A319715 -id:A319715 - cp's OEIS frontend

A333623 Numbers k such that k is divisible by the sum of digits of all the divisors of k in primorial base (A319715).

1, 2, 3, 4, 14, 22, 40, 64, 90, 104, 120, 160, 169, 175, 182, 220, 272, 275, 338, 360, 500, 550, 640, 646, 752, 775, 792, 858, 928, 930, 1120, 1230, 1280, 1332, 1496, 1710, 2050, 2204, 2303, 2368, 2475, 2584, 2632, 2640, 2806, 2838, 2886, 2898, 3002, 3174, 3192

Offset: 1

Amiram Eldar, Mar 29 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A049345, A093705, A276150, A319715, A333617, A333619, A333620, A333622.

max = 5; bases = Prime@Range[max, 1, -1]; nmax = Times @@ bases - 1; primDigSum[n_] := Plus @@ IntegerDigits[n, MixedRadix[bases]]; primDivDigDum[n_] := DivisorSum[n, primDigSum[#] &]; Select[Range[nmax], Divisible[#, primDivDigDum[#]] &]

14 is a term since its divisors are {1, 2, 7, 14}, their representations in primorial base (A049345) are {1, 10, 101, 210}, and their sum of sums of digits is 1 + (1 + 0) + (1 + 0 + 1) + (2 + 1 + 0) = 7 which is a divisor of 14.

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

A319713 Sum of A276150(d) over proper divisors d of n, where A276150 gives the sum of digits in primorial base.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 4, 3, 5, 1, 7, 1, 4, 6, 6, 1, 8, 1, 10, 5, 6, 1, 11, 4, 5, 6, 9, 1, 15, 1, 10, 7, 7, 6, 15, 1, 6, 6, 16, 1, 15, 1, 13, 13, 8, 1, 19, 3, 13, 8, 12, 1, 17, 8, 17, 7, 9, 1, 24, 1, 4, 13, 12, 7, 17, 1, 12, 9, 17, 1, 23, 1, 5, 15, 11, 7, 17, 1, 24, 12, 7, 1, 28, 9, 6, 10, 19, 1, 27, 6, 15, 5, 8, 8, 25, 1, 12, 13, 24, 1, 19, 1, 20, 21

Offset: 1

Antti Karttunen, Oct 02 2018

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

Cf. A276150, A319711, A319715.

d[n_] := Module[{k = n, p = 2, s = 0, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, s += r; p = NextPrime[p]]; s]; a[n_] := DivisorSum[n, d[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Mar 05 2024 *)

A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A319713(n) = sumdiv(n,d,(dA276150(d));

a(n) = Sum_{d|n, dA276150(d).

a(n) = A319715(n) - A276150(n).

A319712 Sum of A034968(d) over divisors d of n, where A034968 gives the sum of digits in factorial base.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 3, 6, 6, 8, 5, 9, 4, 7, 10, 10, 6, 11, 5, 14, 10, 11, 7, 12, 6, 7, 9, 12, 5, 17, 4, 13, 11, 11, 11, 18, 5, 10, 11, 21, 7, 19, 6, 18, 19, 14, 8, 18, 6, 13, 12, 13, 6, 17, 12, 18, 12, 11, 7, 29, 6, 10, 19, 19, 14, 23, 7, 19, 16, 25, 9, 24, 5, 10, 17, 17, 13, 19, 6, 30, 15, 14, 8, 31, 15, 13, 14, 27, 9, 35, 13, 23, 14, 17, 17

Offset: 1

Antti Karttunen, Oct 02 2018

Inverse Möbius transform of A034968.

Antti Karttunen, Table of n, a(n) for n = 1..40320
Index entries for sequences related to factorial base representation.

Cf. A034968, A319711, A319715.

d[n_] := Module[{k = n, m = 2, s = 0, r}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, s += r; m++]; s]; a[n_] := DivisorSum[n, d[#] &]; Array[a, 100] (* Amiram Eldar, Feb 14 2024 *)

A034968(n) = { my(s=0, b=2, d); while(n, d = (n%b); s += d; n = (n-d)/b; b++); (s); };
A319712(n) = sumdiv(n,d,A034968(d));

a(n) = Sum_{d|n} A034968(d).

a(n) = A319711(n) + A034968(n).

A355686 Dirichlet inverse of A276150, where A276150(n) is the sum of digits when n is written in primorial base.

Original entry on oeis.org

1, -1, -2, -1, -3, 3, -2, 1, 1, 3, -4, 2, -3, 1, 8, -1, -5, -5, -4, 5, 3, 3, -6, -7, 4, 1, -2, 2, -7, -11, -2, 3, 13, 7, 8, 4, -3, 5, 8, -5, -5, -1, -4, 10, -7, 7, -6, 8, -1, -4, 14, 7, -7, 9, 18, 1, 9, 7, -8, -16, -3, 1, 4, -1, 13, -17, -4, 7, 19, -3, -6, 16, -5, 1, -16, 4, 9, -7, -6, 3, 6, 3, -8, -5, 23, 1, 20

Offset: 1

Antti Karttunen, Jul 14 2022

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

Cf. A276150, A355687.

Cf. also A319715.

A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
memoA355686 = Map();
A355686(n) = if(1==n,1,my(v); if(mapisdefined(memoA355686,n,&v), v, v = -sumdiv(n,d,if(dA276150(n/d)*A355686(d),0)); mapput(memoA355686,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA276150(n/d) * a(d).

a(n) = A355687(n) - A276150(n).

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A333623 Numbers k such that k is divisible by the sum of digits of all the divisors of k in primorial base (A319715).

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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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A319713 Sum of A276150(d) over proper divisors d of n, where A276150 gives the sum of digits in primorial base.

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A319712 Sum of A034968(d) over divisors d of n, where A034968 gives the sum of digits in factorial base.

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A355686 Dirichlet inverse of A276150, where A276150(n) is the sum of digits when n is written in primorial base.

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