A377876 -id:A377876 - cp's OEIS frontend

A377872 Numbers k for which A276085(k) is a multiple of 27, where A276085 is fully additive with a(p) = p#/p.

1, 55, 95, 115, 155, 174, 187, 203, 232, 265, 282, 297, 323, 325, 329, 335, 376, 391, 396, 438, 462, 474, 511, 513, 515, 527, 528, 539, 553, 584, 606, 616, 621, 632, 649, 654, 678, 684, 704, 707, 745, 763, 791, 798, 808, 828, 837, 872, 901, 904, 906, 912, 913, 931, 966, 978, 1002, 1057, 1064, 1073, 1074, 1075, 1104, 1105

Offset: 1

Antti Karttunen, Nov 10 2024

nonn

A multiplicative semigroup; if m and n are in the sequence then so is m*n.
From Antti Karttunen, Nov 17 2024: (Start)
Question: What is the asymptotic density of this sequence? There are 1, 3, 56, 484, 4899, 50034, 508254 terms <= 10^k, for k=1..7. See also questions in A377869 and in A377878.
If 3*x is a term, then 4*x is also a term, and vice versa.
Contains no even semiprimes (A100484), semiprimes of the form 3*prime (A001748), nor terms of the form 4*prime (A001749).
(End)

Antti Karttunen, Table of n, a(n) for n = 1..12000
Index entries for sequences related to primorial numbers

Cf. A276085, A377869, A377876, A377878.
Subsequence of A339746, and of A377873.
Cf. also A369007, A377875.

isA377872(n) = { my(m=27, 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)); };

{k such that Sum e*A377876(A000720(p)-1) == 0 (mod 27), when k = Product(p^e)}.

A377877 The n-th primorial number reduced modulo 3125.

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 1905, 1135, 2815, 2245, 2605, 2630, 435, 2210, 1280, 785, 980, 1570, 2020, 965, 2890, 1595, 1005, 2165, 2060, 2945, 570, 2460, 720, 355, 2615, 855, 2630, 935, 1840, 2285, 1285, 1745, 60, 645, 2210, 1840, 1790, 1265, 395, 2815, 810, 2160, 430, 735, 2690, 1770, 1155, 230, 1480, 2235, 305, 795

Offset: 0

Antti Karttunen, Nov 13 2024

Antti Karttunen, Table of n, a(n) for n = 0..15625
Index entries for sequences related to primorial numbers

Cf. A002110.

Cf. also A086360, A377876.

Mod[FoldList[Times, 1, Prime[Range[100]]], 3125] (* Paolo Xausa, Nov 13 2024 *)

A002110(n) = prod(i=1,n,prime(i));
A377877(n) = (A002110(n)%3125);

up_to = 15625;
A377877list(up_to_n) = { my(m=5^5, 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); };
v377877 = A377877list(up_to);
A377877(n) = v377877[1+n];

a(n) = A002110(n) mod (5^5).

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

Labos Elemer, Jul 21 2003

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.

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

Antti Karttunen, Table of n, a(n) for n = 0..19683 (terms 1..10000 from Nathaniel Johnston)
Index entries for sequences related to primorial numbers

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

Cf. also A377876, A377877.

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

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}]

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

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

a(n) = A010888(A002110(n)).

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

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A377872 Numbers k for which A276085(k) is a multiple of 27, where A276085 is fully additive with a(p) = p#/p.

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A377877 The n-th primorial number reduced modulo 3125.

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A086360 The n-th primorial number reduced modulo 9.

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