A371091 -id:A371091 - cp's OEIS frontend

A372559 a(n) is the index of the first occurrence of n in A371091.

0, 1, 3, 9, 21, 51, 111, 321, 741, 2001, 4311, 8931, 22791, 52821, 112881, 293061, 803571, 1824591, 4887651, 14587341, 33986721, 92184861, 208581141, 431674011, 877859751, 2216416971, 4893531411, 11363224641, 24302611101, 63120770481, 140757089241, 341317579371, 742438559631, 1945801500411, 4352527381971, 11773265516781

Offset: 0

Antti Karttunen, May 11 2024

The pattern in the primorial base expansion (A049345) of the terms is constructed recursively, so that the digit-positions of the primorial base expansion are successively filled with the positive terms of this sequence (1, 3, 9, 21, ...), up to that term that still fits to the position, i.e., is less than prime(i), for the positions i >= 1 indexed from the least significant end of the expansion. The nonleading digits are "frozen", and only the most significant digit keeps on increasing from a(1) to the maximal allowed a(x) for its position, after which the next term's expansion is obtained by prepending 1 to the front. See the examples.

			   n,      a(n)     in primorial base
   0,         0 =             0
   1,         1 =             1
   2,         3 =            11
   3,         9 =           111
   4,        21 =           311 (3 is less than prime(3)=5, so can be used now)
   5,        51 =          1311 (9 cannot yet be used, so append 1 to the front)
   6,       111 =          3311 (and then replace by next higher term that fits)
   7,       321 =         13311
   8,       741 =         33311
   9,      2001 =         93311 (9 is less than prime(5)=11, so can be used now)
  10,      4311 =        193311
  11,      8931 =        393311
  12,     22791 =        993311
  13,     52821 =       1993311
  14,    112881 =       3993311
  15,    293061 =       9993311
  16,    803571 =      19993311
  17,   1824591 =      39993311
  18,   4887651 =      99993311
  19,  14587341 =     199993311
  20,  33986721 =     399993311
  21,  92184861 =     999993311
  22, 208581141 =  {21}99993311 (21 is less than prime(9)=23, so can be used now)
  23, 431674011 = 1{21}99993311
etc.

Antti Karttunen, Table of n, a(n) for n = 0..1001
Index entries for sequences related to primorial base

Positions of records in A371091.

Cf. A002110, A235224, A276153.

A002110(n) = prod(i=1,n,prime(i));
A235224(n) = { my(s=0, p=2); while(n, s++; n = n\p; p = nextprime(1+p)); (s); };
A276153(n) = { my(p=2,d=0); while(n, d = n%p; n = n\p; p = nextprime(1+p)); (d); };
memoA372559 = Map();
A372559(n) = if(n<=2, n+(n>1), my(v); if(mapisdefined(memoA372559,n,&v), v, my(prev=A372559(n-1), hi=A235224(prev), hd=A276153(prev),k=0,u); while(A372559(k)A372559(1+k); v = if(u>=prime(hi), prev+A002110(hi), prev+((u-hd)*A002110(hi-1))); mapput(memoA372559,n,v); (v)));

For n >= 0, A371091(a(n)) = n, and for all k < a(n), A371091(k) < n.

A276150 Sum of digits when n is written in primorial base (A049345); minimal number of primorials (A002110) that add to n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 22 2016

The sum of digits of n in primorial base is odd if n is 1 or 2 (mod 4) and even if n is 0 or 3 (mod 4). Proof: primorials are 1 or 2 (mod 4) and a(n) can be constructed via the greedy algorithm. So if n = 4k + r where 0 <= r < 4, 4k needs an even number of primorials and r needs hammingweight(r) = A000120(r) primorials. Q.E.D. - David A. Corneth, Feb 27 2019

			For n=24, which is "400" in primorial base (as 24 = 4*(3*2*1) + 0*(2*1) + 0*1, see A049345), the sum of digits is 4, thus a(24) = 4.

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

Cf. A000120, A001222, A002110, A049345, A053589, A235168, A260188, A267263, A276084, A276086, A276151, A277022, A278226, A283477, A319713, A319715 (inverse Möbius transform), A321683, A324342, A324382, A324383, A324386, A324387, A371091, A373605, A373606, A373607.
Cf. A333426 [k such that a(k)|k], A339215 [numbers not of the form x+a(x) for any x], A358977 [k such that gcd(k, a(k)) = 1].
Cf. A338835, A366693.
Cf. A014601, A042963 (positions of even and odd terms), A343048 (positions of records).
Differs from analogous A034968 for the first time at n=24.

nn = 120; b = MixedRadix[Reverse@ Prime@ NestWhileList[# + 1 &, 1, Times @@ Prime@ Range[# + 1] <= nn &]]; Table[Total@ IntegerDigits[n, b], {n, 0, nn}] (* Version 10.2, or *)
nn = 120; 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[Total@ f@ n, {n, 0, 120}] (* Michael De Vlieger, Aug 26 2016 *)

A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); }; \\ Antti Karttunen, Feb 27 2019

from sympy import prime, primefactors
def Omega(n): return 0 if n==1 else Omega(n//primefactors(n)[0]) + 1
def a276086(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
def a(n): return Omega(a276086(n))
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 23 2017

a(n) = 1 + a(A276151(n)) = 1 + a(n-A002110(A276084(n))), a(0) = 0.
or for n >= 1: a(n) = 1 + a(n-A260188(n)).
Other identities and observations. For all n >= 0:
a(n) = A001222(A276086(n)) = A001222(A278226(n)).
a(n) >= A371091(n) >= A267263(n).
From Antti Karttunen, Feb 27 2019: (Start)
a(n) = A000120(A277022(n)).
a(A283477(n)) = A324342(n).
(End)
a(n) = A373606(n) + A373607(n). - Antti Karttunen, Jun 19 2024

A371090 Additive with a(p^1) = 1, a(p^e) = a(A276086(e)) for e > 1, where A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 3, 3, 2, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 2, 2, 2, 3, 1, 2

Offset: 1

Antti Karttunen, Mar 31 2024

nonn

Used to construct A371091.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A001221, A001222, A276086, A371091.
Differs from A064547 for the first time at n=63, where a(64) = 1, while A064547(64) = 2.
Differs from A058061 for the first time at n=128, where a(128) = 2, while A058061(128) = 3.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A371090(n) = vecsum(apply(e->if(1==e,1,A371090(A276086(e))),factor(n)[, 2]));

Additive with a(p^1) = 1, a(p^e) = A371091(e) for e > 1.

For all n >= 1, A001221(n) <= a(n) <= A001222(n).

cp's OEIS Frontend

A372559 a(n) is the index of the first occurrence of n in A371091.

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A276150 Sum of digits when n is written in primorial base (A049345); minimal number of primorials (A002110) that add to n.

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A371090 Additive with a(p^1) = 1, a(p^e) = a(A276086(e)) for e > 1, where A276086 is the primorial base exp-function.

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