A143293 -id:A143293 - cp's OEIS frontend

A276086 Primorial base exp-function: digits in primorial base representation of n become the exponents of successive prime factors whose product a(n) is.

1, 2, 3, 6, 9, 18, 5, 10, 15, 30, 45, 90, 25, 50, 75, 150, 225, 450, 125, 250, 375, 750, 1125, 2250, 625, 1250, 1875, 3750, 5625, 11250, 7, 14, 21, 42, 63, 126, 35, 70, 105, 210, 315, 630, 175, 350, 525, 1050, 1575, 3150, 875, 1750, 2625, 5250, 7875, 15750, 4375, 8750, 13125, 26250, 39375, 78750, 49, 98, 147, 294, 441, 882, 245, 490, 735, 1470, 2205, 4410, 1225, 2450

Offset: 0

Antti Karttunen, Aug 21 2016

Prime product form of primorial base expansion of n.
Sequence is a permutation of A048103. It maps the smallest prime not dividing n to the smallest prime dividing n, that is, A020639(a(n)) = A053669(n) holds for all n >= 1.
The sequence satisfies the exponential function identity, a(x + y) = a(x) * a(y), whenever A329041(x,y) = 1, that is, when adding x and y together will not generate any carries in the primorial base. Examples of such pairs of x and y are A328841(n) & A328842(n), and also A328770(n) (when added with itself). - Antti Karttunen, Oct 31 2019
From Antti Karttunen, Feb 18 2022: (Start)
The conjecture given in A327969 asks whether applying this function together with the arithmetic derivative (A003415) in some combination or another can eventually transform every positive integer into zero.
Another related open question asks whether there are any other numbers than n=6 such that when starting from that n and by iterating with A003415, one eventually reaches a(n). See comments in A351088.
This sequence is used in A351255 to list the terms of A099308 in a different order, by the increasing exponents of the successive primes in their prime factorization. (End)
From Bill McEachen, Oct 15 2022: (Start)
From inspection, the least significant decimal digits of a(n) terms form continuous chains of 30 as follows. For n == i (mod 30), i=0..5, there are 6 ordered elements of these 8 {1,2,3,6,9,8,7,4}. Then for n == i (mod 30), i=6..29, there are 12 repeated pairs = {5,0}.
Moreover, when the individual elements of any of the possible groups of 6 are transformed via (7*digit) (mod 10), the result matches one of the other 7 groupings (not all 7 may be seen). As example, {1,2,3,6,9,8} transforms to {7,4,1,2,3,6}. (End)
The least significant digit of a(n) in base 4 is given by A353486, and in base 6 by A358840. - Antti Karttunen, Oct 25 2022, Feb 17 2024

			For n = 24, which has primorial base representation (see A049345) "400" as 24 = 4*A002110(2) + 0*A002110(1) + 0*A002110(0) = 4*6 + 0*2 + 0*1, thus a(24) = prime(3)^4 * prime(2)^0 * prime(1)^0 = 5^4 = 625.
For n = 35 = "1021" as 35 = 1*A002110(3) + 0*A002110(2) + 2*A002110(1) + 1*A002110(0) = 1*30 + 0*6 + 2*2 + 1*1, thus a(35) = prime(4)^1 * prime(2)^2 * prime(1) = 7 * 3*3 * 2 = 126.

Antti Karttunen, Table of n, a(n) for n = 0..2310
Antti Karttunen, Program in LODA-assembly
Antti Karttunen, Program in LODA-assembly [Cached copy]
Index entries for sequences related to primorial base

Cf. A276085 (a left inverse) and also A276087, A328403.
Cf. A000040, A001221, A001222, A002110, A020639, A049345, A053669, A055396, A057588, A071178, A143293, A257993, A267263, A276084, A276088, A276092, A276093, A276147, A276150, A276151, A276153, A276156, A283477, A324198 (= gcd(n, a(n))), A328584 (= lcm(n, a(n))), A324646, A324289, A328386, A328403, A328475, A328571, A328572, A328578, A328612, A328613, A328620, A328624, A328627, A328763, A328766, A328828, A328835, A328841, A328842, A328843, A328844, A329041, A324580 [= n*a(n)], A324895 (largest proper divisor of a(n)), A351252, A353486 (reduced modulo 4), A358840 (modulo 6), A353489, A353516.
Cf. A048103 (terms sorted into ascending order), A100716 (natural numbers not present in this sequence).
Cf. A278226 (associated filter-sequence), A286626 (and its rgs-version), A328477.
Cf. A328316 (iterates started from zero).
Cf. A327858, A327859, A327860, A327963, A328097, A328098, A328099, A328110, A328112, A328382 for various combinations with arithmetic derivative (A003415).
Cf. also A327167, A329037.
Cf. A019565 and A054842 for base-2 and base-10 analogs and A276076 for the analogous "factorial base exp-function", from which this differs for the first time at n=24, where a(24)=625 while A276076(24)=7.
Cf. A327969, A351088, A351458 for sequences with conjectures involving this sequence.

b = MixedRadix[Reverse@ Prime@ Range@ 12]; Table[Function[k, Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]@ IntegerDigits[n, b], {n, 0, 51}] (* Michael De Vlieger, Aug 23 2016, Version 10.2 *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Prime@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Prime@ Range[# + 1] <= n &]; Rest[a][[All, 1]]]; Table[Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, Reverse@ f@ n], {n, 0, 73}] (* Michael De Vlieger, Aug 30 2016, Pre-Version 10 *)
a[n0_] := Module[{m = 1, i = 1, n = n0, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 01 2021, after Antti Karttunen's Sage code *)

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; }; \\ Antti Karttunen, May 12 2017

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); }; \\ (Better than above one, avoids unnecessary construction of primorials). - Antti Karttunen, Oct 14 2019

from sympy import prime
def a(n):
    i=0
    m=pr=1
    while n>0:
        i+=1
        N=prime(i)*pr
        if n%N!=0:
            m*=(prime(i)**((n%N)/pr))
            n-=n%N
        pr=N
    return m # Indranil Ghosh, May 12 2017, after Antti Karttunen's PARI code

from sympy import nextprime
def a(n):
    m, p = 1, 2
    while n > 0:
        n, r = divmod(n, p)
        m *= p**r
        p = nextprime(p)
    return m
print([a(n) for n in range(74)])  # Peter Luschny, Apr 20 2024

def A276086(n):
    m=1
    i=1
    while n>0:
        p = sloane.A000040(i)
        m *= (p**(n%p))
        n = floor(n/p)
        i += 1
    return (m)
# Antti Karttunen, Oct 14 2019, after Indranil Ghosh's Python code above, and my own leaner PARI code from Oct 14 2019. This avoids unnecessary construction of primorials.

(define (A276086 n) (let loop ((n n) (t 1) (i 1)) (if (zero? n) t (let* ((p (A000040 i)) (d (modulo n p))) (loop (/ (- n d) p) (* t (expt p d)) (+ 1 i))))))

(definec (A276086 n) (if (zero? n) 1 (* (expt (A053669 n) (A276088 n)) (A276086 (A276093 n))))) ;; Needs macro definec from http://oeis.org/wiki/Memoization#Scheme

(definec (A276086 n) (if (zero? n) 1 (* (A053669 n) (A276086 (- n (A002110 (A276084 n))))))) ;; Needs macro definec from http://oeis.org/wiki/Memoization#Scheme

a(0) = 1; for n >= 1, a(n) = A053669(n) * a(A276151(n)) = A053669(n) * a(n-A002110(A276084(n))).
a(0) = 1; for n >= 1, a(n) = A053669(n)^A276088(n) * a(A276093(n)).
a(n) = A328841(a(n)) + A328842(a(n)) = A328843(n) + A328844(n).
a(n) = a(A328841(n)) * a(A328842(n)) = A328571(n) * A328572(n).
a(n) = A328475(n) * A328580(n) = A328476(n) + A328580(n).
a(A002110(n)) = A000040(n+1). [Maps primorials to primes]
a(A143293(n)) = A002110(n+1). [Maps partial sums of primorials to primorials]
a(A057588(n)) = A276092(n).
a(A276156(n)) = A019565(n).
a(A283477(n)) = A324289(n).
a(A003415(n)) = A327859(n).
Here the text in brackets shows how the right hand side sequence is a function of the primorial base expansion of n:
A001221(a(n)) = A267263(n). [Number of nonzero digits]
A001222(a(n)) = A276150(n). [Sum of digits]
A067029(a(n)) = A276088(n). [The least significant nonzero digit]
A071178(a(n)) = A276153(n). [The most significant digit]
A061395(a(n)) = A235224(n). [Number of significant digits]
A051903(a(n)) = A328114(n). [Largest digit]
A055396(a(n)) = A257993(n). [Number of trailing zeros + 1]
A257993(a(n)) = A328570(n). [Index of the least significant zero digit]
A079067(a(n)) = A328620(n). [Number of nonleading zeros]
A056169(a(n)) = A328614(n). [Number of 1-digits]
A056170(a(n)) = A328615(n). [Number of digits larger than 1]
A277885(a(n)) = A328828(n). [Index of the least significant digit > 1]
A134193(a(n)) = A329028(n). [The least missing nonzero digit]
A005361(a(n)) = A328581(n). [Product of nonzero digits]
A072411(a(n)) = A328582(n). [LCM of nonzero digits]
A001055(a(n)) = A317836(n). [Number of carry-free partitions of n in primorial base]
Various number theoretical functions applied:
A000005(a(n)) = A324655(n). [Number of divisors of a(n)]
A000203(a(n)) = A324653(n). [Sum of divisors of a(n)]
A000010(a(n)) = A324650(n). [Euler phi applied to a(n)]
A023900(a(n)) = A328583(n). [Dirichlet inverse of Euler phi applied to a(n)]
A069359(a(n)) = A329029(n). [Sum a(n)/p over primes p dividing a(n)]
A003415(a(n)) = A327860(n). [Arithmetic derivative of a(n)]
Other identities:
A276085(a(n)) = n.          [A276085 is a left inverse]
A020639(a(n)) = A053669(n). [The smallest prime not dividing n -> the smallest prime dividing n]
A046523(a(n)) = A278226(n). [Least number with the same prime signature as a(n)]
A246277(a(n)) = A329038(n).
A181819(a(n)) = A328835(n).
A053669(a(n)) = A326810(n), A326810(a(n)) = A328579(n).
A257993(a(n)) = A328570(n), A328570(a(n)) = A328578(n).
A328613(a(n)) = A328763(n), A328620(a(n)) = A328766(n).
A328828(a(n)) = A328829(n).
A053589(a(n)) = A328580(n). [Greatest primorial number which divides a(n)]
A276151(a(n)) = A328476(n). [... and that primorial subtracted from a(n)]
A111701(a(n)) = A328475(n).
A328114(a(n)) = A328389(n). [Greatest digit of primorial base expansion of a(n)]
A328389(a(n)) = A328394(n), A328394(a(n)) = A328398(n).
A235224(a(n)) = A328404(n), A328405(a(n)) = A328406(n).
a(A328625(n)) = A328624(n), a(A328626(n)) = A328627(n). ["Twisted" variants]
a(A108951(n)) = A324886(n).
a(n) mod n = A328386(n).
a(a(n)) = A276087(n), a(a(a(n))) = A328403(n). [2- and 3-fold applications]
a(2n+1) = 2 * a(2n). - Antti Karttunen, Feb 17 2022

Name edited and new link-formulas added by Antti Karttunen, Oct 29 2019

Name changed again by Antti Karttunen, Feb 05 2022

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

A024451 a(n) is the numerator of Sum_{i = 1..n} 1/prime(i).

Original entry on oeis.org

0, 1, 5, 31, 247, 2927, 40361, 716167, 14117683, 334406399, 9920878441, 314016924901, 11819186711467, 492007393304957, 21460568175640361, 1021729465586766997, 54766551458687142251, 3263815694539731437539, 201015517717077830328949, 13585328068403621603022853

Offset: 0

N. J. A. Sloane, Clark Kimberling

Arithmetic derivative of p#: a(n) = A003415(A002110(n)). - Reinhard Zumkeller, Feb 25 2002
(n-1)-st elementary symmetric functions of first n primes; see Mathematica section. - Clark Kimberling, Dec 29 2011
Denominators of the harmonic mean of the first n primes; A250130 gives the numerators. - Colin Barker, Nov 14 2014
Let Pn(n) = A002110 denote the primorial function. The average number of distinct prime factors <= prime(n) in the natural numbers up to Pn(n) is equal to Sum_{i = 1..n} 1/prime(i). - Jamie Morken, Sep 17 2018
Conjecture: All terms are squarefree numbers. - Nicolas Bělohoubek, Apr 13 2022
The above conjecture would imply that for n > 0, gcd(a(n), A369651(n)) = 1. See corollary 2 on the page 4 of Ufnarovski-Åhlander paper. - Antti Karttunen, Jan 31 2024
Apart from the initial 0, a subsequence of A048103. Proof: For all primes p, when i >= A000720(p), neither p itself nor p^p divides a(i) [implied by Henry Bottomley's Sep 27 2006 formula], but neither does p^p divide a(i) when 0 < i < A000720(p), as then p^p > a(i). See A074107, which gives an upper bound for this sequence. - Antti Karttunen, Nov 19 2024

			0/1, 1/2, 5/6, 31/30, 247/210, 2927/2310, 40361/30030, 716167/510510, 14117683/9699690, ...

S. R. Finch, Mathematical Constants, Cambridge, 2003, Sect. 2.2.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Sect. VII.28.

Alois P. Heinz, Table of n, a(n) for n = 0..350 (terms n = 1..100 from T. D. Noe)
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.
Index entries for sequences related to primorial numbers

Denominators are A002110.
See also A106830/A034386, A241189/A241190, A241191/A241192, A061015/A061742, A115963/A115964, A250133/A296358, and A096795/A051451, A354417/A354418, A354859/A354860.
Row sums of A077011 and A258566.
Subsequence of A048103 (after the initial 0).
Cf. A003415, A002110, A070826, A143293, A203008, A250130, A327860, A348301, A369651.
Cf. A053144 (a lower bound), A074107 (an upper bound).
Cf. A109628 (indices k where a(k) is prime), A244622 (corresponding primes), A244621 (a(n) mod 12).
Cf. A369972 (k where prime(1+k)|a(k)), A369973 (corresponding primorials), A293457 (corresponding primes), A377992 (antiderivatives of the terms > 1 of this sequence).
Cf. also A223037, A260615, A274070, A327978, A353299, A353534, A356253.

[ Numerator(&+[ NthPrime(k)^-1: k in [1..n]]): n in [1..18] ];  // Bruno Berselli, Apr 11 2011

h:= n-> add(1/(ithprime(i)),i=1..n);
t1:=[seq(h(n),n=0..50)];
t1a:=map(numer,t1); # A024451
t1b:=map(denom,t1); # A002110 - N. J. A. Sloane, Apr 25 2014

a[n_] := Numerator @ Sum[1/Prime[i], {i, n}]; Array[a,18]  (* Jean-François Alcover, Apr 11 2011 *)
f[k_] := Prime[k]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A024451 *)
(* Clark Kimberling, Dec 29 2011 *)
Numerator[Accumulate[1/Prime[Range[20]]]] (* Harvey P. Dale, Apr 11 2012 *)

a(n) = numerator(sum(i=1, n, 1/prime(i))); \\ Michel Marcus, Sep 18 2018

from sympy import prime
from fractions import Fraction
def a(n): return sum(Fraction(1, prime(k)) for k in range(1, n+1)).numerator
print([a(n) for n in range(20)]) # Michael S. Branicky, Feb 12 2021

from math import prod
from sympy import prime
def A024451(n):
    q = prod(plist:=tuple(prime(i) for i in range(1,n+1)))
    return sum(q//p for p in plist) # Chai Wah Wu, Nov 03 2022

Limit_{n->oo} (Sum_{p <= n} 1/p - log log n) = 0.2614972... = A077761.
a(n) = (Product_{i=1..n} prime(i))*(Sum_{i=1..n} 1/prime(i)). - Benoit Cloitre, Jan 30 2002
(n+1)-st elementary symmetric function of the first n primes.
a(n) = a(n-1)*A000040(n) + A002110(n-1). - Henry Bottomley, Sep 27 2006
From Antti Karttunen, Jan 31 2024, Feb 08 2024 and Nov 19 2024: (Start)
a(0) = 0, for n > 0, a(n) = 2*A203008(n-1) + A070826(n).
For n > 0, a(n) = A327860(A143293(n-1)).
For n > 0, a(n) = A348301(n) + A002110(n).
For n = 3..175, a(n) = A356253(A002110(n)). [See comments in A356253.]
For n >= 0, A053144(n) <= a(n) <= A074107(n) < A070826(1+n).
(End)

a(0)=0 prepended by Alois P. Heinz, Jun 26 2015

A276087 a(n) = A276086(A276086(n)).

Original entry on oeis.org

2, 3, 6, 5, 30, 125, 18, 45, 150, 7, 1050, 343, 1250, 2625, 7350, 16807, 1650, 847, 43218, 3465, 27731550, 3195731, 1183724850, 435930295269007, 17794411250, 7105308412125, 3782866198615350, 2709490941157, 6237907125450, 161696170950365051, 10, 75, 750, 175, 294, 12005, 126, 2205, 51450, 11, 565950, 1331, 21008750, 9904125, 6225450, 161051, 1002614947950, 190333

Offset: 0

Antti Karttunen, Aug 21 2016

Antti Karttunen, Table of n, a(n) for n = 0..210

Cf. A000040, A002110, A143293, A276086.

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12], f}, f[n_] := Times @@ Power @@@ # &@ Transpose@{Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[n, b]; Array[Nest[f, #, 2] &, 48, 0]] (* Michael De Vlieger, Oct 15 2019 *)

(define (A276087 n) (A276086 (A276086 n)))

a(n) = A276086(A276086(n)).
Other identities. For all n >= 0:
a(A143293(n)) = A000040(2+n).

A327969 The length of a shortest path from n to zero when using the transitions x -> A003415(x) and x -> A276086(x), or -1 if no zero can ever be reached from n.

Original entry on oeis.org

0, 1, 2, 2, 5, 2, 3, 2, 6, 4, 3, 2, 5, 2, 5, 6, 6, 2, 5, 2, 7, 4, 3, 2

Offset: 0

Antti Karttunen, Oct 07 2019

The terms of this sequence are currently known only up to n=23, with the value of a(24) still being uncertain. For the tentative values of the later terms, see sequence A328324 which gives upper bounds for these terms, many of which are very likely also exact values for them.
As A051903(A003415(n)) >= A051903(n)-1, it means that it takes always at least A051903(n) steps to a prime if iterating solely with A003415.
Some known values and upper bounds from n=24 onward:
a(24) <= 11.
a(25) = 4.
a(26) = 7.
a(27) <= 22.
a(33) = 4.
a(39) = 4.
a(40) = 5.
a(42) = 3.
a(44) <= 10.
a(45) = 5.
a(46) = 5.
a(48) = 9.
a(49) = 6.
a(50) = 6.
a(55) = 7.
a(74) = 5.
a(77) = 6.
a(80) <= 18.
a(111) = 6.
a(112) = 8.
a(125) <= 9.
a(240) = 7.
a(625) <= 10.
a(875) = 8.
From Antti Karttunen, Feb 20 2022: (Start)
a(2556) <= 20.
a(5005) <= 19.
What is the value of a(128), and is A328324(128) well-defined?
When I created this sequence, I conjectured that by applying two simple arithmetic operations "arithmetic derivative" (A003415) and "primorial base exp-function" (A276086) in some combination, and starting from any positive integer, we could always reach zero (via a prime and 1).
At the first sight it seems almost certain that the conjecture holds, as it is always possible at every step to choose from two options (which very rarely meet, see A351088), leading to an exponentially growing search tree, and also because A276086 always jumps out of any dead-end path with p^p-factors (dead-end from the arithmetic derivative's point of view). However, it should be realized that one can reach the terms of either A157037 or A327978 with a single step of A003415 only from squarefree numbers (or respectively, cubefree numbers that are not multiples of 4, see A328234), and in general, because A003415 decreases the maximal exponent of the prime factorization (A051903) at most by one, if the maximal exponent in the prime factorization of n is large, there is a correspondingly long path to traverse if we take only A003415-steps in the iteration, and any step could always lead with certain probability to a p^p-number. Note that the antiderivatives of primorials with a square factor seem quite rare, see A351029.
And although taking a A276086-step will always land us to a p^p-free number (which a priori is not in the obvious dead-end path of A003415, although of course it might eventually lead to one), it (in most cases) also increases the magnitude of number considerably, that tends to make the escape even harder. Particularly, in the majority of cases A276086 increases the maximal exponent (which in the preimage is A328114, "maximal digit value used when n is written in primorial base"), so there will be even a longer journey down to squarefree numbers when using A003415. See the sequences A351067 and A351071 for the diminishing ratios suggesting rapidly diminishing chances of successfully reaching zero from larger terms of A276086. Also, the asymptotic density of A276156 is zero, even though A351073 may contain a few larger values.
On the other hand, if we could prove that by (for example) continuing upwards with any p^p-path of A003415 we could eventually reach with a near certainty a region of numbers with low values of A328114 (i.e., numbers with smallish digits in primorial base, like A276156), then the situation might change (see also A351089). However, a few empirical runs seemed to indicate otherwise.
For all of the above reasons, I now conjecture that there are natural numbers from which it is not possible to reach zero with any combination of steps. For example 128 or 5^5 = 3125.
(End)

			Let -A> stand for an application of A003415 and -B> for an application of A276086, then, we have for example:
a(8) = 6 as we have 8 -A>  12 -B>  25 -A> 10 -A>  7 -A> 1 -A> 0, six transitions in total (and there are no shorter paths).
a(15) = 6 as we have 15 -B> 150 -A> 185 -A> 42 -A> 41 -A> 1 -A> 0, six transitions in total (and there are no shorter paths).
a(20) = 7, as 20 -B> 375 -A> 350 -A> 365 -A> 78 -A> 71 -A> 1 -A> 0, and there are no shorter paths.
For n=112, we know that a(112) cannot be larger than eight, as A328099^(8)(112) = 0, so we have a path of length 8 as 112 -A> 240 -B> 77 -A> 18 -A> 21 -A> 10 -A> 7 -A> 1 -A> 0. Checking all 32 combinations of the paths of lengths of 5 starting from 112 shows that none of them or their prefixes ends with a prime, thus there cannot be any shorter path, and indeed a(112) = 8.
a(24) <= 11 as A328099^(11)(24) = 0, i.e., we have 24 -A> 44 -A> 48 -A> 112 -A> 240 -B> 77 -A> 18 -A> 21 -A> 10 -A> 7 -A> 1 -A> 0. On the other hand, 24 -B> 625 -B> 17794411250 -A> 41620434625 -A> 58507928150 -A> 86090357185 -A> 54113940517 -A> 19982203325 -A> 12038411230 -A> 8426887871 -A> 1 -A> 0, thus offering another path of length 11.

Cf. A003415, A046099, A051674, A051903, A068346, A276086, A276087, A327859, A327860, A328099, A328112, A328114, A328116, A328307.
Cf. A328324 (a sequence giving upper bounds, computed with restricted search space).
Sequences for whose terms k, value a(k) has a guaranteed constant upper bound: A000040, A002110, A143293, A157037, A192192, A327978, A328232, A328233, A328239, A328240, A328243, A328249, A328313.
Sequences for whose terms k, it is guaranteed that a(k) has finite value > 0, even if not bound by a constant: A099308, A328116.
Cf. also A256750, A327966, A328110, A351029, A351088, A351067, A351071, A351073, A351089 and A351255, A351256, A351257, A351258, A351261.

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); };
A327969(n,searchlim=0) = if(!n,n,my(xs=Set([n]),newxs,a,b,u); for(k=1,oo, print("n=", n, " k=", k, " xs=", xs); newxs=Set([]); for(i=1,#xs,u = xs[i]; a = A003415(u); if(0==a, return(k)); if(isprime(a), return(k+2)); b = A276086(u); if(isprime(b), return(k+1+(u>2))); newxs = setunion([a],newxs); if(!searchlim || (b<=searchlim),newxs = setunion([b],newxs))); xs = newxs));

a(0) = 0, a(p^p) = 1 + a(A276086(p^p)) for primes p, and for other numbers, a(n) = 1+min(a(A003415(n)), a(A276086(n))).
a(p) = 2 for all primes p.
For all n, a(n) <= A328324(n).
Let A stand the transition x -> A003415(x), and B stand for x -> A276086(x). The following sequences give some constant upper limits, because it is guaranteed that the combination given in brackets (the leftmost A or B is applied first) will always lead to a prime:
For all n, a(A157037(n)) = 3.  [A]
For n > 1, a(A002110(n)) = 3.  [B]
For all n, a(A192192(n)) <= 4. [AA]
For all n, a(A327978(n)) = 4.  [AB]
For all n, a(A328233(n)) <= 4. [BA]
For all n, a(A143293(n)) <= 4. [BB]
For all n, a(A328239(n)) <= 5. [AAA]
For all n, a(A328240(n)) <= 5. [BAA]
For all n, a(A328243(n)) <= 5. [ABB]
For all n, a(A328313(n)) <= 5. [BBB]
For all n, a(A328249(n)) <= 6. [BAAA]
For all k in A046099, a(k) >= 4, and if A328114(k) > 1, then certainly a(k) > 4.

A327978 Numbers whose arithmetic derivative (A003415) is a primorial number (A002110) > 1.

Original entry on oeis.org

9, 161, 209, 221, 2189, 2561, 3281, 3629, 5249, 5549, 6401, 7181, 7661, 8321, 8909, 9089, 9869, 10001, 10349, 10541, 10961, 11009, 11021, 29861, 38981, 52601, 66149, 84101, 93029, 97481, 132809, 150281, 158969, 163301, 197669, 214661, 227321, 235721, 285449, 321989, 338021, 357881, 369701, 381449, 385349, 416261, 420089, 442889

Offset: 1

Antti Karttunen, Oct 09 2019

nonn

Numbers n such that A327859(n) = A276086(A003415(n)) is an odd prime.
Composite terms in A328232.
Although it first might seem that the numbers whose arithmetic derivative is A002110(k) all appear before any of those whose arithmetic derivative is A002110(k+1), that is not true, as for example, we have a(56) = 570149, and A003415(570149) = 2310, a(57) = 570209, and A003415(570209) = 30030, but then a(58) = 573641 with A003415(573641) = 2310 again.
Because this is a subsequence of A327862 (all primorials > 1 are of the form 4k+2), only odd numbers are present.
Conjecture: No multiples of 5 occur in this sequence, and no multiples of 3 after the initial 9.
Of the first 10000 terms, all others are semiprimes (with 9 the only square one), except 1547371 = 7^2 * 23 * 1373 and 79332523 = 17^2 * 277 * 991, the latter being the only known term whose decimal expansion ends with 3. If all solutions were semiprimes p*q such that p+q = A002110(k) for some k > 1 (see A002375), it would be a sufficient reason for the above conjecture to hold. - David A. Corneth and Antti Karttunen, Oct 11 2019
In any case, the solutions have to be of the form "odd numbers with an even number of prime factors with multiplicity" (see A235992), and terms must also be cubefree (A004709), as otherwise the arithmetic derivative would not be squarefree.
Sequence A366890 gives the non-Goldbachian solutions, i.e., numbers that are not semiprimes. See also A368702. - Antti Karttunen, Jan 17 2024

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1078 terms from Antti Karttunen)
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to primorial numbers

Cf. A002110, A002375, A002620, A003415, A024451, A143293, A157037, A276150, A327859, A327969, A328233, A328243.
Cf. A351029 (number of k for which k' = A002110(n)).
Cf. A368703, A368704 (the least and the greatest k for which k' = A002110(n)).
Cf. A366890 (terms that are not semiprimes), A368702 (numbers k such that k' is one of the terms of this sequence).
Subsequence of following sequences: A004709, A189553, A327862, A328232, A328234.

ad[n_] := n * Total @ (Last[#]/First[#] & /@ FactorInteger[n]); primQ[n_] := Max[(f = FactorInteger[n])[[;;,2]]] == 1 && PrimePi[f[[-1,1]]] == Length[f]; Select[Range[10^4], primQ[ad[#]] &] (* Amiram Eldar, Oct 11 2019 *)

A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
isA327978flat(n) = { my(u=A003415(n)); ((u>1)&&(1==A276150(u))); }; \\ Slow!
k=0; for(n=1,A002620(30030),if(isA327978flat(n), k++; write("b327978.txt", k, " ", n)));

A327969(a(n)) = 4 for all n.

A328389 Maximal digit value in primorial base expansion of A276086(n): a(n) = A328114(A276086(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 2, 2, 2, 1, 2, 3, 4, 3, 2, 5, 2, 2, 4, 2, 5, 4, 5, 10, 6, 6, 8, 6, 5, 9, 1, 2, 3, 2, 2, 4, 2, 2, 3, 1, 3, 3, 5, 4, 3, 5, 7, 4, 4, 8, 3, 3, 4, 9, 9, 8, 7, 11, 4, 8, 3, 3, 4, 4, 3, 4, 2, 2, 3, 7, 10, 10, 5, 4, 6, 3, 8, 9, 7, 5, 10, 10, 10, 8, 5, 5, 8, 6, 9, 7, 4, 4, 6, 9, 4, 7, 8, 5, 3, 5, 7, 4, 7, 7, 11, 9

Offset: 0

Antti Karttunen, Oct 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..32589
Index entries for sequences related to primorial base

Cf. A002110, A051903, A143293, A276086, A276087, A328114, A328322, A328390, A328391, A328392, A328395.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328114(n) = { my(s=0, p=2); while(n, s = max(s,(n%p)); n = n\p; p = nextprime(1+p)); (s); };
A328389(n) = A328114(A276086(n));

a(n) = A328114(A276086(n)).

a(n) = A051903(A276087(n)).

A060389 a(1)=p_1, a(2)=p_1 + p_1p_2, a(3)=p_1 + p_1p_2 + p_1p_2p_3, ... where p_i is the i-th prime.

Original entry on oeis.org

2, 8, 38, 248, 2558, 32588, 543098, 10242788, 233335658, 6703028888, 207263519018, 7628001653828, 311878265181038, 13394639596851068, 628284422185342478, 33217442899375387208, 1955977793053588026278

Offset: 1

Jason Earls, Apr 04 2001

nonn

Partial sums of the primorials A002110 starting from 2. - Charles R Greathouse IV, Feb 05 2014
All terms are even. From a(98) on, all terms are multiples of 523. - Charles R Greathouse IV, Feb 05 2014
The only values of n for which a(n)/2 is prime are: 3, 5, 7, 11, 15, 47, 49. The corresponding 7 primes are: 19, 1279, 271549, 103631759509, 314142211092671239, 826811434211869939736645732264127163964562391958563586838409421490271014424018927729, 41839936239750050346953677118447851613901200239299781782205558511980130628486398081201749. - Amiram Eldar, May 04 2017

			a(4) = 248 because p_1 + p_1*p_2 + p_1*p_2*p_3 + p_1*p_2*p_3*p_4 = 2 + 6 + 30 + 210 = 248.
a(5) = 2558: A002110(3) = 30, A286624(4) = 85, a(2) = 8; 30*85 + 8 = 2558. - _Bob Selcoe_, Oct 12 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for sequences related to primorial numbers

Cf. A002110, A070826, A084737, A143293, A203008 [= A327860(a(n))], A276085, A286624, A373158.

for n from 1 to 30 do printf(`%d,`,sum(product(ithprime(i), i=1..j), j=1..n)) od:

Accumulate[Denominator[Accumulate[1/Prime[Range[20]]]]] (* Alonso del Arte, Mar 21 2013 *)
Accumulate@ FoldList[Times, Prime@ Range@ 17] (* Michael De Vlieger, May 04 2017 *)

a(n)=my(s,P=1); forprime(p=2,prime(n),s+=P*=p); s \\ Charles R Greathouse IV, Feb 05 2014

a(n) = A002110(n-2)*A286624(n-1) + a(n-3), n >= 4. - Bob Selcoe, Oct 12 2017

a(n) = A276085(A070826(1+n)) = A084737(2+n)-2 = A373158(A002110(n)). - Antti Karttunen, Feb 06 2024, Oct 28 2024

More terms from James Sellers, Apr 05 2001

A328570 Index of the least significant zero digit in the primorial base expansion of n, when the rightmost digit is in the position 1.

Original entry on oeis.org

1, 2, 1, 3, 1, 3, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 1, 3, 1, 3, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 3, 1, 3, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 3, 1, 3, 1, 2, 1, 5, 1, 5, 1, 2, 1, 5

Offset: 0

Antti Karttunen, Oct 20 2019

Index of the least prime not dividing A276086(n), where A276086 converts the primorial base expansion of n into its prime product form.

Starting from x = n, repeatedly divide x by prime(1) (discarding the remainder), and set x to the integer quotient floor(x/prime(1)), then divide x with prime(2) (again discarding the remainder, and setting x to the integer quotient), etc., stopping as soon one of the primes is a divisor of the previous integer quotient (leaving zero remainder). a(n) is then the index of that prime, equal to 1 + the number of iterations done.

			For n = 2, we divide it with A000040(1) = 2, and it leaves zero remainder, so we have finished on the first round (needing no actual iterations), and thus a(2) = 1. Note that 2 in primorial base (A049345) is written as "10", and indeed the first zero from the right occurs at the position 1.
For n = 5, we first divide 5 with prime(1) = 2, and discarding the remainder, we are left with floor(5/2) = 2. Then we divide that 2 with prime(2) = 3, leaving floor(2/3) = 0 and remainder 2. And finally we divide 0 with prime(3) = 5, and that doesn't leave any remainder, thus we are finished on the third round, and a(5) = 3. Note that 5 in primorial base is written as "21", and allowing here a leading zero, written as "021", we see that it is in this case the least significant zero occurring at position 3 from the right.
For n = 43, we first divide it with prime(1) = 2, leaving a remainder 1 and integer quotient 21. Then we divide 21 with prime(2) = 3, which doesn't leave any remainder, thus we are finished on the second round, and a(43) = 2. Note that 43 is written as "1201" in primorial base, with the least significant zero occurring in the position 2.

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base

Cf. A000720, A002110, A049345, A053669, A055396, A143293, A257993, A276086, A276087, A276088, A326810, A328475, A328476, A328569, A328578, A328580.

A328570(n) = { my(i=1, p=2); while(n && (n%p), i++; n = n\p; p = nextprime(1+p)); (i); };

a(n) = A000720(A326810(n)) = A257993(A276086(n)) = A055396(A276087(n)).
For all n >= 0, A002110(a(n)) = A328580(n), a(A276086(n)) = A328578(n).
For all odd n, A000040(a(n)) = A326810(n) > A276088(n).
For all n >= 0, A276086(n)/A002110(a(n)-1) = A328475(n) and A276086(n)-A002110(a(n)-1) = A328476(n).

cp's OEIS Frontend

A276086 Primorial base exp-function: digits in primorial base representation of n become the exponents of successive prime factors whose product a(n) is.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Python

Sage

Scheme

Scheme

Scheme

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A024451 a(n) is the numerator of Sum_{i = 1..n} 1/prime(i).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Python

Formula

Extensions

A276087 a(n) = A276086(A276086(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

A327969 The length of a shortest path from n to zero when using the transitions x -> A003415(x) and x -> A276086(x), or -1 if no zero can ever be reached from n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A327978 Numbers whose arithmetic derivative (A003415) is a primorial number (A002110) > 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

A060389 a(1)=p_1, a(2)=p_1 + p_1p_2, a(3)=p_1 + p_1p_2 + p_1p_2p_3, ... where p_i is the i-th prime.