A317836 -id:A317836 - 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

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.

A317826 Number of partitions of n with carry-free sum in factorial base.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 1, 2, 2, 5, 4, 11, 2, 4, 4, 11, 9, 26, 3, 7, 7, 21, 16, 52, 1, 2, 2, 5, 4, 11, 2, 5, 5, 15, 11, 36, 4, 11, 11, 36, 26, 92, 7, 21, 21, 74, 52, 198, 2, 4, 4, 11, 9, 26, 4, 11, 11, 36, 26, 92, 9, 26, 26, 92, 66, 249, 16, 52, 52, 198, 137, 560, 3, 7, 7, 21, 16, 52, 7, 21, 21, 74, 52, 198, 16, 52, 52, 198, 137, 560, 31, 109

Offset: 0

Antti Karttunen, Aug 08 2018

nonn

"Carry-free sum" in this context means that when the digits of summands (written in factorial base, see A007623) are lined up (right-justified), then summing up of each column will not result in carries to any columns left of that column, that is, the sum of digits of the k-th column from the right (with the rightmost as column 1) over all the summands is the same as the k-th digit of n, thus at most k. Among other things, this implies that in any solution, at most one of the summands may be odd. Moreover, such an odd summand is present if and only if n is odd.
a(n) is the number of set partitions of the multiset that contains d copies of each number k, collected over all k in which digit-positions (the rightmost being k=1) there is a nonzero digit d in true factorial base representation of n, where also digits > 9 are allowed.
Distinct terms are the distinct terms in A050322, that is, A045782. - David A. Corneth & Antti Karttunen, Aug 10 2018

			  n  in fact.base  a(n) carry-free partitions
------------------------------
  0     "0"         1   {}    (unique empty partition, thus a(0) = 1)
  1     "1"         1   {1}
  2    "10"         1   {2}
  3    "11"         2   {2, 1} and {3}, in fact.base: {"10", "1"} and {"11"}
  4    "20"         2   {2, 2} and {4}, in fact.base: {"10" "10"} and {"20"}
  5    "21"         4   {2, 2, 1}, {3, 2}, {4, 1} and {5},
    in factorial base:  {"10", "10", "1"}, {"11", "10"}, {"20", "1"} and {"21"}.

Cf. A001055, A007623, A025487, A045782 (range of this sequence), A050322, A276076, A278236.
Cf. A317827 (positions of records), A317828 (record values), A317829.
Cf. also A227154, A317836.

fcnt(n, m) = {local(s); s=0; if(n == 1, s=1, fordiv(n, d, if(d > 1 & d <= m, s=s+fcnt(n/d, d)))); s};
A001055(n) = fcnt(n, n); \\ From A001055
A276076(n) = { my(i=0,m=1,f=1,nextf); while((n>0),i=i+1; nextf = (i+1)*f; if((n%nextf),m*=(prime(i)^((n%nextf)/f));n-=(n%nextf));f=nextf); m; };
A317826(n) = A001055(A276076(n));

\\ Slightly faster, memoized version:
memA001055 = Map();
A001055(n) = {my(v); if(mapisdefined(memA001055,n), v = mapget(memA001055,n), v = fcnt(n, n); mapput(memA001055,n,v); (v));}; \\ Cached version.
A276076(n) = { my(i=0,m=1,f=1,nextf); while((n>0),i=i+1; nextf = (i+1)*f; if((n%nextf),m*=(prime(i)^((n%nextf)/f));n-=(n%nextf));f=nextf); m; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A317826(n) = A001055(A046523(A276076(n)));

a(n) = A001055(A276076(n)) = A001055(A278236(n)).
a(A000142(n)) = 1.
a(A001563(n)) = A000041(n).
a(A033312(n+1)) = A317829(n) for n >= 1.

A358236 Number of factorizations of n where the sum of the factors is carryfree when the addition is done in the primorial base.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1, 1, 5, 1, 2, 1, 4, 1, 3, 1, 1, 1, 1, 1, 5, 1, 2, 1, 2, 1, 4, 1, 1, 1, 1, 1, 5, 1, 2, 1, 4, 1, 4, 1, 2, 1, 1, 1, 5, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 9, 1, 2, 1, 4, 1, 4, 1, 1, 1, 1, 1, 5, 1, 2, 1, 2, 1, 5, 1, 1, 1, 1, 1, 8, 1, 3, 1, 4, 1, 3, 1, 2, 1

Offset: 1

Antti Karttunen, Nov 29 2022

			36 has in total 9 = A001055(36) factorizations:
  factors in decimal  in primorial base   Do they generate carries when summed?
  [3, 3, 2, 2]        [11, 11, 10, 10]    Yes, as A049345(3+3+2+2) = "120".
  [4, 3, 3]           [20, 11, 11]        Yes, in the least significant place.
  [6, 3, 2]           [100, 11, 10]       No, 6+3+2 = 11 = "121".
  [6, 6]              [100, 100]          No, 6+6 = 12 = "200".
  [9, 2, 2]           [111, 10, 10]       Yes, in the second place from right.
  [9, 4]              [111, 20]           Ditto.
  [12, 3]             [200, 11]           No, 12+3 = 15 = "211".
  [18, 2]             [300, 10]           No, 18+2 = 20 = "310".
  [36]                [1100]              No, as a single factor never does.
Thus only five of the sums are carryfree, and a(36) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to primorial base

Cf. A001055, A049345, A276086, A327936, A358233.

Cf. also A317836.

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); };
A358236(n, m=n, facs=List([])) = if(1==n, 1==A327936(factorback(apply(A276086,Vec(facs)))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A358236(n/d, d, newfacs))); (s));

For all n >= 1, a(2n-1) = 1, a(4n-2) = A358233(4n-2).

For all n >= 1, A358233(n) <= a(n) <= A001055(n).

A380475 a(n) is the least term in A380468 that has exactly n prime factors.

Original entry on oeis.org

1, 2, 6, 186, 4686

Offset: 0

Antti Karttunen, Feb 03 2025

If it exists, a(5) > 2^35. - Antti Karttunen, Feb 19 2025

Conjecture: Sequence is finite and a(4) is the last term. This is equivalent to the claim that no arithmetic derivative (A003415) of a product of five or more distinct primes, (i.e., the value of the (n-1)-st elementary symmetric polynomial formed from those n distinct primes) can be formed as a carry-free sum of those n summands in primorial base (A049345). See also A380476 and A380528, A380530.

			186 = 2*3*31 and A276086(186/2) = 2058 = 2 * 3 * 7^3, A276086(186/3) = 3 * 7^2, A276086(186/31) = 5, whose product =  2^1 * 3^2 * 5^1 * 7^5 = 1512630 = A380459(186), and as all the exponents are less than the corresponding primes, the product is in A048103, and because there are no any smaller number with three prime factors satisfying the same condition (of A380468), 186 is the term a(3) of this sequence. Note that A049345(A003415(186)) = 5121, where the digits are the exponents in the product read from the largest to the smallest prime factor.
See also the example in A380476 about 4686.

Cf. A001221, A001222, A003415, A048103, A049345, A276086, A380459, A380468, A380476 (terms of A380468 with more than three prime factors).
Cf. A047247, A047257.
Cf. also A317836, A358235, A380525, A380528, A380530.

a(n) = Min_{k in A380468} for which A001221(k) = n.

A380476 Numbers k with at least 4 prime factors such that A380459(k) is in A048103, i.e., has no divisors of the form p^p.

Original entry on oeis.org

4686, 32406, 184866, 209166, 388086, 1099626, 1714866, 2111406, 2166846, 2356206, 3081606, 3303366, 6445806, 11366106, 21621606, 23022366, 39824466, 39826986, 42882846, 43197846, 46043826, 58216686, 61265886, 63603546, 66496506, 66611166, 87941706, 88968246, 92086746, 97117026, 101108706, 103367886, 118743306, 119658066

Offset: 1

Antti Karttunen, Feb 04 2025

nonn

Numbers m with four or more distinct prime factors such that their arithmetic derivative (A003415) can be formed as a carryless (or "carry-free") sum (in the primorial base, A049345) of the respective summands. See the example.
The terms are all squarefree and even (see A380468 and A380478 to find out why). Moreover, they are all multiples of six, because A380459(n) = Product_{d|n} A276086(n/d)^A349394(d) applied to a product of 2*p*q*r, with p, q, r three odd primes > 3 would yield three subproducts which would be multiples of 3 (consider A047247), so the 3-adic valuation of the whole product would be >= 3; hence the second smallest prime factor must be 3. For a similar reason, with terms that are product of four primes, the two remaining prime factors are either both of the form 6m+1 (A002476), or they are both of the form 6m-1 (A007528).
It is conjectured that there are no terms with more than four prime factors. See A380475 and A380528, A380530, also A380526.

			4686 = 2*3*11*71 and taking subproducts of three primes at time, we obtain 2*3*11 = 66, 2*3*71 = 426, 2*11*71 = 1562, 3*11*71 = 2343. Then A380459(4686) = A276086(66) * A276086(426) * A276086(1562) * A276086(2343) = 1622849599205985150 = 2^1 * 3^2 * 5^2 * 7^6 * 11^9 * 13^1, and because all the exponents are less than the corresponding primes, the product is in A048103.
Considering the primorial base expansions of the same summands (subproducts), we obtain
    2100  = A049345(66)
   20100  = A049345(426)
   73010  = A049345(1562)
  101011  = A049345(2343)
  ------
  196221  = A049345(A003415(4686)), with the summands adding together cleanly without any carries.
Note how the primorial base digits at the bottom are the exponents in the product A380459(4686) given above, read from the largest to the smallest prime factor

Antti Karttunen, Table of n, a(n) for n = 1..433; all terms <= 2^35
Index entries for sequences related to primorial base.

Intersection of A033987 and A380468.
Subsequence of A005117, A358673, A380478.
Conjectured to be a subsequence of A046386.
Cf. A003415, A001222, A048103, A276086, A349394, A380459, A380467.
Cf. A002476, A007528, A047247, A047257.
Cf. also A317836, A358235, A380475, A380526, A380528, A380530.

is_A380476(n) = (issquarefree(n) && (omega(n)>=4) && A380467(n)); \\ Note that issquarefree here is just an optimization as A380467(n) = 1 implies squarefreeness of 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.

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

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A317826 Number of partitions of n with carry-free sum in factorial base.

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A358236 Number of factorizations of n where the sum of the factors is carryfree when the addition is done in the primorial base.

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A380475 a(n) is the least term in A380468 that has exactly n prime factors.

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A380476 Numbers k with at least 4 prime factors such that A380459(k) is in A048103, i.e., has no divisors of the form p^p.

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A359778 Number of factorizations of n into factors not divisible by p^p for any prime p (terms of A048103).

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