A377872 -id:A377872 - 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.

A377869 Numbers k such that A276085(k) has no divisors of the form p^p, where A276085 is fully additive with a(p) = p#/p.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 17, 18, 19, 22, 23, 24, 26, 27, 29, 30, 31, 32, 34, 36, 37, 38, 40, 41, 42, 43, 45, 46, 47, 48, 50, 53, 54, 56, 58, 59, 60, 61, 62, 63, 64, 66, 67, 70, 71, 72, 73, 74, 75, 78, 79, 80, 82, 83, 84, 86, 88, 89, 90, 94, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 109

Offset: 1

Antti Karttunen, Nov 10 2024

nonn

Numbers k for which A276085(k) is in A048103, i.e., in the range of A276086.
Numbers k such that A276086(A276085(A276085(k))) is equal to A276085(k).
This is a subsequence of A369003 (numbers k for which A276085(k) is not a multiple of 4), from which it differs for the first time at n=122, where a(122) = 175, as A369003(122) = 174 is not included in this sequence.
From Antti Karttunen, Nov 17 2024: (Start)
More generally, this is equal to setwise difference A000027 \ (A369002 U A377872 U A377878 U ...).
Even semiprimes (A100484) is a subsequence, but the odd semiprimes (A046315) are all in the complement (A377873), because they are included in A369002.
For k=1..6, there are 8, 70, 656, 6531, 64773, 645301 terms <= 10^k. Question: What is the asymptotic density of this sequence, if it has one?
(End)

			A276085(11) = 210 = 2*3*5*7, which has no divisor of the form p^p, therefore 11 is included in this sequence.
A276085(15) = 8 = 2^2 * 2, which has a divisor of the form p^p, therefore 15 is NOT included in this sequence.
A276085(25) = 12 = 2^2 * 3, which has a divisor of the form p^p, therefore 25 is NOT included.
A276085(34) = 30031 = A002110(1-1)+A002110(7-1) (as 34 = 2*17 = prime(1)*prime(7)), and because 30031 = 59*509 (an odd semiprime), 34 is included.
A276085(60) = 10 = 2*5, which has no divisors of the form p^p, therefore 60 is included.
A276085(102) = 30033 = 3^2 * 47 * 71, which has no p^p divisors, therefore 102 is included.
A276085(174) = 223092873 = 3^3 * 3 * 1063 * 2591, which thus has a divisor of the form p^p, and therefore 174 is NOT included in this sequence.

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

Cf. A046315, A048103, A276085, A276086, A377868 (characteristic function), A377873 (complement).
Setwise difference A369003 \ A377875.
Cf. A000040, A100484, A377871, A377989 (subsequences).
Cf. A369002, A377872, A377878.

PARI
```
\\ See A377868.
```

A369007 Numbers k for which k' / gcd(k,k') is a multiple of 27, where k' stands for the arithmetic derivative of k, A003415(k).

Original entry on oeis.org

1, 158, 160, 208, 212, 275, 329, 374, 410, 428, 473, 482, 515, 533, 629, 698, 700, 707, 713, 785, 910, 914, 952, 1055, 1058, 1067, 1072, 1076, 1078, 1080, 1183, 1204, 1238, 1346, 1404, 1431, 1445, 1454, 1490, 1562, 1661, 1691, 1724, 1813, 1841, 1865, 1888, 1937, 1994, 2102, 2219, 2291, 2320, 2372, 2426, 2432, 2570

Offset: 1

Antti Karttunen, Jan 14 2024

nonn

Question: What is the asymptotic density of these terms? 0.0253...?

A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Nov 11 2024

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

Positions of multiples of 27 in A083345.
Subsequence of A369003 and of A369644.
Cf. A003415, A007949.
Cf. also A377872.

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
isA369007(n) = !(A083345(n)%27);

For all n >= 1, A007949(a(n)) = 3*k, for k >= 0. - Antti Karttunen, Nov 12 2024

A377873 Numbers k such that A276085(k) has at least one divisor of the form p^p, where A276085 is fully additive with a(p) = p#/p.

Original entry on oeis.org

1, 9, 12, 15, 16, 20, 21, 25, 28, 33, 35, 39, 44, 49, 51, 52, 55, 57, 65, 68, 69, 76, 77, 81, 85, 87, 91, 92, 93, 95, 108, 111, 115, 116, 119, 121, 123, 124, 129, 133, 135, 141, 143, 144, 145, 148, 155, 159, 161, 164, 169, 172, 174, 177, 180, 183, 185, 187, 188, 189, 192, 201, 203, 205, 209, 212, 213, 215, 217, 219

Offset: 1

Antti Karttunen, Nov 10 2024

nonn

Numbers k for which A276085(k) is in A100716.
1 is included as A276085(1) = 0, which is a multiple of all numbers of the form p^p.
Differs from A369002 for the first at a(53) = 174, the value which is not present at A369002.

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

Cf. A100716, A276085, A377868, A377869 (complement).

Subsequences: A369002, A377872.

PARI
```
isA377873(n) = !A377868(n);
```

A377875 Numbers k for which A276085(k) is not a multiple of 4 and has at least one divisor of the form p^p, with p an odd prime, where A276085 is fully additive with a(p) = p#/p.

Original entry on oeis.org

174, 232, 282, 325, 376, 438, 462, 474, 539, 584, 606, 616, 632, 654, 678, 798, 808, 872, 904, 906, 931, 966, 978, 1002, 1064, 1074, 1075, 1105, 1127, 1182, 1208, 1288, 1302, 1304, 1336, 1398, 1432, 1506, 1519, 1576, 1626, 1662, 1736, 1755, 1842, 1864, 1866, 2008, 2168, 2216, 2226, 2340, 2425, 2442, 2456, 2488, 2514

Offset: 1

Antti Karttunen, Nov 11 2024

nonn

Among the initial 15275 terms, in 110 cases A276085(k) is a multiple of 5^5, and for no cases a multiple of 7^7.

			A276085(55) = 216 = 2^3 * 3^3, which although it has a divisor of the form p^p, with p an odd prime, it is also a multiple of 4, and therefore 55 is NOT included in this sequence.
A276085(174) = 223092873 = 3^3 * 3 * 1063 * 2591, which has a divisor of the form p^p, with p an odd prime, thus 174 is included in this sequence.
A276085(4823) = 614889782588493750 =  2 * 3 * 5^5 * 13 * 2522624749081, thus 4823 is included.
A276085(1104299) = 11231250 = 2 * 3 * 5^5 * 599, thus 1104299 is included.

Antti Karttunen, Table of n, a(n) for n = 1..15275

Setwise difference A369003 \ A377869.
Setwise difference A377873 \ A369002.
Cf. A083345, A276085, A377868, A377872, A377874.

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
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); };
A377868(n) = if(isprime(n), 1, my(x=A276085(n),pp); forprime(p=2,, pp = p^p; if(!(x%pp), return(0)); if(pp > x, return(1))));
isA377875(n) = ((A083345(n)%2) && !A377868(n));

{k such that A377868(k) < A377874(k)}.

A377878 Numbers k for which A276085(k) is a multiple of 3125, where A276085 is fully additive with a(p) = p#/p.

Original entry on oeis.org

1, 4823, 8267, 9553, 15623, 15833, 15929, 20633, 23393, 28417, 33079, 34027, 36941, 37129, 37939, 42599, 43249, 44431, 47291, 49374, 60097, 65832, 66323, 69287, 69749, 70613, 74063, 74281, 74333, 74999, 77231, 83881, 86191, 86551, 87776, 88727, 99683, 106481, 108673, 111366, 113922, 115729, 118517, 124841, 126054, 129337

Offset: 1

Antti Karttunen, Nov 13 2024

nonn

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

Question: Does this sequence have asymptotic density? See also questions in A377872 and A377869.

Antti Karttunen, Table of n, a(n) for n = 1..15711
Index entries for sequences related to primorial numbers

Cf. A000720, A276085, A377869, A377877.
Subsequence of A373140, and of A377873.
Cf. also A377872.

isA377878(n) = { my(m=5^5, f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= Mod(prime(i),m); i++); s += f[k, 2]*pr); (0==lift(s)); };

{k such that Sum e*A377877(A000720(p)-1) == 0 (mod 5^5), when k = Product(p^e)}.

A377876 The n-th primorial number reduced modulo 27.

Original entry on oeis.org

1, 2, 6, 3, 21, 15, 6, 21, 21, 24, 21, 3, 3, 15, 24, 21, 6, 3, 21, 3, 24, 24, 6, 12, 15, 24, 21, 3, 24, 24, 12, 12, 6, 12, 21, 24, 6, 24, 24, 12, 24, 3, 3, 6, 24, 3, 3, 12, 3, 6, 24, 3, 15, 24, 3, 15, 3, 24, 24, 6, 12, 21, 24, 24, 12, 3, 6, 15, 6, 3, 21, 15, 12, 3, 12, 12, 6, 12, 12, 6, 24, 12, 3, 24, 24, 6, 12, 15

Offset: 0

Antti Karttunen, Nov 12 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..19683
Index entries for sequences related to primorial numbers

Cf. A002110, A377872.

Cf. also A086360, A377877.

R:= 1: p:= 1: v:= 1:
for i from 1 to 100 do
  p:= nextprime(p); v:= p*v mod 27;
  R:= R,v;
od:
R; # Robert Israel, Nov 12 2024

Mod[FoldList[Times,1,Prime[Range[87]]],27] (* James C. McMahon, Nov 12 2024 *)

A002110(n) = prod(i=1,n,prime(i));
A377876(n) = (A002110(n)%27);

up_to = 105;
A377876list(up_to_n) = { my(m=27, v=vector(1+up_to_n), pr=1); v[1] = 1; for(n=1, up_to_n, pr *= Mod(prime(n),m); v[1+n] = lift(pr)); (v); };
v377876 = A377876list(up_to);
A377876(n) = v377876[1+n];

from functools import reduce
from sympy import primerange, prime
def A377876(n): return reduce(lambda x,y:x*y%27,primerange(prime(n)+1)) if n else 1 # Chai Wah Wu, Nov 12 2024

a(n) = A002110(n) mod 27.

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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A377869 Numbers k such that A276085(k) has no divisors of the form p^p, where A276085 is fully additive with a(p) = p#/p.

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A369007 Numbers k for which k' / gcd(k,k') is a multiple of 27, where k' stands for the arithmetic derivative of k, A003415(k).

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A377873 Numbers k such that A276085(k) has at least one divisor of the form p^p, where A276085 is fully additive with a(p) = p#/p.

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A377875 Numbers k for which A276085(k) is not a multiple of 4 and has at least one divisor of the form p^p, with p an odd prime, where A276085 is fully additive with a(p) = p#/p.

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A377878 Numbers k for which A276085(k) is a multiple of 3125, where A276085 is fully additive with a(p) = p#/p.

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A377876 The n-th primorial number reduced modulo 27.

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