Antti Karttunen, Jun 30 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000045, A113177.
Cf. A373585 (antiparity of terms), A373586 (indices of even terms), A373587 (of odd terms), A374052 (of multiples of 3).
Cf. also A372576, A374123, A374125.

A113177(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])));
A374124(n) = (A113177(n)%360);

A374125 a(n) = A328845(n) mod 360, where A328845 is the first Fibonacci-based variant of the arithmetic derivative.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 7, 13, 12, 12, 15, 89, 20, 233, 33, 25, 32, 157, 33, 221, 40, 53, 189, 217, 52, 50, 119, 54, 80, 149, 65, 229, 80, 289, 331, 100, 84, 17, 101, 5, 100, 301, 127, 77, 40, 105, 97, 73, 128, 182, 125, 145, 264, 293, 135, 140, 188, 341, 327, 161, 160, 161, 129, 201, 192, 150, 251, 293, 336, 337, 235, 289

Offset: 0

Antti Karttunen, Jun 30 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A000045, A328845.
Cf. A374045 (antiparity of terms), A374046 (indices of even terms), A374047 (of odd terms), A374122 (of multiples of 3).
Cf. also A372576, A374123, A374124.

A374125[n_] := If[n <= 1, 0, Mod[n*Total[MapApply[#2*Fibonacci[#]/# &, FactorInteger[n]]], 360]];
Array[A374125, 100, 0] (* Paolo Xausa, Dec 16 2024 *)

A328845(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])/f[i, 1]));
A374125(n) = (A328845(n)%360);

A374123 a(n) = A328768(n) mod 360, where A328768 is the first primorial based variant of the arithmetic derivative.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 7, 30, 12, 12, 17, 210, 20, 150, 67, 28, 32, 150, 33, 30, 44, 104, 71, 210, 52, 60, 313, 54, 148, 150, 71, 30, 80, 292, 317, 192, 84, 210, 79, 116, 108, 210, 229, 330, 164, 114, 83, 150, 128, 60, 145, 124, 292, 210, 135, 36, 324, 128, 329, 330, 172, 30, 91, 354, 192, 108, 257, 30, 308, 316, 59, 210

Offset: 0

Antti Karttunen, Jun 30 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A002110, A328768.
Cf. A042965 (indices of even terms), A016825 (of odd terms), A152822 (antiparity of terms), A373992 (indices of multiples of 3).
Cf. also A372576, A374124, A374125.

A002110(n) = prod(i=1,n,prime(i));
A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));
A374123(n) = (A328768(n)%360);

A372575 Lexicographically earliest infinite sequence such that a(i) = a(j) => A276085(i) = A276085(j), for all i, j >= 1, where A276085 is the primorial base log-function.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 25 2024

nonn

Restricted growth sequence transform of A276085, where A276085 is fully additive with a(p) = A002110(A000720(p)-1).
For all i, j >= 1:
  a(i) = a(j) => A035263(i) = A035263(j),
  a(i) = a(j) => A369001(i) = A369001(j),
  a(i) = a(j) => A372576(i) = A372576(j).
A048103(n) gives the position of the initial occurrence of each n, while A100716 gives the indices of the further occurrences.

Cf. A000720, A002110, A276085, A035263, A369001, A372576.
Cf. A048103 (positions of records and initial occurrences), A100716.
Cf. also A322815.

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; };
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)); };
v372575 = rgs_transform(vector(up_to, n, A276085(n)));
A372575(n) = v372575[n];

For all n >= 1, a(A048103(n)) = n.

cp's OEIS Frontend

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

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A339746 Positive integers of the form 2^i*3^j*k, gcd(k,6)=1, and i == j (mod 3).

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A374124 a(n) = A113177(n) mod 360, where A113177 is fully additive with a(p) = Fibonacci(p).

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A374125 a(n) = A328845(n) mod 360, where A328845 is the first Fibonacci-based variant of the arithmetic derivative.

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A374123 a(n) = A328768(n) mod 360, where A328768 is the first primorial based variant of the arithmetic derivative.

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A372575 Lexicographically earliest infinite sequence such that a(i) = a(j) => A276085(i) = A276085(j), for all i, j >= 1, where A276085 is the primorial base log-function.

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A339746 Positive integers of the form 2^i3^jk, gcd(k,6)=1, and i == j (mod 3).