A267263 -id:A267263 - cp's OEIS frontend

A324341 If 2n = 2^e1 + ... + 2^ek [e1 .. ek distinct], then a(n) is the number of nonzero digits when A002110(e1) * ... * A002110(ek) is written in primorial base.

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

Offset: 0

Antti Karttunen, Feb 23 2019

nonn

Number of nonzero digits when A283477(n) is represented in primorial base, A049345.

Number of distinct prime factors in A324289(n).

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to primorial base

Cf. A001221, A002110, A049345, A267263, A276086, A283477, A324289, A324342.

A002110(n) = prod(i=1,n,prime(i));
A030308(n,k) = bittest(n,k);
A283477(n) = prod(i=0,#binary(n),if(0==A030308(n,i),1,A030308(n,i)*A002110(1+i)));
A267263(n) = { my(s=0); forprime(p=2, , if(n%p, s++, if(n==0, return(s))); n\=p); }; \\ From A267263
A324341(n) = A267263(A283477(n));

a(n) = A267263(A283477(n)).
a(n) = A001221(A324289(n)) = A001221(A276086(A283477(n))).
a(n) <= A324342(n).

A328614 Number of 1-digits in primorial base expansion of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 22 2019

			In primorial base (A049345), 87 is written as "2411" because 87 = 2*A002110(3) + 4*A002110(2) + 1*A002110(1) + 1*A002110(0) = 2*30 + 4*6 + 1*2 + 1*1. Only two of these digits are "1"'s, thus a(87) = 2.

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

Cf. A049345, A056169, A267263, A276086, A328615, A328616.
Cf. A143293 (positions of records after initial zero).
Cf. also A257511.

a[n_] := Module[{k = n, p = 2, s = 0, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, If[r == 1, s++]; p = NextPrime[p]]; s]; Array[a, 100, 0] (* Amiram Eldar, Mar 13 2024 *)

A328614(n) = { my(s=0, p=2); while(n, s += (1==(n%p)); n = n\p; p = nextprime(1+p)); (s); };

a(n) = A056169(A276086(n)).
a(n) = A267263(n) - A328615(n).
For n >= 1, a(A143293(n-1)) = n. [This is the first occurrence of each n]

A328318 Number of nonzero digits in representation of A328316(n) in primorial base; Number of distinct prime factors in A328316(1+n).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 5, 16, 104, 7447

Offset: 0

Antti Karttunen, Oct 14 2019

Index entries for sequences related to primorial base

Cf. A001221, A267263, A276086, A328316, A328317, A328319.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328316(n) = if(!n,0,A276086(A328316(n-1)));
A267263(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += !!d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A328318(n) = A267263(A328316(n));
\\ Or alternatively, more slowly as:
A328318(n) = omega(A328316(1+n));

a(n) = A267263(A328316(n)).

a(n) = A001221(A328316(1+n)).

A328482 Number of distinct terms required when n is expressed as a greedy sum of terms of A129912 (number of nonzero digits when n is expressed in greedy A129912-base).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 19 2019

nonn

			Terms of A129912 (numbers that are products of distinct primorial numbers) begin as: 1, 2, 6, 12, 30, 60, 180, 210, 360, 420, 1260, ...
Number 5 is expressed as 5 = 2 + 2 + 1 = 2*2 + 1*1, when always choosing the largest term which is <= {what is remaining of the original number}. Thus a(5) = 2 (number of distinct terms used, 1 and 2).
Number 21 is expressed as 21 = 12 + 6 + 2 + 1, thus a(21) = 4.

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

Cf. A002110, A129912, A328481, A328483.

Cf. also A267263.

isA129912(n) = { my(o=valuation(n, 2), t); if(o<1||n<2, return(n==1)); n>>=o; forprime(p=3, , t=valuation(n, p); n/=p^t; if(t>o || tA129912
prepare_A129912_upto(n) = { my(xs=List([]), k=0); while(kA129912(k), listput(xs,k))); List(Vecrev(xs)); };
number_of_distinct_terms_in_greedy_sum(n,terms) = { my(c=0); while(n,if(terms[1] > n, listpop(terms,1), c++; n %= terms[1])); (c); };
A328482(n) = number_of_distinct_terms_in_greedy_sum(n,prepare_A129912_upto(n));

a(A129912(n)) = a(A002110(n)) = 1.

For all n, a(n) <= A328481(n).

A328615 Number of digits larger than 1 in primorial base expansion of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 22 2019

			In primorial base (A049345), 87 is written as "2411" because 87 = 2*A002110(3) + 4*A002110(2) + 1*A002110(1) + 1*A002110(0) = 2*30 + 4*6 + 1*2 + 1*1. Only the digits 2 and 4 of these are larger than one, thus a(87) = 2.

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

Cf. A001221, A002110, A049345, A267263, A276086, A328114, A328572, A328614, A328616.

a[n_] := Module[{k = n, p = 2, s = 0, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, If[r > 1, s++]; p = NextPrime[p]]; s]; Array[a, 100, 0] (* Amiram Eldar, Mar 13 2024 *)

A328615(n) = { my(s=0, p=2); while(n, s += (1<(n%p)); n = n\p; p = nextprime(1+p)); (s); };

a(n) = A267263(n) - A328614(n).

a(n) = A001221(A328572(n)).

A342461 Number of nonzero digits when A329886(n) is written in primorial base, where A329886 is the primorial inflation of Doudna-tree.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 15 2021

nonn

Cf. A001221, A005940, A108951, A267263, A329886, A342456, A342457, A342462.

Cf. also A324341, A329040.

A342461(n) = omega(A342456(n)); \\ Other code as in A342456.

a(n) = A001221(A342456(n)) = A001221(A342457(n)).
a(n) = A267263(A329886(n)) = A329040(A005940(1+n)).
a(n) <= A342462(n).
For n >= 0, a(2^n) = 1.

A346470 a(n) = psi(A276086(n)), where psi is Dedekind psi function A001615, and A276086 is the prime product form of primorial base expansion of n.

Original entry on oeis.org

1, 3, 4, 12, 12, 36, 6, 18, 24, 72, 72, 216, 30, 90, 120, 360, 360, 1080, 150, 450, 600, 1800, 1800, 5400, 750, 2250, 3000, 9000, 9000, 27000, 8, 24, 32, 96, 96, 288, 48, 144, 192, 576, 576, 1728, 240, 720, 960, 2880, 2880, 8640, 1200, 3600, 4800, 14400, 14400, 43200, 6000, 18000, 24000, 72000, 72000, 216000, 56, 168, 224, 672

Offset: 0

Antti Karttunen, Jul 21 2021

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

Cf. A001615.
Other number-theoretical functions similarly applied to A276086: A267263 (omega), A276150 (bigomega), A324650 (phi), A324653 (sigma), A324655 (tau), A327860 (arithmetic derivative).
Cf. also A346471, A346475.

A346470(n) = { my(m=1, p=2, e); while(n, e = (n%p); if(e, m *= (p+1)*(p^(e-1))); n = n\p; p = nextprime(1+p)); (m); };

a(n) = A001615(A276086(n)).

A371091 Number of 1's in the recursive decomposition of primorial base expansion of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 31 2024

Take the primorial base expansion of n (A049345), and then replace any digit larger than 1 with its own primorial base expansion, and do this recursively until no digits larger than 1 remain. a(n) is then the number of 1's in the completed decomposition. (See the examples). This decomposition offers a way to design a natural primorial based numeral system that does not require an infinite number of arbitrary glyphs for its digits, but instead suffices with just two graphically distinct subfigures whose exact positions in the whole hierarchically organized composite glyph determines the numerical value of that glyph, a bit like in Maya numerals or Babylonian cuneiform digits, but based on a primorial number system instead of vigesimal or sexagesimal.

			     n  A049345(n)     recursive              a(n) = number of 1's
                       decomposition          in the decomposition
--------------------------------------------------------------------
     0         0         ()                             0
     1         1         (1)                            1
     2        10         (1 0)                          1
     3        11         (1 1)                          2
     4        20         ((1 0) 0)                      1
     5        21         ((1 0) 1)                      2
     6       100         (1 0 0)                        1
     7       101         (1 0 1)                        2
     8       110         (1 1 0)                        2
     9       111         (1 1 1)                        3
    10       120         (1 (1 0) 0)                    2
    11       121         (1 (1 0) 1)                    3
    12       200         ((1 0) 0 0)                    1
    ..
    21       311         ((1 1) 1 1)                    4
    ..
    24       400         (((1 0) 0) 0 0)                1
    ..
    29       421         (((1 0) 0) (1 0) 1)            3
    30      1000         (1 0 0 0)                      1
    ..
    51      1311         (1 (1 1) 1 1)                  5
    ..
    59      1421         (1 ((1 0) 0) (1 0) 1)          4
    60      2000         ((1 0) 0 0 0)                  1
    ..
   111      3311         ((1 1) (1 1) 1 1)              6
   ...
   360     15000         (1 ((1 0) 1) 0 0 0)            3
   ...
  2001     93311         ((1 1 1) (1 1) (1 1) 1 1)      9
  ....
  4311    193311         (1 (1 1 1) (1 1) (1 1) 1 1)   10.
29 is decomposed in piecemeal fashion as: A049345(29) = 421 --> ("20" "10" "1") --> (((1 0) 0) (1 0) 1).

Antti Karttunen, Table of n, a(n) for n = 0..60060
Index entries for sequences related to primorial base

Cf. A049345, A267263, A276086, A276150, A371090.
Cf. A372559 (positions of records and the first occurrence of n).
Differs from A328482 for the first time at n=360, where a(360) = 3, while A328482(360) = 1.

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,A371091(e)),factor(n)[, 2]));
A371091(n) = A371090(A276086(n));

a(n) = A371090(A276086(n)).

For all n, A267263(n) <= a(n) <= A276150(n).

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

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jan 12 2020

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

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

Cf. A053669, A267263, A289234.

Cf. also A286626, A328477, A330740, A331171.

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

a(n) = min(n, A289234(n)).

A365460 Number of distinct primorials in the greedy sum of primorials adding to A181821(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 4, 1, 2, 2, 3, 1, 2, 2, 2, 2, 3, 2, 4, 1, 3, 3, 2, 1, 5, 3, 3, 1, 3, 2, 3, 2, 2, 3, 5, 1, 3, 2, 3, 2, 5, 2, 2, 1, 3, 4, 5, 2, 5, 4, 3, 1, 3, 2, 6, 2, 4, 2, 6, 1, 6, 4, 3, 3, 4, 3, 6, 1, 3, 3, 7, 2, 4, 4, 4, 2, 8, 2, 4, 3, 4, 4, 4, 1, 6, 3, 3, 2, 6, 3, 7, 2, 3

Offset: 1

Antti Karttunen, Sep 15 2023

nonn

Cf. A122111, A181821, A267263, A329040.

Cf. also A365461.

A181821(n) = { my(f=factor(n),p=0,m=1); forstep(i=#f~,1,-1,while(f[i,2], f[i,2]--; m *= (p=nextprime(p+1))^primepi(f[i,1]))); (m); };
A267263(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += !!d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A365460(n) = A267263(A181821(n));

a(n) = A267263(A181821(n)).
a(n) = A329040(A122111(n)).
a(n) <= A365461(n).

cp's OEIS Frontend

A324341 If 2n = 2^e1 + ... + 2^ek [e1 .. ek distinct], then a(n) is the number of nonzero digits when A002110(e1) * ... * A002110(ek) is written in primorial base.

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A328614 Number of 1-digits in primorial base expansion of n.

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A328318 Number of nonzero digits in representation of A328316(n) in primorial base; Number of distinct prime factors in A328316(1+n).

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A328482 Number of distinct terms required when n is expressed as a greedy sum of terms of A129912 (number of nonzero digits when n is expressed in greedy A129912-base).

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A328615 Number of digits larger than 1 in primorial base expansion of n.

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A342461 Number of nonzero digits when A329886(n) is written in primorial base, where A329886 is the primorial inflation of Doudna-tree.

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A346470 a(n) = psi(A276086(n)), where psi is Dedekind psi function A001615, and A276086 is the prime product form of primorial base expansion of n.

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A371091 Number of 1's in the recursive decomposition of primorial base expansion of n.

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