A351233 -id:A351233 - 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

A351231 Denominator of A003415(n) / A276086(n), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 1, 3, 6, 9, 18, 1, 10, 5, 5, 45, 90, 25, 50, 25, 75, 225, 450, 125, 250, 125, 75, 1125, 2250, 625, 125, 125, 1250, 5625, 11250, 7, 14, 21, 3, 63, 21, 7, 70, 5, 105, 315, 630, 175, 350, 175, 350, 63, 3150, 125, 125, 175, 525, 1125, 15750, 4375, 4375, 13125, 13125, 39375, 78750, 49, 98, 49, 98, 147, 49, 245, 490, 245

Offset: 0

Antti Karttunen, Feb 05 2022

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003415, A276086, A327858, A351230 (numerators), A351232, A351233.

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); };
A351231(n) = denominator(A003415(n) / A276086(n));

a(n) = A276086(n) / A327858(n) = A276086(n) / gcd(A003415(n), A276086(n)).

a(n) = A276086(A351233(n)).

A351234 a(n) = A276085(gcd(A003415(n), A276086(n))).

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 6, 0, 2, 3, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 2, 7, 0, 0, 0, 7, 8, 2, 0, 0, 0, 0, 0, 31, 0, 3, 6, 0, 32, 1, 0, 0, 0, 0, 2, 2, 12, 0, 30, 31, 8, 7, 30, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 2, 2, 5, 0, 0, 2, 1, 0, 0, 0, 0, 2, 6, 6, 5, 0, 0, 0, 3, 0, 0, 0, 1, 8, 1, 36, 0, 0, 1, 2, 1, 60, 3, 0, 0, 30, 8, 36, 0, 30

Offset: 0

Antti Karttunen, Feb 05 2022

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003415, A276085, A276086, A327858, A351233.

Cf. also A351254.

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)); };
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));
A351234(n) = A276085(A327858(n));

a(n) = A276085(A327858(n)) = A276085(gcd(A003415(n), A276086(n))).

a(n) = n - A351233(n).

A351253 a(n) = A276085(A351251(n)).

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 6, 7, 8, 7, 4, 11, 12, 13, 14, 7, 16, 17, 18, 19, 14, 19, 22, 23, 24, 13, 26, 25, 28, 29, 30, 31, 32, 31, 34, 5, 36, 37, 38, 37, 34, 41, 12, 43, 44, 37, 46, 47, 48, 19, 38, 49, 52, 53, 54, 49, 26, 55, 58, 59, 60, 61, 62, 31, 64, 65, 66, 67, 68, 67, 34, 71, 72, 73, 74, 61, 76, 47, 78, 79, 74, 79

Offset: 0

Antti Karttunen, Feb 05 2022

Antti Karttunen, Table of n, a(n) for n = 0..30030

Cf. A276085, A276086, A324198, A351251, A351254.

Cf. also A351233.

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)); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A351251(n) = { my(u=A276086(n)); (u/gcd(n, u)); };
A351253(n) = A276085(A351251(n));

a(n) = A276085(A351251(n)) = A276085(A276086(n) / A324198(n)).

a(n) = n - A351254(n).

A369970 Numbers k such that A003415(k) is a multiple of A276086(k), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 6, 2315, 510510

Offset: 1

Antti Karttunen, Feb 07 2024

For the general dynamics of this phenomenon, see the scatter plots of A351231 and A351233.
Question: Are the terms by necessity all squarefree?
As a subsequence this sequence includes all primorials with indices k such that A024451(k) is a multiple of A000040(1+k). See A369972 and A369973.
872415232 < a(6) <= 13082761331670030 [= A369973(4)].

			2315 is included as A003415(2315) = 5+463 = 468 = 2^2 * 3^2 * 13 (note that 2315 is a semiprime = 5*463, thus its arithmetic derivative is the sum of its two prime factors), and because that 468 is a multiple of A276086(2315) = 234 = 2 * 3^2 * 13 [the exponents of primes are here read from the primorial base expansion of 2315, A049345(2315) = 100021].
510510 is included because A003415(510510) = 19*37693, which is a multiple of A276086(510510) = 19.

Index entries for sequences related to primorial base

Cf. A000040, A003415, A024451, A276086, A369972, A369973 (subsequence).
Positions of 1's in A351231, positions of 0's in A351233 and in A369971.
After the two initial terms, a subsequence of A351228.
Cf. also A358221.

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); };
isA369970(n) = !(A003415(n)%A276086(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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A351231 Denominator of A003415(n) / A276086(n), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

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A351234 a(n) = A276085(gcd(A003415(n), A276086(n))).

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A351253 a(n) = A276085(A351251(n)).

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A369970 Numbers k such that A003415(k) is a multiple of A276086(k), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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