A289234 -id:A289234 - cp's OEIS frontend

A344760 "Inverted variant" of A342002: a(n) = A342002(A289234(n)).

0, 1, 1, 5, 2, 7, 1, 7, 8, 31, 13, 41, 3, 11, 14, 43, 19, 53, 2, 9, 11, 37, 16, 47, 4, 13, 17, 49, 22, 59, 1, 9, 10, 41, 17, 55, 12, 59, 71, 247, 106, 317, 26, 87, 113, 331, 148, 401, 19, 73, 92, 289, 127, 359, 33, 101, 134, 373, 169, 443, 4, 15, 19, 59, 26, 73, 27, 89, 116, 337, 151, 407, 41, 117, 158, 421, 193, 491, 34

Offset: 0

Antti Karttunen, Jul 10 2021

Antti Karttunen, Table of n, a(n) for n = 0..13860
Index entries for sequences related to primorial base

Cf. A003415, A003557, A276086, A342001, A342002, A289234, A346101.

Cf. also A344761, A344762.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A346101(n) = { my(pr=1, p=2, m=1); while(n>0, my(d=n%p); if(d>0, m *= p^lift(1/Mod(d, p))); pr *= p; n \= p; p = nextprime(1+p)); return(m); };
A344760(n) = A342001(A346101(n));

A289234(n) = { my(pr=1, p=2, v=0); while(n>0, my (d=n%p); if(d>0, v += pr * lift(1/Mod(d, p))); pr *= p; n \= p; p = nextprime(1+p)); return(v); }; \\ From A289234.
A342002(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= p^(e>0); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
A344760(n) = A342002(A289234(n));

a(n) = A342001(A346101(n)) = A342002(A289234(n)).

A346101 "Inverted variant" of A276086: a(n) = A276086(A289234(n)).

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 5, 10, 15, 30, 45, 90, 125, 250, 375, 750, 1125, 2250, 25, 50, 75, 150, 225, 450, 625, 1250, 1875, 3750, 5625, 11250, 7, 14, 21, 42, 63, 126, 35, 70, 105, 210, 315, 630, 875, 1750, 2625, 5250, 7875, 15750, 175, 350, 525, 1050, 1575, 3150, 4375, 8750, 13125, 26250, 39375, 78750, 2401, 4802, 7203, 14406

Offset: 0

Antti Karttunen, Jul 10 2021

Antti Karttunen, Table of n, a(n) for n = 0..16170
Index entries for sequences related to primorial base

Cf. A276086, A289234, A344760.

Cf. also A328624, A328627, A346102.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A289234(n) = { my(pr=1, p=2, v=0); while(n>0, my (d=n%p); if(d>0, v += pr * lift(1/Mod(d, p))); pr *= p; n \= p; p = nextprime(1+p)); return(v); }; \\ From A289234.
A346101(n) = A276086(A289234(n));

A346101(n) = { my(p=2, m=1); while(n>0, my(d=n%p); if(d>0, m *= p^lift(1/Mod(d, p))); n \= p; p = nextprime(1+p)); return(m); };

a(n) = A276086(A289234(n)).

A331172 a(n) = min(n, A289234(n)), where A289234 is primorial base "reciprocal" flip.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 12, 13, 14, 15, 16, 17, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 42, 43, 44, 45, 46, 47, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 0

Antti Karttunen, Jan 12 2020

For all i, j:
  a(i) = a(j) => A267263(i) = A267263(j).
For all i, j > 0:
  a(i) = a(j) => A053669(i) = A053669(j).

Antti Karttunen, Table of n, a(n) for n = 0..30029
Index entries for sequences related to primorial base

Cf. A053669, A267263, A289234.

Cf. also A286626, A328477, A330740, A331171.

A289234(n) = { my(pr=1, p=2, v=0); while(n>0, my (d=n%p); if(d>0, v += pr * lift(1/Mod(d, p))); pr *= p; n \= p; p = nextprime(p+1)); return(v); }; \\ From A289234.
A331172(n) = min(n, A289234(n));

a(n) = min(n, A289234(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

A342002 Čiurlionis sequence: Arithmetic derivative without its inherited divisor applied to the primorial base exp-function: a(n) = A342001(A276086(n)).

Original entry on oeis.org

0, 1, 1, 5, 2, 7, 1, 7, 8, 31, 13, 41, 2, 9, 11, 37, 16, 47, 3, 11, 14, 43, 19, 53, 4, 13, 17, 49, 22, 59, 1, 9, 10, 41, 17, 55, 12, 59, 71, 247, 106, 317, 19, 73, 92, 289, 127, 359, 26, 87, 113, 331, 148, 401, 33, 101, 134, 373, 169, 443, 2, 11, 13, 47, 20, 61, 17, 69, 86, 277, 121, 347, 24, 83, 107, 319, 142, 389, 31

Offset: 0

Antti Karttunen, Feb 28 2021

The scatter plot shows an interesting structure.
The terms are essentially the "wild" or "unherited" part of the arithmetic derivative (A003415) of those natural numbers (A048103) that are not immediately beyond all hope of reaching zero by iteration (as the terms of A100716 are), ordered by the primorial base expansion of n as in A276086. Sequence A342018 shows the positions of the terms here that have just moved to the "no hope" region, while A342019 shows how many prime powers in any term have breached the p^p limit. Note that the results are same as for A327860(n), as the division by "regular part", A328572(n) does not affect the "wild part" of the arithmetic derivative of A276086(n). - Antti Karttunen, Mar 12 2021
I decided to name this sequence in honor of Lithuanian artist Mikalojus Čiurlionis, 1875 - 1911, as the scatter plot of this sequence reminds me thematically of his work "Pyramid sonata", with similar elements: fractal repetition in different scales and high tension present, discharging as lightning. Like Čiurlionis's paintings, this sequence has many variations, see the Formula and Crossrefs sections. - Antti Karttunen, Apr 30 2022

Antti Karttunen, Table of n, a(n) for n = 0..11550
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
Wikimedia, Čiurlionis: Piramidziu sonata, Allegro (a medium resolution scan of the painting "Pyramid Sonata, Allegro")
Wikipedia, Mikalojus Konstantinas Čiurlionis
Index entries for sequences related to primorial base

Cf. A002110 (positions of 1's), A003415, A003557, A083345, A085731, A276086, A289234, A327860, A328571, A328572, A342001, A342005, A342006, A342016, A342022 (rgs-transform), A342417, A342419.
Cf. A342463 [= a(A329886(n))], A342920 [= a(A108951(n))], A342921 [= a(A276156(n))], A342017 [= A342007(a(n))], A342019 [= A129251(a(n))].
Cf. A166486 (a(n) mod 2, parity of terms, see comment in A327860), A353640 (a(n) mod 4).
Cf. A344760, A344761, A344762, A346252, A346253 and A345930, A353572, A353574 for permuted and other variants.
Cf. A351952 (similar definition, but using factorial base, with quite a different look).

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A342002(n) = A342001(A276086(n)); \\ Uses also code from A342001.

A342002(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= p^(e>0); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); }; \\ Antti Karttunen, Mar 12 2021

A342002(n) = { my(s=0, p=2, e); while(n, e = (n%p); s += (e/p); n = n\p; p = nextprime(1+p)); numerator(s); }; \\ (Taking denominator instead would give A328571) - Antti Karttunen, Mar 12 2021

a(n) = A342001(A276086(n)) = A083345(A276086(n)).
a(n) = A327860(n) / A328572(n) = A003415(A276086(n)) / A003557(A276086(n)).
From Antti Karttunen, Jul 18 2021: (Start)
There are several permutations of this sequence. The following formulas show the relations:
a(n) = A344760(A289234(n)).
a(n) = A346252(A328623(n)) = A346253(A328622(n)).
a(n) = A344761(A328626(n)) = A344762(A328625(n)).
(End)

Sequence renamed as "Čiurlionis sequence" to honor Lithuanian artist Mikalojus Čiurlionis - Antti Karttunen, Apr 30 2022

A328625 In primorial base representation of n, multiply all other digits except the least significant with 1+{their right hand side neighbor}, and reduce each modulo prime(k) (with k > 1) to get a new digit for the position k (the least significant digit stays as it is), then convert back to decimal.

Original entry on oeis.org

0, 1, 2, 5, 4, 3, 6, 7, 14, 17, 22, 21, 12, 13, 26, 29, 10, 9, 18, 19, 8, 11, 28, 27, 24, 25, 20, 23, 16, 15, 30, 31, 32, 35, 34, 33, 66, 67, 74, 77, 82, 81, 102, 103, 116, 119, 100, 99, 138, 139, 128, 131, 148, 147, 174, 175, 170, 173, 166, 165, 60, 61, 62, 65, 64, 63, 126, 127, 134, 137, 142, 141, 192, 193, 206, 209, 190, 189, 48, 49, 38, 41, 58, 57, 114

Offset: 0

Antti Karttunen, Oct 23 2019

nonn

In primorial base (see A049345) we keep the least significant digit (0 or 1) intact, and replace each digit d(i) left of that (for i >= 2) with a new digit value computed as d(i)*(1+d(i-1)) mod prime(i). a(n) is then the newly constructed primorial expansion converted back to decimal.

Because for all primes p, Z_p is a field (not just a ring), this sequence is a permutation of nonnegative integers, and roughly speaking, offers a kind of analog of A003188 for primorial base system. Note however that it is the digit neighbor on the right (not left) hand side that affects here what will be the new digit at each position.

			In primorial base (A049345) 199 is written as "6301" because 6*A002110(3) + 3*A002110(2) + 0*A002110(1) + 1*A002110(0) = 6*30 + 3*6 + 0*2 + 1*1 = 199. Multiplying each digit left of the least significant by 1+{digit one step right}, and reducing modulo the corresponding prime yields 4*6 mod 7, 1*3 mod 5, 2*0 mod 3, (with the least significant 1 staying the same), so we get "3301", which is the primorial base expansion of 109, thus a(199) = 109.

Cf. A002110, A049345, A276085, A328620, A328624, A328626 (inverse permutation), A328628.

Cf. also A003188, A289234, A328622.

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)); };
A328624(n) = { my(m=1, p=2, e, g=1); while(n, e = (n%p); m *= (p^((g*e)%p)); g = e+1; n = n\p; p = nextprime(1+p)); (m); };
A328625(n) = A276085(A328624(n));

a(n) = A276085(A328624(n)).

For all n, A328620(a(n)) = A328620(n).

A328622 In primorial base representation of n, multiply by 2 all other digits except the least significant, and reduce each such product modulo prime(k) (to get the new digit), where k > 1 is the position of the digit, then convert back to decimal.

Original entry on oeis.org

0, 1, 4, 5, 2, 3, 12, 13, 16, 17, 14, 15, 24, 25, 28, 29, 26, 27, 6, 7, 10, 11, 8, 9, 18, 19, 22, 23, 20, 21, 60, 61, 64, 65, 62, 63, 72, 73, 76, 77, 74, 75, 84, 85, 88, 89, 86, 87, 66, 67, 70, 71, 68, 69, 78, 79, 82, 83, 80, 81, 120, 121, 124, 125, 122, 123, 132, 133, 136, 137, 134, 135, 144, 145, 148, 149, 146, 147, 126, 127, 130, 131

Offset: 0

Antti Karttunen, Oct 23 2019

			In primorial base (A049345) 199 is written as "6301" because 6*A002110(3) + 3*A002110(2) + 0*A002110(1) + 1*A002110(0) = 6*30 + 3*6 + 0*2 + 1*1 = 199. Multiplying each digit except the least significant by 2, and then reducing them modulo the corresponding prime leaves us with 2*6 mod 7, 2*3 mod 5, 2*0 mod 3, (with the least significant 1 staying the same), so we get "5101", which is the primorial base expansion of 157, thus a(199) = 157.
For 157, the new "doubled and reduced" expansion is 2*5 mod 7, 2*1 mod 5, 2*0 mod 3 and the trailing 1 stays intact, so we get "3201", which is the primorial base expansion of 103, thus a(157) = 103.

Cf. A328623 (inverse), and also A289234.

Cf. A002110, A049345, A276085, A276086, A328618.

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); };
A328618(n) = { my(f = factor(n), m, q); for(k=1, #f~, q = (f[k, 2]\f[k, 1]); m = (f[k, 2]%f[k, 1]); if(m&&(f[k,1]!=2), f[k, 2] = q*f[k, 1] + ((2*f[k, 2])%f[k, 1]))); factorback(f); };
A328622(n) = A276085(A328618(A276086(n)));

a(n) = A276085(A328618(A276086(n))).

A328626 Inverse permutation to A328625.

Original entry on oeis.org

0, 1, 2, 5, 4, 3, 6, 7, 20, 17, 16, 21, 12, 13, 8, 29, 28, 9, 18, 19, 26, 11, 10, 27, 24, 25, 14, 23, 22, 15, 30, 31, 32, 35, 34, 33, 126, 127, 80, 167, 166, 81, 162, 163, 128, 119, 118, 129, 78, 79, 116, 131, 130, 117, 114, 115, 164, 83, 82, 165, 60, 61, 62, 65, 64, 63, 36, 37, 140, 107, 106, 141, 102, 103, 38, 209, 208, 39, 138, 139, 206, 41, 40

Offset: 0

Antti Karttunen, Oct 25 2019

nonn

Cf. A328625 (inverse permutation).
Cf. A049345, A276085, A328620, A328627, A328629.
Cf. also A289234, A328623.

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)); };
A328627(n) = { my(m=1, p=2, d=0); while(n, d = lift(Mod(n,p)/(d+1)); m *= (p^d); n = n\p; p = nextprime(1+p)); (m); };
A328626(n) = A276085(A328627(n));

a(n) = A276085(A328627(n)).

For all n, A328620(a(n)) = A328620(n).

A328623 Inverse permutation to A328622.

Original entry on oeis.org

0, 1, 4, 5, 2, 3, 18, 19, 22, 23, 20, 21, 6, 7, 10, 11, 8, 9, 24, 25, 28, 29, 26, 27, 12, 13, 16, 17, 14, 15, 120, 121, 124, 125, 122, 123, 138, 139, 142, 143, 140, 141, 126, 127, 130, 131, 128, 129, 144, 145, 148, 149, 146, 147, 132, 133, 136, 137, 134, 135, 30, 31, 34, 35, 32, 33, 48, 49, 52, 53, 50, 51, 36, 37, 40, 41, 38, 39, 54, 55, 58, 59, 56, 57, 42

Offset: 0

Antti Karttunen, Oct 23 2019

Cf. A276085, A276086, A328619.

Cf. A328622 (inverse), and also A289234.

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); };
A328619(n) = { my(f = factor(n), m, q); for(k=1, #f~, q = (f[k, 2]\f[k, 1]); m = (f[k, 2]%f[k, 1]); if(m&&(f[k,1]!=2), f[k, 2] = q*f[k, 1] + lift(Mod(m,f[k, 1])/2))); factorback(f); };
A328623(n) = A276085(A328619(A276086(n)));

a(n) = A276085(A328619(A276086(n))).

A305458 In primorial base: a(n) is obtained by replacing each nonzero digit of n with its product with the nonzero digits at lower indices (See Comments for precise definition).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 12, 13, 14, 15, 28, 29, 18, 19, 20, 21, 10, 11, 24, 25, 26, 27, 22, 23, 30, 31, 32, 33, 64, 65, 36, 37, 38, 39, 76, 77, 72, 73, 74, 75, 148, 149, 108, 109, 110, 111, 190, 191, 144, 145, 146, 147, 52, 53, 60, 61, 62, 63

Offset: 0

Rémy Sigrist, Jun 01 2018

For a number n >= 0, let d_k, ..., d_0 be the digits of n in primorial base (n = Sum_{i=0..k} d_i * A002110(i), and for i = 0..k, 0 <= d_i < prime(i+1)); the digits of a(n) in primorial base, say e_k, ..., e_0, satisfy: for i = 0..k:
- if d_i = 0, then e_i = 0,
- if d_i > 0, then e_i == Product_{j=0..i and d_j > 0} d_j (mod prime(i+1)).
This sequence is a permutation of the nonnegative integers with inverse A305463.
The fixed points of this sequence (A305462) correspond to the numbers with all digits, except possibly the leading digit, equal to zero or one in primorial base.
Like A289234, this sequence preserves the number of digits and the number of nonzero digits in primorial base.
For any prime number p:
- we can build an analog of this sequence, say f_p, for the base p,
- in particular, f_2 = A001477,
- f_p is a permutation of the nonnegative integers,
- f_p preserves the number of digits and the number of nonzero digits in base p,
- the fixed points of f_p correspond to the numbers with all digits, except possibly the leading digit, equal to zero or one in base p.

			The digits of 7772 in primorial base are 3,4,0,0,1,0.
Also:
- 1 == 1 (mod prime(2)),
- 4 * 1 == 4 (mod prime(5)),
- 3 * 4 * 1 == 12 (mod prime(6)).
Hence the digits of a(7772) in primorial base are 12,4,0,0,1,0, and a(7772) = 28562.

Cf. A001477, A002110, A049345, A235224, A267263, A289234, A305462 (fixed points), A305463 (inverse).

a(n) = my (v=0, k=1, r=2, p=1); while (n, my (d=n % r); if (d, k *= d; v += p * lift(Mod(k, r))); n \= r; p *= r; r = nextprime(r+1)); return (v)

A235224(a(n)) = A235224(n).

A267263(a(n)) = A267263(n).

cp's OEIS Frontend

A344760 "Inverted variant" of A342002: a(n) = A342002(A289234(n)).

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A346101 "Inverted variant" of A276086: a(n) = A276086(A289234(n)).

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A331172 a(n) = min(n, A289234(n)), where A289234 is primorial base "reciprocal" flip.

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

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A342002 Čiurlionis sequence: Arithmetic derivative without its inherited divisor applied to the primorial base exp-function: a(n) = A342001(A276086(n)).

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A328622 In primorial base representation of n, multiply by 2 all other digits except the least significant, and reduce each such product modulo prime(k) (to get the new digit), where k > 1 is the position of the digit, then convert back to decimal.

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A328626 Inverse permutation to A328625.

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