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 1-3 of 3 results.

A377878 Numbers k for which A276085(k) is a multiple of 3125, where A276085 is fully additive with a(p) = p#/p.

Original entry on oeis.org

1, 4823, 8267, 9553, 15623, 15833, 15929, 20633, 23393, 28417, 33079, 34027, 36941, 37129, 37939, 42599, 43249, 44431, 47291, 49374, 60097, 65832, 66323, 69287, 69749, 70613, 74063, 74281, 74333, 74999, 77231, 83881, 86191, 86551, 87776, 88727, 99683, 106481, 108673, 111366, 113922, 115729, 118517, 124841, 126054, 129337
Offset: 1

Views

Author

Antti Karttunen, Nov 13 2024

Keywords

Comments

A multiplicative semigroup; if m and n are in the sequence then so is m*n.
Question: Does this sequence have asymptotic density? See also questions in A377872 and A377869.

Links

Crossrefs

Cf. A000720, A276085, A377869, A377877.
Subsequence of A373140, and of A377873.
Cf. also A377872.

Programs

  • PARI
    isA377878(n) = { my(m=5^5, f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= Mod(prime(i),m); i++); s += f[k, 2]*pr); (0==lift(s)); };

Formula

{k such that Sum e*A377877(A000720(p)-1) == 0 (mod 5^5), when k = Product(p^e)}.

A377876 The n-th primorial number reduced modulo 27.

Original entry on oeis.org

1, 2, 6, 3, 21, 15, 6, 21, 21, 24, 21, 3, 3, 15, 24, 21, 6, 3, 21, 3, 24, 24, 6, 12, 15, 24, 21, 3, 24, 24, 12, 12, 6, 12, 21, 24, 6, 24, 24, 12, 24, 3, 3, 6, 24, 3, 3, 12, 3, 6, 24, 3, 15, 24, 3, 15, 3, 24, 24, 6, 12, 21, 24, 24, 12, 3, 6, 15, 6, 3, 21, 15, 12, 3, 12, 12, 6, 12, 12, 6, 24, 12, 3, 24, 24, 6, 12, 15
Offset: 0

Views

Author

Antti Karttunen, Nov 12 2024

Keywords

Links

Crossrefs

Cf. A002110, A377872.
Cf. also A086360, A377877.

Programs

  • Maple
    R:= 1: p:= 1: v:= 1:
for i from 1 to 100 do
  p:= nextprime(p); v:= p*v mod 27;
  R:= R,v;
od:
R; # Robert Israel, Nov 12 2024
  • Mathematica
    Mod[FoldList[Times,1,Prime[Range[87]]],27] (* James C. McMahon, Nov 12 2024 *)
  • PARI
    A002110(n) = prod(i=1,n,prime(i));
A377876(n) = (A002110(n)%27);
  • PARI
    up_to = 105;
A377876list(up_to_n) = { my(m=27, v=vector(1+up_to_n), pr=1); v[1] = 1; for(n=1, up_to_n, pr *= Mod(prime(n),m); v[1+n] = lift(pr)); (v); };
v377876 = A377876list(up_to);
A377876(n) = v377876[1+n];
  • Python
    from functools import reduce
from sympy import primerange, prime
def A377876(n): return reduce(lambda x,y:x*y%27,primerange(prime(n)+1)) if n else 1 # Chai Wah Wu, Nov 12 2024

Formula

a(n) = A002110(n) mod 27.

A086360 The n-th primorial number reduced modulo 9.

Original entry on oeis.org

1, 2, 6, 3, 3, 6, 6, 3, 3, 6, 3, 3, 3, 6, 6, 3, 6, 3, 3, 3, 6, 6, 6, 3, 6, 6, 3, 3, 6, 6, 3, 3, 6, 3, 3, 6, 6, 6, 6, 3, 6, 3, 3, 6, 6, 3, 3, 3, 3, 6, 6, 3, 6, 6, 3, 6, 3, 6, 6, 6, 3, 3, 6, 6, 3, 3, 6, 6, 6, 3, 3, 6, 3, 3, 3, 3, 6, 3, 3, 6, 6, 3, 3, 6, 6, 6, 3, 6
Offset: 0

Views

Author

Labos Elemer, Jul 21 2003

Keywords

Comments

a(n) is the fixed point reached by decimal-digit-sum-function (A007953), when starting the iteration from the value of the n-th primorial, A002110(n). - The (edited) original definition of the sequence, which is equal to a simple definition a(n) = A002110(n) mod 9, because taking the decimal digit sum preserves congruence modulo 9. - Antti Karttunen, Nov 14 2024
Only a(0)=1 and a(1)=2; each subsequent term is either a 3 or a 6.

Examples

			For n=7, 7th primorial = 510510, list of iterated digit sums is {510510,12,3}, thus a(7)=3.

Links

Crossrefs

Cf. A002110, A007953, A010878, A010888, A029898, A038194, A086353-A086361.
Cf. also A377876, A377877.

Programs

  • Maple
    A086360 := proc(n) option remember: if(n=1)then return 2:fi: return ithprime(n)*procname(n-1) mod 9: end: seq(A086360(n), n=1..100); # Nathaniel Johnston, May 04 2011
  • Mathematica
    sud[x_] := Apply[Plus, DeleteCases[IntegerDigits[x], 0]] q[x_] := Apply[Times, Table[Prime[w], {w, 1, x}]] Table[FixedPoint[sud, q[w]], {w, 1, 128}]
  • PARI
    up_to = 19683;
A086360list(up_to_n) = { my(m=9, v=vector(1+up_to_n), pr=1); v[1] = 1; for(n=1, up_to_n, pr = (pr*prime(n))%m; v[1+n] = pr); (v); };
v086360 = A086360list(up_to);
A086360(n) = v086360[1+n]; \\ Antti Karttunen, Nov 14 2024

Formula

a(n) = A010878(A002110(n)) = A002110(n) mod 9.
a(n) = A010888(A002110(n)).

Extensions

Term a(0)=1 prepended, old definition moved to comments and replaced with one of the formulas, keyword:base removed because not really base-dependent - Antti Karttunen, Nov 14 2024
Showing 1-3 of 3 results.

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