A324382 -id:A324382 - cp's OEIS frontend

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

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

A329902 Primorial deflation of the n-th highly composite number: the unique integer k such that A108951(k) = A002182(n).

Original entry on oeis.org

1, 2, 4, 3, 6, 12, 9, 24, 10, 20, 15, 40, 30, 60, 28, 21, 56, 42, 84, 63, 168, 126, 336, 140, 66, 189, 280, 132, 99, 264, 198, 528, 220, 396, 297, 440, 792, 156, 117, 312, 234, 624, 260, 468, 351, 520, 936, 390, 1040, 1872, 780, 585, 306, 1560, 340, 612, 459, 680, 1224, 510, 1360, 2448, 1020, 765, 342, 2040, 1530, 684, 513

Offset: 1

Antti Karttunen, Dec 22 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (computed from the b-file of A002182)
Michael De Vlieger, List of a(n) for n = 1..779674 (gzipped text file, prepared from Flammenkamp's data)
Michael De Vlieger, List of a(n) for n = 1..500000 (noncompressed text file, an abridged version of above)

Cf. A002182, A002183, A056239, A108951, A112778, A112779, A306802, A319626, A324381, A324382, A324581, A324582, A324888, A329040, A329605, A329900, A330743, A330748.

Subsequence of A181815.

Map[Times @@ Prime@(TakeWhile[Reap[FixedPointList[Block[{k = 1}, While[Mod[#, Prime@ k] == 0, k++]; Sow[k - 1]; #/Product[Prime@ i, {i, k - 1}]] &, #]][[-1, 1]], # > 0 &]) &, Take[Import["https://oeis.org/b002182.txt", "Data"][[All, -1]], 69] ] (* Michael De Vlieger, Jan 13 2020, imports b-file at A002182 *)

a(n) = A329900(A002182(n)) = A319626(A002182(n)).
a(n) = A181815(A306802(n)).
A108951(a(n)) = A002182(n). [Highly composite numbers (undeflated)]
A056239(a(n)) = A112778(n). [Number of prime factors, counted with multiplicity]
A001222(a(n)) = A112779(n). [Largest exponent in the prime factorization]
A329605(a(n)) = A002183(n). [Number of divisors]
A329040(a(n)) = A324381(n).
A324888(a(n)) = A324382(n).
a(A330748(n)) = A330743(n).

More linking formulas added by Antti Karttunen, Jan 13 2020

A324386 a(n) = A324383(A006068(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 4, 4, 4, 2, 2, 6, 6, 1, 2, 4, 8, 4, 4, 6, 12, 2, 8, 6, 10, 6, 22, 10, 8, 1, 4, 4, 6, 2, 8, 6, 8, 4, 6, 12, 14, 2, 16, 10, 16, 2, 8, 16, 4, 6, 14, 8, 24, 6, 30, 18, 20, 6, 26, 18, 26, 1, 6, 8, 8, 4, 12, 12, 6, 8, 12, 14, 18, 4, 20, 20, 20, 4, 16, 16, 8, 12, 28, 16, 10, 12, 22, 26, 14, 12, 34, 20, 22, 2, 12

Offset: 0

Antti Karttunen, Feb 27 2019

nonn

This is most likely equal to A276150(A086141(n)), apart from the different offset used in A086141.

The same comments about the parity of terms as in A324383 and A324387 apply also here, except here 1's occur at positions given by 2^k - 1.

Cf. A002110, A006068, A025487, A086141, A276150, A322827, A324342, A324382.

Cf. also A324383, A324387 (permutations of this sequence) and A324380, A324390.

A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
A276150(n) = { my(s=0,m); forprime(p=2, , if(!n, return(s)); m = n%p; s += m; n = (n-m)/p); };
A322827(n) = if(!n,1,my(bits = Vecrev(binary(n)), rl=1, o = List([])); for(i=2,#bits,if(bits[i]==bits[i-1], rl++, listput(o,rl))); listput(o,rl); my(es=Vecrev(Vec(o)), m=1); for(i=1,#es,m *= prime(i)^es[i]); (m));
A324383(n) = A276150(A322827(n));
A324386(n) = A324383(A006068(n));

a(n) = A324383(A006068(n)) = A276150(A322827(A006068(n))).

a(A000225(n)) = 1 for all n.

A324387 Minimal number of primorials (A002110) that add to the n-th number that is a product of primorials: a(n) = A276150(A025487(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 4, 4, 1, 2, 2, 4, 2, 4, 4, 4, 4, 6, 8, 6, 8, 1, 2, 2, 6, 6, 6, 10, 2, 4, 4, 6, 8, 6, 10, 4, 8, 6, 8, 12, 6, 10, 6, 8, 12, 10, 8, 12, 12, 10, 16, 12, 20, 1, 2, 6, 8, 10, 6, 10, 8, 10, 16, 14, 20, 2, 4, 12, 10, 10, 14, 10, 16, 12, 20, 6, 6, 10, 8, 10, 12, 20, 4, 8, 14, 14, 20, 14, 10, 16, 14, 24, 6, 12, 12

Offset: 1

Antti Karttunen, Feb 27 2019

nonn

A098719 gives the positions of ones in this sequence. See also comments in A324383.

Antti Karttunen, Table of n, a(n) for n = 1..10001 (based on the b-file of A025487 as computed by Will Nicholes and Franklin T. Adams-Watters)
Index entries for sequences related to primorial base
Index entries for sequences related to primorial numbers

Cf. A002110, A025487, A098719 (positions of ones), A276150, A324342.

Cf. A324382 for a subsequence, and A324383, A324386 for permutations of this sequence.

A276150(n) = { my(s=0,m); forprime(p=2, , if(!n, return(s)); m = n%p; s += m; n = (n-m)/p); };
A324387(n) = A276150(A025487(n));

a(n) = A276150(A025487(n)).

A324383 a(n) is the minimal number of primorials that add to A322827(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 2, 2, 6, 1, 6, 4, 2, 4, 4, 8, 6, 6, 10, 8, 1, 10, 22, 4, 6, 2, 12, 8, 4, 4, 2, 8, 16, 6, 4, 24, 6, 8, 14, 26, 18, 1, 26, 20, 6, 18, 30, 6, 12, 2, 14, 16, 2, 10, 16, 8, 6, 4, 8, 6, 2, 4, 4, 12, 14, 14, 18, 18, 12, 16, 32, 42, 28, 6, 22, 32, 24, 24, 42, 46, 32, 18, 20, 30, 1, 24, 54, 38, 26, 14, 44, 34, 8

Offset: 0

Antti Karttunen, Feb 27 2019

nonn

a(n) is odd if and only if n is one of the terms of A000975: 1, 2, 5, 10, 21, 42, 85, ..., in which case A322827(n) will be one of primorials (A002110), and a(n) = 1. This happens because A276150 is even for all multiples of four, and a product of two or more primorials > 1 is always a multiple of 4. Note that the same property does not hold in factorial system: 36 = 3!*3!, but A034968(36) = 3 as 36 = 4!+3!+3!.

Cf. A000975 (positions of ones), A002110, A003188, A025487, A276150, A322827, A324342, A324382.

Cf. also A324386, A324387 (permutations of this sequence).

A276150(n) = { my(s=0,m); forprime(p=2, , if(!n, return(s)); m = n%p; s += m; n = (n-m)/p); };
A322827(n) = if(!n,1,my(bits = Vecrev(binary(n)), rl=1, o = List([])); for(i=2,#bits,if(bits[i]==bits[i-1], rl++, listput(o,rl))); listput(o,rl); my(es=Vecrev(Vec(o)), m=1); for(i=1,#es,m *= prime(i)^es[i]); (m));
A324383(n) = A276150(A322827(n));

a(n) = A276150(A322827(n)).

a(n) = A324386(A003188(n)).

A324582 a(n) = A002182(n) * A324581(n) = A002182(n) * A276086(A002182(n)).

Original entry on oeis.org

2, 6, 36, 30, 300, 15000, 1260, 42000, 2940, 288120, 21176820, 18480, 66555720, 328703760, 12298440, 2232166860, 360122920080, 360360, 103062960, 22107004920, 4215068938080, 129290917072196880, 3525159950945805332160, 90107494796113466546674800, 645822919595173320, 72532204477502449680, 1648012277067163992784800

Offset: 1

Antti Karttunen, Mar 09 2019

nonn

Note that gcd(A002182(n), A324581(n)) = A324198(A002182(n)) = 1 for all n because each term of A002182 is a product of primorial numbers (A002110).

A324381 Number of nonzero digits when the n-th highly composite number is written in primorial base: a(n) = A267263(A002182(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 26 2019

nonn

			For n=12, A002182(12) = 240, which is written as "11000" in primorial base (A049345) because 240 = 1*A002110(4) + 1*A002110(3) = 210+30, thus a(12) = 2, as there are two nonzero digits.
For n=18, A002182(18) = 2520 = "110000" in primorial base because 2520 = 1*A002110(5) + 1*A002110(4) = 2310+210, thus a(18) = 2.
For n=26, A002182(26) = 45360 = "1670000" in primorial base because 45360 = 1*A002110(6) + 6*A002110(5) + 7*A002110(4), thus a(26) = 3, as there are three nonzero digits.

Cf. A002100, A002182, A049345, A267263, A324382.

Cf. also A324341.

A267263(n) = { my(s=0); forprime(p=2, , if(n%p, s++, if(n==0, return(s))); n\=p); }; \\ From A267263
A324381(n) = A267263(A002182(n));

a(n) = A267263(A002182(n)).

a(n) <= A324382(n).

A324581 a(n) = A276086(A002182(n)).

Original entry on oeis.org

2, 3, 9, 5, 25, 625, 35, 875, 49, 2401, 117649, 77, 184877, 456533, 14641, 1771561, 214358881, 143, 20449, 2924207, 418161601, 8550986578849, 174859124550883201, 3575694237941010577249, 23298085122481, 1599034490244763, 32698656291015158587, 30466726698629, 39841104144361, 52099905419549, 89093921102069, 152355876914189, 260537564663909

Offset: 1

Antti Karttunen, Mar 09 2019

nonn

Note that gcd(a(n), A002182(n)) = A324198(A002182(n)) = 1 for all n because each term of A002182 is a product of primorial numbers (A002110).

Antti Karttunen, Table of n, a(n) for n = 1..670
Michael De Vlieger, Small chart of first terms.
Michael De Vlieger, 2400 x 3600 chart of first terms.
Index entries for sequences related to primorial base
Index entries for sequences related to primorial numbers

Subsequence of A324576.

Cf. A002110, A002182, A276086, A324198, A324381, A324382, A324582.

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 20], s = DivisorSigma[0, Range[10^5]], t}, t = Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]; Array[Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[(*a002182[[#]]*)t[[#]], b] &, Length@ t]] (* Michael De Vlieger, Mar 18 2019 *)

\\ A002182 assumed to be precomputed
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; };
A324581(n) = A276086(A002182(n));

a(n) = A276086(A002182(n)).
a(n) = A324582(n)/A002182(n).
A001221(a(n)) = A324381(n).
A001222(a(n)) = A324382(n).

cp's OEIS Frontend

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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A329902 Primorial deflation of the n-th highly composite number: the unique integer k such that A108951(k) = A002182(n).

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A324386 a(n) = A324383(A006068(n)).

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A324387 Minimal number of primorials (A002110) that add to the n-th number that is a product of primorials: a(n) = A276150(A025487(n)).

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A324383 a(n) is the minimal number of primorials that add to A322827(n).

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A324582 a(n) = A002182(n) * A324581(n) = A002182(n) * A276086(A002182(n)).

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A324381 Number of nonzero digits when the n-th highly composite number is written in primorial base: a(n) = A267263(A002182(n)).

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A324581 a(n) = A276086(A002182(n)).

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