A328114 -id:A328114 - cp's OEIS frontend

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

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)).

A328390 Maximal digit value in primorial base expansion of the arithmetic derivative of n: a(n) = A328114(A003415(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..30030
Index entries for sequences related to primorial base

Cf. A003415, A051903, A327859, A328114, A328389, A328391, A328392.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A328114(n) = { my(s=0, p=2); while(n, s = max(s,(n%p)); n = n\p; p = nextprime(1+p)); (s); };
A328390(n) = A328114(A003415(n));

a(n) = A328114(A003415(n)).

a(n) = A051903(A327859(n)).

A328392 Maximal digit value in primorial base expansion of A327860(n): a(n) = A328114(A327860(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 4, 2, 2, 4, 6, 1, 3, 2, 2, 4, 5, 6, 7, 3, 7, 10, 7, 6, 9, 1, 1, 2, 2, 3, 5, 2, 4, 2, 1, 3, 4, 3, 5, 2, 6, 9, 4, 3, 10, 3, 6, 8, 2, 8, 5, 7, 7, 5, 10, 2, 2, 3, 4, 2, 6, 4, 2, 6, 9, 3, 4, 4, 5, 6, 9, 6, 7, 5, 7, 9, 5, 11, 10, 4, 3, 5, 11, 16, 7, 4, 3, 5, 2, 5, 6, 5, 7, 4, 6, 8, 10, 5, 9, 12, 11

Offset: 0

Antti Karttunen, Oct 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..30030
Index entries for sequences related to primorial base

Cf. A002110, A003415, A276086, A327859, A327860, A328114, A328389, A328390, A328391, A328396.

A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
A328114(n) = { my(s=0, p=2); while(n, s = max(s,(n%p)); n = n\p; p = nextprime(1+p)); (s); };
A328392(n) = A328114(A327860(n));

a(n) = A328114(A327860(n)) = A328390(A276086(n)) = A328114(A003415(A276086(n))).

A328394 Maximal digit value in primorial base expansion of A276087(n): a(n) = A328114(A276087(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 3, 2, 5, 1, 5, 4, 6, 3, 3, 7, 7, 4, 7, 5, 16, 6, 12, 27, 21, 35, 28, 23, 31, 28, 2, 2, 4, 5, 4, 5, 4, 10, 9, 2, 11, 6, 7, 10, 12, 7, 30, 6, 10, 15, 14, 7, 23, 37, 26, 32, 28, 33, 24, 28, 8, 3, 17, 11, 3, 5, 6, 11, 7, 12, 30, 21, 28, 15, 28, 11, 24, 30, 14, 16, 43, 17, 52, 26, 19, 29, 27, 33, 46, 27, 12, 15, 28, 28, 24, 27, 11, 20, 16, 20

Offset: 0

Antti Karttunen, Oct 15 2019

nonn

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

Cf. A051903, A276086, A276087, A328114, A328322, A328389, A328403.

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 120], f}, f[n_] := Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[n, b]; Array[Max@ IntegerDigits[Nest[f, #, 2], b] &, 100, 0]] (* Michael De Vlieger, Oct 15 2019 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A276087(n) = A276086(A276086(n));
A328114(n) = { my(s=0, p=2); while(n, s = max(s,(n%p)); n = n\p; p = nextprime(1+p)); (s); };
A328394(n) = A328114(A276087(n));

a(n) = A328389(A276086(n)) = A328114(A276087(n)) = A051903(A328403(n)).

A350076 Numbers k for which the maximal digit in their primorial base expansion (A328114) is greater than or equal to the maximal exponent in the prime factorization of k (A051903).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92

Offset: 1

Antti Karttunen, Feb 01 2022

Numbers k for which the maximal prime exponent of A276086(k) is greater than or equivalent to the maximal prime exponent of k, A051903(k).

Index entries for sequences related to primorial base

Cf. A051903, A276086, A328114, A350070 (characteristic function), A350075 (complement).
Positions of nonnegative terms in A350074.
Cf. also A342006.

A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA350076(n) = (A051903(A276086(n)) >= A051903(n));

A369645 Numbers k for which the difference A051903(k) - A328114(k) reaches a new maximum in range 1..k, where A051903 is the maximal exponent in the prime factorization of n, and A328114 is the maximal digit in the primorial base expansion of n.

Original entry on oeis.org

1, 2, 8, 32, 256, 2560, 30720, 32768, 4194304, 20971520, 58720256, 234881024, 536870912, 1342177280

Offset: 1

Antti Karttunen, Feb 01 2024

			           k   factorization   max.exp.  in primorial  max digit  diff
                                             base
           1                       0,            1,       1,      -1
           2 = 2^1,                1,           10,       1,       0
           8 = 2^3,                3,          110,       1,       2
          32 = 2^5,                5,         1010,       1,       4
         256 = 2^8,                8,        11220,       2,       6
        2560 = 2^9 * 5^1,          9,       111120,       2,       7
       30720 = 2^11 * 3^1 * 5^1,  11,      1032000,       3,       8
       32768 = 2^15,              15,      1120110,       2,      13
     4194304 = 2^22,              22,     83876020,       8,      14
    20971520 = 2^22 * 5^1,        22,    231462310,       6,      16
    58720256 = 2^23 * 7^1,        23,    610501410,       6,      17
   234881024 = 2^25 * 7^1,        25,   1141710210,       7,      18
   536870912 = 2^29,              29,   296AA71010,      10,      19
  1342177280 = 2^28 * 5^1,        28,   6071712310,       7,      21.
On the penultimate row, letter "A" in the primorial base expansion stands for ten (10 in decimal), as 2^29 = 0*prime(0)# + 1*prime(1)# + 0*prime(2)# + 1*prime(3)# + 7*prime(4)# + 10*prime(5)# + 10*prime(6)# + 6*prime(7)# + 9*prime(8)# + 2*prime(9)#, where prime(n)# = A002110(n).

Index entries for sequences related to primorial base

Positions of records for -A350074(n).
Cf. A002110, A049345, A051903, A328114.
Cf. also A369646, A369647.
After the initial 1, subsequence of A351038, after the two initial terms, subsequence of A350075.

A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328114(n) = { my(s=0, p=2); while(n, s = max(s, (n%p)); n = n\p; p = nextprime(1+p)); (s); };
A350074(n) = (A328114(n) - A051903(n));
m=A350074(1); print1(1,", "); for(n=2,oo,x=A350074(n); if(x

A369646 Numbers k such that the difference A051903(k) - A328114(A003415(k)) reaches a new maximum in range 1..k, where A051903 is the maximal exponent in the prime factorization of n, A328114 is the maximal digit in the primorial base expansion of n, and A003415 is the arithmetic derivative.

Original entry on oeis.org

1, 8, 16, 832, 1024, 95232, 131072, 2097152, 1006632960, 1090519040

Offset: 1

Antti Karttunen, Feb 02 2024

			           k   factorization    max.exp.  k' in primorial  max digit  diff
                                                  base
           1                        0,              0,        0,       0
           8 = 2^3,                 3,            200,        2,       1
          16 = 2^4,                 4,           1010,        1,       3
         832 = 2^6 * 13^1,          6,         111120,        2,       4
        1024 = 2^10,               10,         222310,        3,       7
       95232 = 2^10 * 3^1 * 31^1,  10,       10021220,        2,       8
      131072 = 2^17,               17,       23132010,        3,      14
     2097152 = 2^21,               21,      252354100,        5,      16
  1006632960 = 2^26 * 3^1 * 5^1,   26,    23194866010,        9,      17
  1090519040 = 2^24 * 5^1 * 13^1,  24,    22053155300,        5,      19.
Here k' stands for the arithmetic derivative of k, A003415(k). Primorial base expansion is obtained with A049345.

Index entries for sequences related to primorial base

Positions of records for -A351097(n).
After the initial 1, a subsequence of A351098.
Cf. A002110, A003415, A049345, A051903, A328114.
Cf. also A369645, A369647.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328114(n) = { my(s=0, p=2); while(n, s = max(s, (n%p)); n = n\p; p = nextprime(1+p)); (s); };
A351097(n) = (A328114(A003415(n))-A051903(n));
m=A351097(1); print1(1,", "); for(n=2,oo,x=A351097(n); if(x

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

Original entry on oeis.org

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

A108951 Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 210, 8, 36, 60, 2310, 24, 30030, 420, 180, 16, 510510, 72, 9699690, 120, 1260, 4620, 223092870, 48, 900, 60060, 216, 840, 6469693230, 360, 200560490130, 32, 13860, 1021020, 6300, 144, 7420738134810, 19399380, 180180, 240, 304250263527210, 2520

Offset: 1

Paul Boddington, Jul 21 2005

This sequence is a permutation of A025487.
And thus also a permutation of A181812, see the formula section. - Antti Karttunen, Jul 21 2014
A previous description of this sequence was: "Multiplicative with a(p^e) equal to the product of the e-th powers of all primes at most p" (see extensions), Giuseppe Coppoletta, Feb 28 2015

			a(12) = a(2^2) * a(3) = (2#)^2 * (3#) = 2^2 * 6 = 24
a(45) = (3#)^2 * (5#) = (2*3)^2 * (2*3*5) = 1080 (as 45 = 3^2 * 5).

Amiram Eldar, Table of n, a(n) for n = 1..2370 (terms 1..256 from Antti Karttunen)
Index to divisibility sequences.
Index entries for sequences computed from indices in prime factorization.
Index entries for sequences related to primorial numbers.

Cf. A319626, A329900 (left inverses).

Cf. A034386, A002110, A025487, A048673, A064216, A064989, A085082, A122111, A124859, A161360, A181811, A181812, A181814, A181815, A181817, A181819, A181822, A238690, A283477, A283478, A307035, A324886, A324887, A324888, A324896, A325226, A329040, A329046, A329047, A329344, A329348, A329349, A329378, A329382, A329600, A329602, A329605, A329607, A329615, A329616, A329617, A329619, A329622, A319627, A329647, A331292, A337474, A346108, A346109, A344698, A344699.

a[n_] := a[n] = Module[{f = FactorInteger[n], p, e}, If[Length[f]>1, Times @@ a /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]; a[1] = 1; Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Feb 24 2015 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}], {n, 42}] (* Michael De Vlieger, Mar 18 2017 *)

primorial(n)=prod(i=1,primepi(n),prime(i))
a(n)=my(f=factor(n)); prod(i=1,#f~, primorial(f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Jun 28 2015

from sympy import primerange, factorint
from operator import mul
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, May 14 2017

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
def p(f):
    return sharp_primorial(f[0])^f[1]
[prod(p(f) for f in factor(n)) for n in range (1,51)]
# Giuseppe Coppoletta, Feb 07 2015

Dirichlet g.f.: 1/(1-2*2^(-s))/(1-6*3^(-s))/(1-30*5^(-s))...
Completely multiplicative with a(p_i) = A002110(i) = prime(i)#. [Franklin T. Adams-Watters, Jun 24 2009; typos corrected by Antti Karttunen, Jul 21 2014]
From Antti Karttunen, Jul 21 2014: (Start)
a(1) = 1, and for n > 1, a(n) = n * a(A064989(n)).
a(n) = n * A181811(n).
a(n) = A002110(A061395(n)) * A331188(n). - [added Jan 14 2020]
a(n) = A181812(A048673(n)).
Other identities:
A006530(a(n)) = A006530(n). [Preserves the largest prime factor of n.]
A071178(a(n)) = A071178(n). [And also its exponent.]
a(2^n) = 2^n. [Fixes the powers of two.]
A067029(a(n)) = A007814(a(n)) = A001222(n). [The exponent of the least prime of a(n), that prime always being 2 for n>1, is equal to the total number of prime factors in n.]
(End)
From Antti Karttunen, Nov 19 2019: (Start)
Further identities:
a(A307035(n)) = A000142(n).
a(A003418(n)) = A181814(n).
a(A025487(n)) = A181817(n).
a(A181820(n)) = A181822(n).
a(A019565(n)) = A283477(n).
A001221(a(n)) = A061395(n).
A001222(a(n)) = A056239(n).
A181819(a(n)) = A122111(n).
A124859(a(n)) = A181821(n).
A085082(a(n)) = A238690(n).
A328400(a(n)) = A329600(n). (smallest number with the same set of distinct prime exponents)
A000188(a(n)) = A329602(n). (square root of the greatest square divisor)
A072411(a(n)) = A329378(n). (LCM of exponents of prime factors)
A005361(a(n)) = A329382(n). (product of exponents of prime factors)
A290107(a(n)) = A329617(n). (product of distinct exponents of prime factors)
A000005(a(n)) = A329605(n). (number of divisors)
A071187(a(n)) = A329614(n). (smallest prime factor of number of divisors)
A267115(a(n)) = A329615(n). (bitwise-AND of exponents of prime factors)
A267116(a(n)) = A329616(n). (bitwise-OR of exponents of prime factors)
A268387(a(n)) = A329647(n). (bitwise-XOR of exponents of prime factors)
A276086(a(n)) = A324886(n). (prime product form of primorial base expansion)
A324580(a(n)) = A324887(n).
A276150(a(n)) = A324888(n). (digit sum in primorial base)
A267263(a(n)) = A329040(n). (number of distinct nonzero digits in primorial base)
A243055(a(n)) = A329343(n).
A276088(a(n)) = A329348(n). (least significant nonzero digit in primorial base)
A276153(a(n)) = A329349(n). (most significant nonzero digit in primorial base)
A328114(a(n)) = A329344(n). (maximal digit in primorial base)
A062977(a(n)) = A325226(n).
A097248(a(n)) = A283478(n).
A324895(a(n)) = A324896(n).
A324655(a(n)) = A329046(n).
A327860(a(n)) = A329047(n).
A329601(a(n)) = A329607(n).
(End)
a(A181815(n)) = A025487(n), and A319626(a(n)) = A329900(a(n)) = n. - Antti Karttunen, Dec 29 2019
From Antti Karttunen, Jul 09 2021: (Start)
a(n) = A346092(n) + A346093(n).
a(n) = A346108(n) - A346109(n).
a(A342012(n)) = A004490(n).
a(A337478(n)) = A336389(n).
A336835(a(n)) = A337474(n).
A342002(a(n)) = A342920(n).
A328571(a(n)) = A346091(n).
A328572(a(n)) = A344592(n).
(End)
Sum_{n>=1} 1/a(n) = A161360. - Amiram Eldar, Aug 04 2022

More terms computed by Antti Karttunen, Jul 21 2014
The name of the sequence was changed for more clarity, in accordance with the above remark of Franklin T. Adams-Watters (dated Jun 24 2009). It is implicitly understood that a(n) is then uniquely defined by completely multiplicative extension. - Giuseppe Coppoletta, Feb 28 2015
Name "Primorial inflation" (coined by Matthew Vandermast in A181815) prefixed to the name by Antti Karttunen, Jan 14 2020

A327860 Arithmetic derivative of the primorial base exp-function: a(n) = A003415(A276086(n)).

Original entry on oeis.org

0, 1, 1, 5, 6, 21, 1, 7, 8, 31, 39, 123, 10, 45, 55, 185, 240, 705, 75, 275, 350, 1075, 1425, 3975, 500, 1625, 2125, 6125, 8250, 22125, 1, 9, 10, 41, 51, 165, 12, 59, 71, 247, 318, 951, 95, 365, 460, 1445, 1905, 5385, 650, 2175, 2825, 8275, 11100, 30075, 4125, 12625, 16750, 46625, 63375, 166125, 14, 77, 91, 329, 420

Offset: 0

Antti Karttunen, Sep 30 2019

Are there any other fixed points after 0, 1, 7, 8 and 2556? (A328110, see also A351087 and A351088).
Out of the 30030 initial terms, 19220 are multiples of 5. (See A327865).
Proof that a(n) is even if and only if n is a multiple of 4: Consider Charlie Neder's Feb 25 2019 comment in A235992. As A276086 is never a multiple of 4, and as it toggles the parity, we only need to know when A001222(A276086(n)) = A276150(n) is even. The condition for that is given in the latter sequence by David A. Corneth's Feb 27 2019 comment. From this it also follows that A166486 gives similarly the parity of terms of A342002, A351083 and A345000. See also comment in A327858. - Antti Karttunen, May 01 2022

			2556 has primorial base expansion [1,1,1,1,0,0] as 1*A002110(5) + 1*A002110(4) + 1*A002110(3) + 1*A002110(2) = 2310 + 210 + 30 + 6 = 2556. That in turn is converted by A276086 to 13^1 * 11^1 * 7^1 * 5^1 = 5005, whose arithmetic derivative is 5' * 1001 + 1001' * 5 = 1*1001 + 311*5 = 2556, thus 2556 is one of the rare fixed points (A328110) of this sequence.

Antti Karttunen, Table of n, a(n) for n = 0..2310
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..30030
Index entries for sequences related to primorial base

Cf. A002110 (positions of 1's), A003415, A048103, A276086, A327858, A327859, A327865, A328110 (fixed points), A328233 (positions of primes), A328242 (positions of squarefree terms), A328388, A328392, A328571, A328572, A329031, A329032, A329041, A342002.
Cf. A345000, A351074, A351075, A351076, A351077, A351080, A351083, A351084, A351087 (numbers k such that a(k) is a multiple of k), A351088.
Coincides with A329029 on positions given by A276156.
Cf. A166486 (a(n) mod 2), A353630 (a(n) mod 4).
Cf. A267263, A276150, A324650, A324653, A324655 for omega, bigomega, phi, sigma and tau applied to A276086(n).
Cf. also A351950 (analogous sequence).

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12]}, Array[Function[k, If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]] ] &@ Abs[Times @@ Power @@@ # &@ Transpose@{Prime@ Range@ Length@ k, Reverse@ k}]]@ IntegerDigits[#, b] &, 65, 0]] (* Michael De Vlieger, Mar 12 2021 *)

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
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; };
A327860(n) = A003415(A276086(n));

A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); }; \\ (Standalone version) - Antti Karttunen, Nov 07 2019

a(n) = A003415(A276086(n)).
a(A002110(n)) = 1 for all n >= 0.
From Antti Karttunen, Nov 03 2019: (Start)
Whenever A329041(x,y) = 1, a(x + y) = A003415(A276086(x)*A276086(y)) = a(x)*A276086(y) + a(y)*A276086(x). For example, we have:
a(n) = a(A328841(n)+A328842(n)) = A329031(n)*A328572(n) + A329032(n)*A328571(n).
A051903(a(n)) = A328391(n).
A328114(a(n)) = A328392(n).
(End)
From Antti Karttunen, May 01 2022: (Start)
a(n) = A328572(n) * A342002(n).
For all n >= 0, A000035(a(n)) = A166486(n). [See comments]
(End)

Verbal description added to the definition by Antti Karttunen, May 01 2022

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A328390 Maximal digit value in primorial base expansion of the arithmetic derivative of n: a(n) = A328114(A003415(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A328392 Maximal digit value in primorial base expansion of A327860(n): a(n) = A328114(A327860(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A328394 Maximal digit value in primorial base expansion of A276087(n): a(n) = A328114(A276087(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A350076 Numbers k for which the maximal digit in their primorial base expansion (A328114) is greater than or equal to the maximal exponent in the prime factorization of k (A051903).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A369645 Numbers k for which the difference A051903(k) - A328114(k) reaches a new maximum in range 1..k, where A051903 is the maximal exponent in the prime factorization of n, and A328114 is the maximal digit in the primorial base expansion of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A369646 Numbers k such that the difference A051903(k) - A328114(A003415(k)) reaches a new maximum in range 1..k, where A051903 is the maximal exponent in the prime factorization of n, A328114 is the maximal digit in the primorial base expansion of n, and A003415 is the arithmetic derivative.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

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