A324351 -id:A324351 - 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

A324198 a(n) = gcd(n, A276086(n)), where A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 15, 1, 1, 1, 1, 5, 3, 1, 1, 1, 25, 1, 3, 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 1, 3, 5, 1, 7, 1, 1, 15, 1, 1, 1, 7, 25, 3, 1, 1, 1, 5, 7, 3, 1, 1, 1, 1, 1, 21, 1, 1, 1, 1, 1, 3, 35, 1, 1, 1, 1, 75, 1, 7, 1, 1, 5, 3, 1, 1, 7, 5, 1, 3, 1, 1, 1, 7, 1, 3, 1, 1, 1, 1, 49, 3, 5, 1, 1, 1, 1, 105

Offset: 0

Antti Karttunen, Feb 25 2019

nonn

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

Cf. A003989, A276086, A323878, A324284, A324350, A324351, A324384, A324577, A324580, A324646, A327858, A328323, A327168, A328316, A328323, A328386, A328584, A346242 (Dirichlet inverse), A351254.
Cf. A324583 (positions of ones), A324584 (and terms larger than one).
Cf. A371098 (odd bisection), A371099 [= a(36n+9)].
Cf. also A328231.

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[#, m]] &, 106, 0] (* Michael De Vlieger, Feb 04 2022 *)

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; };
A324198(n) = gcd(n,A276086(n));

A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p,valuation(orgn,p))); n = n\p; p = nextprime(1+p)); (m); }; \\ Antti Karttunen, Oct 21 2019

a(n) = gcd(n, A276086(n)).
From Antti Karttunen, Oct 21 2019: (Start)
A000005(a(n)) = A327168(n).
a(A328316(n)) = A328323(n).
a(n) = A324580(n) / A328584(n).
(End)

A329041 Square array read by antidiagonals: A(n, k) = A327936(A276086(n) * A276086(k)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 3, 2, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 2, 1, 6, 3, 6, 1, 2, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 1, 1, 3, 3, 3, 3, 1, 1, 3, 3, 3, 3, 1, 1

Offset: 0

Antti Karttunen, Nov 03 2019

Array A(n, k) is symmetric, and is read as (n,k) = (0, 0), (1, 0), (0, 1), (2, 0), (1, 1), (0, 2), (3, 0), (2, 1), (1, 2), (0, 3), ...

If A(n, k) is 1, it tells that adding of n and k do not generate any carries, when done in primorial base (A049345). If A(n, k) is larger than one, then its prime factors indicate in which specific moduli (digit positions) the sum was larger than allowed for that position.

			The top left corner of the array:
        0  1  2  3  4  5  6  7  8  9 10 11 12
      +--------------------------------------
   0: | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   1: | 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...
   2: | 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 1, ...
   3: | 1, 2, 1, 2, 3, 6, 1, 2, 1, 2, 3, 6, 1, ...
   4: | 1, 1, 3, 3, 3, 3, 1, 1, 3, 3, 3, 3, 1, ...
   5: | 1, 2, 3, 6, 3, 6, 1, 2, 3, 6, 3, 6, 1, ...
   6: | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   7: | 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...
   8: | 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 1, ...
   9: | 1, 2, 1, 2, 3, 6, 1, 2, 1, 2, 3, 6, 1, ...
  10: | 1, 1, 3, 3, 3, 3, 1, 1, 3, 3, 3, 3, 1, ...
  11: | 1, 2, 3, 6, 3, 6, 1, 2, 3, 6, 3, 6, 1, ...
  12: | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
...
A(11,25) = A(25,11) = 10 because 11 is written in primorial base representation (A049345) as "121" and 25 as "401", and when these are added together digit by digit, we see that the maximal allowed digits "421" for the rightmost three positions are exceeded in positions 1 and 3, with the 1st and 3rd primes 2 and 5 as their moduli, thus A(11,25) = 2*5 = 10.

Cf. A049345, A276086, A327936.

Cf. also A317836, A324351, A328770, A328841, A328842.

up_to = 105;
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327936(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]>=f[k,1])); factorback(f); };
A329041sq(row,col) = A327936(A276086(row)*A276086(col));
A329041list(up_to) = { my(v = vector(up_to), i=0); for(a=0,oo, for(col=0,a, if(i++ > up_to, return(v)); v[i] = A329041sq(a-col,col))); (v); };
v329041 = A329041list(up_to);
A329041(n) = v329041[1+n];

A(n, k) = A327936(A276086(n) * A276086(k)).

For all n, A(A328841(n), A328842(n)) = 1 and A(A328770(n), A328770(n)) = 1.

A324350 Square array read by antidiagonals: A(x,y) = gcd(A276086(x),A276086(y)), for x, y >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 3, 3, 1, 1, 1, 2, 3, 6, 3, 2, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 2, 1, 6, 9, 6, 1, 2, 1, 1, 1, 1, 1, 9, 9, 1, 1, 1, 1, 1, 2, 3, 2, 1, 18, 1, 2, 3, 2, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 1, 1, 1, 2, 3, 6, 3, 2, 5, 2, 3, 6, 3, 2, 1, 1, 1, 3, 3, 3, 3, 5, 5, 3, 3, 3, 3, 1, 1, 1, 2, 1, 6, 9, 6, 5, 10, 5, 6, 9, 6, 1, 2, 1

Offset: 0

Antti Karttunen, Feb 25 2019

			The array A begins:
       0   1   2   3   4   5   6   7   8   9  10  11  12
  x/y  ------------------------------------------------------
   0:  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   1:  1,  2,  1,  2,  1,  2,  1,  2,  1,  2,  1,  2,  1, ...
   2:  1,  1,  3,  3,  3,  3,  1,  1,  3,  3,  3,  3,  1, ...
   3:  1,  2,  3,  6,  3,  6,  1,  2,  3,  6,  3,  6,  1, ...
   4:  1,  1,  3,  3,  9,  9,  1,  1,  3,  3,  9,  9,  1, ...
   5:  1,  2,  3,  6,  9, 18,  1,  2,  3,  6,  9, 18,  1, ...
   6:  1,  1,  1,  1,  1,  1,  5,  5,  5,  5,  5,  5,  5, ...
   7:  1,  2,  1,  2,  1,  2,  5, 10,  5, 10,  5, 10,  5, ...
   8:  1,  1,  3,  3,  3,  3,  5,  5, 15, 15, 15, 15,  5, ...
   9:  1,  2,  3,  6,  3,  6,  5, 10, 15, 30, 15, 30,  5, ...
  10:  1,  1,  3,  3,  9,  9,  5,  5, 15, 15, 45, 45,  5, ...
  11:  1,  2,  3,  6,  9, 18,  5, 10, 15, 30, 45, 90,  5, ...
  12:  1,  1,  1,  1,  1,  1,  5,  5,  5,  5,  5,  5, 25, ...

Antti Karttunen, Table of n, a(n) for n = 0..7259 (the first 120 antidiagonals of the array)
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65702
Index entries for sequences related to primorial base

Cf. A003989, A276086 (central diagonal), A324198, A324351.

up_to = 65703; \\ = binomial(362+1,2)
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; };
A324350sq(row,col) = gcd(A276086(row),A276086(col));
A324350list(up_to) = { my(v = vector(up_to), i=0); for(a=0,oo, for(col=0,a, if(i++ > up_to, return(v)); v[i] = A324350sq(a-col,col))); (v); };
v324350 = A324350list(up_to);
A324350(n) = v324350[1+n];

A(x,y) = gcd(A276086(x), A276086(y)).

A(x,y) = A276086(A324351(x,y)).

A324384 a(n) = gcd(n, A276154(n)), where A276154 is the primorial base left shift.

Original entry on oeis.org

0, 1, 2, 1, 4, 1, 6, 1, 4, 1, 2, 11, 12, 1, 2, 1, 8, 1, 18, 1, 4, 7, 2, 1, 24, 1, 2, 1, 4, 1, 30, 1, 8, 1, 2, 7, 12, 1, 2, 1, 4, 1, 6, 1, 4, 1, 2, 1, 12, 1, 2, 1, 52, 1, 6, 1, 56, 1, 2, 1, 60, 1, 2, 1, 16, 1, 6, 1, 4, 1, 14, 1, 24, 1, 2, 1, 4, 1, 6, 1, 4, 1, 2, 1, 12, 1, 2, 1, 8, 1, 90, 1, 4, 1, 2, 1, 12, 1, 2, 1, 4, 1, 6, 1, 8, 1

Offset: 0

Antti Karttunen, Feb 26 2019

nonn

For a very few primes, a(p) > 1 (then by necessity a(p) = p). In range 2 .. 2^25 there are three: 2, 11, 119039.

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

Cf. A002110, A049345, A276085, A276086, A276151, A276152, A276154, A323879, A324198, A324350, A324351.

A276151(n) = { my(s=1); forprime(p=2, , if(n%p, return(n-s), s *= p)); };
A276152(n) = { my(s=1); forprime(p=2, , if(n%p, return(s*p), s *= p)); };
A276154(n) = if(!n,n,(A276152(n) + A276154(A276151(n))));
\\ Alternatively, A276154 can be defined with A276085, A276086 and A003961:
A002110(n) = prod(i=1,n,prime(i));
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
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; };
A276154(n) = A276085(A003961(A276086(n)));
A324384(n) = gcd(n,A276154(n));

a(n) = gcd(n, A276154(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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A324198 a(n) = gcd(n, A276086(n)), where A276086 is the primorial base exp-function.

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A329041 Square array read by antidiagonals: A(n, k) = A327936(A276086(n) * A276086(k)).

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A324350 Square array read by antidiagonals: A(x,y) = gcd(A276086(x),A276086(y)), for x, y >= 0.

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A324384 a(n) = gcd(n, A276154(n)), where A276154 is the primorial base left shift.

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