cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

Antti Karttunen, Aug 22 2016

Keywords

Comments

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

Examples

			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.
		

Crossrefs

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. A014601, A042963 (positions of even and odd terms), A343048 (positions of records).
Differs from analogous A034968 for the first time at n=24.

Programs

  • Mathematica
    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 *)
  • PARI
    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
  • Python
    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
    

Formula

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

A333426 Primorial base Niven numbers: numbers divisible by their sum of digits in primorial base (A276150).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 25, 30, 32, 33, 36, 40, 42, 44, 45, 48, 50, 60, 64, 65, 66, 68, 70, 72, 77, 84, 88, 90, 92, 96, 105, 108, 112, 117, 120, 132, 133, 136, 144, 150, 154, 156, 160, 168, 180, 182, 184, 189, 192, 198, 200, 210, 212, 213, 216, 220
Offset: 1

Views

Author

Amiram Eldar, Mar 20 2020

Keywords

Comments

Numbers k for which A276086(k) is in A373852. - Antti Karttunen, Jun 22 2024

Examples

			1 is a term since A276150(1) = 1 divides 1;
2 is a term since A276150(2) = 1 divides 2;
		

Crossrefs

Programs

  • Mathematica
    max = 5; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; sumdig[n_] := Plus @@ IntegerDigits[n, MixedRadix[bases]]; Select[Range[nmax], Divisible[#, sumdig[#]] &]
  • PARI
    isA333426 = A373834; \\ Antti Karttunen, Jun 22 2024

A373833 a(n) = gcd(n, A276150(n)), where A276150 is the digit sum in the primorial base.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 4, 1, 1, 1, 4, 5, 1, 3, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 4, 5, 1, 1, 4, 1, 5, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 5, 3, 1, 4, 1, 5, 1, 4, 1, 1, 3, 2, 7, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 8, 1, 3, 1, 4, 1, 1, 1, 4, 1, 1, 3, 2, 1, 1, 1, 2, 7
Offset: 0

Views

Author

Antti Karttunen, Jun 19 2024

Keywords

Crossrefs

Cf. A049345, A276086, A276150, A333426 [positive k for which a(k) = A276150(k)], A358977 (indices of 1's), A373832, A373834, A373835, A373838, A373839 (indices of multiples of 3).

Programs

  • PARI
    A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
    A373833(n) = gcd(n, A276150(n));

Formula

a(n) = A373835(A276086(n)).
For n >= 1, a(n) = gcd(A276150(n), A373832(n)).

A358975 Numbers that are coprime to their digital sum in base 3 (A053735).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 49, 51, 53, 55, 59, 61, 67, 69, 71, 73, 79, 81, 83, 85, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 123, 125, 127, 129, 131, 137, 139, 141, 143, 147, 149, 151, 153, 155, 157, 159, 161, 163, 167, 169
Offset: 1

Views

Author

Amiram Eldar, Dec 07 2022

Keywords

Comments

Numbers k such that gcd(k, A053735(k)) = 1.
All the terms are odd since if k is even then A053735(k) is even and so gcd(k, A053735(k)) >= 2.
Olivier (1975, 1976) proved that the asymptotic density of this sequence is 4/Pi^2 = 0.40528... (A185199).
The powers of 3 (A000244) are terms. These are also the only ternary Niven numbers (A064150) in this sequence.
Includes all the odd prime numbers (A065091).

Examples

			3 is a term since A053735(3) = 1, and gcd(3, 1) = 1.
		

Crossrefs

Subsequences: A000244, A065091.
Similar sequences: A094387, A339076, A358976, A358977, A358978.

Programs

  • Mathematica
    q[n_] := CoprimeQ[n, Plus @@ IntegerDigits[n, 3]]; Select[Range[200], q]
  • PARI
    is(n) = gcd(n, sumdigits(n, 3)) == 1;

A358976 Numbers that are coprime to the sum of their factorial base digits (A034968).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 25, 28, 29, 31, 32, 33, 37, 39, 41, 43, 44, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 76, 77, 79, 83, 84, 85, 87, 88, 89, 92, 93, 95, 97, 98, 101, 102, 103, 106, 107, 109, 110
Offset: 1

Views

Author

Amiram Eldar, Dec 07 2022

Keywords

Comments

Numbers k such that gcd(k, A034968(k)) = 1.
The factorial numbers (A000142) are terms. These are also the only factorial base Niven numbers (A118363) in this sequence.
Includes all the prime numbers.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 7, 59, 601, 6064, 60729, 607567, 6083420, 60827602, 607643918, 6079478119, ... . Conjecture: The asymptotic density of this sequence exists and equals 6/Pi^2 = 0.607927... (A059956), the same as the density of A094387.

Examples

			3 is a term since A034968(3) = 2, and gcd(3, 2) = 1.
		

Crossrefs

Subsequences: A000040, A000142.
Similar sequences: A094387, A339076, A358975, A358977, A358978.

Programs

  • Mathematica
    q[n_] := Module[{k = 2, s = 0, m = n, r}, While[m > 0, r=Mod[m,k]; s+=r; m=(m-r)/k; k++]; CoprimeQ[n, s]]; Select[Range[120], q]
  • PARI
    is(n)={my(k=2, s=0, m=n); while(m>0, s+=m%k; m\=k; k++); gcd(s,n)==1;}

A358978 Numbers that are coprime to the number of terms in their Zeckendorf representation (A007895).

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 13, 15, 17, 19, 20, 21, 23, 25, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 47, 49, 50, 51, 52, 53, 55, 57, 59, 61, 62, 63, 64, 65, 67, 70, 71, 73, 75, 77, 79, 83, 85, 87, 88, 89, 91, 95, 97, 98, 100, 101, 103, 104, 107, 109
Offset: 1

Views

Author

Amiram Eldar, Dec 07 2022

Keywords

Comments

First differs from A063743 at n = 22.
Numbers k such that gcd(k, A007895(k)) = 1.
The Fibonacci numbers (A000045) are terms. These are also the only Zeckendorf-Niven numbers (A328208) in this sequence.
Includes all the prime numbers.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 7, 61, 614, 6028, 61226, 606367, 6041106, 61235023, 612542436, 6034626175, 60093287082, 609082612171, ... . Conjecture: The asymptotic density of this sequence exists and equals 6/Pi^2 = 0.607927... (A059956), the same as the density of A094387.

Examples

			3 is a term since A007895(3) = 1, and gcd(3, 1) = 1.
		

Crossrefs

Subsequences: A000040, A000045.
Similar sequences: A094387, A339076, A358975, A358976, A358977.

Programs

  • Mathematica
    z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; Select[Range[120], CoprimeQ[#, z[#]] &] (* after Alonso del Arte at A007895 *)
  • PARI
    is(n) = if(n<4, 1, my(k=2, m=n, s, t); while(fibonacci(k++)<=m, ); while(k && m, t=fibonacci(k); if(t<=m, m-=t; s++); k--); gcd(n, s)==1); \\ after Charles R Greathouse IV at A007895
Showing 1-6 of 6 results.