A100778 -id:A100778 - cp's OEIS frontend

A363455 The number of distinct primorial numbers (A002110) larger than 1 in the representation of A025487(n) as a product of primorial numbers.

0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 3, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 1, 3, 2, 1, 3, 2, 3, 2, 2, 2, 2, 2, 1, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 3, 2, 2, 3, 2, 3, 3

Offset: 1

Amiram Eldar, Jun 03 2023

nonn

The number of distinct exponents in the prime factorization of A025487(n).
The minimal number of powers of primorial numbers (A100778) in the representation of A025487(n) as a product of powers of primorial numbers.
The record values are all the nonnegative integers. The positions of the records are the positions of the terms of the Chernoff sequence (A006939) in A025487, i.e., the first position of k, for k = 0, 1, 2, ..., is A363456(k).

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

Cf. A002110, A006939, A025487, A051282, A071625, A100778, A304886, A363456.

e[1] = 0; e[n_] := Length[Union[FactorInteger[n][[;; , 2]]]]; s = {0}; Do[If[GreaterEqual @@ (f = FactorInteger[n])[[;; , 2]] && PrimePi[f[[-1, 1]]] == Length[f], AppendTo[s, e[n]]], {n, 2, 10000}]; s

a(n) = A071625(A025487(n)).

A343458 Distinct values of the least common multiple of initial segments of numbers of least prime signature (A025487).

Original entry on oeis.org

1, 2, 4, 12, 24, 48, 240, 480, 1440, 2880, 5760, 40320, 120960, 241920, 483840, 2419200, 4838400, 14515200, 29030400, 319334400, 638668800, 1916006400, 3832012800, 7664025600, 38320128000, 498161664000, 996323328000, 6974263296000, 20922789888000, 41845579776000, 83691159552000

Offset: 1

Hal M. Switkay, Apr 15 2021

nonn

The least common multiple of all numbers of least prime signature (A025487) <= c equals the least common multiple of all primorial powers (A100778) <= c, where c is an arbitrary positive real number.
The terms of this sequence are themselves numbers of least prime signature. Write a(n) in its prime factorization, Product_{i=1..k} A000040(i)^e_i. Then e_i is approximately proportional to 1/log_2(A002110(i)).
More precisely, the least common multiple of all numbers of least prime signature (A025487) <= c has prime factorization Product_{i>=1} A000040(i)^e_i, where e_i = floor(log(c)/log(A002110(i))).

			The least common multiple of the numbers of least prime signature up through 36 is equal to the least common multiple of all primorial powers up through 36, including 2^5 = 32, 6^2 = 36, and 30^1 = 30. Thus 2^5 * 3^2 * 5 = 1440 is a term of this sequence.

David A. Corneth, Table of n, a(n) for n = 1..1317

Cf. A025487, A100778.

a(1) = 1, a(n) = lcm(a(n-1), A100778(n)) for n >= 2. - David A. Corneth, Apr 18 2021

More terms from David A. Corneth, Apr 18 2021

A334990 a(1) = 1 and for any n > 1 with prime factorization Product_{k = 1..m} prime(k)^e_k (where prime(k) denotes the k-th prime number and e_m > 0), a(n) = Product_{k = 1..m-1} prime(k)^(e_k XOR e_{k+1}) (where XOR denotes the bitwise XOR operator).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 5, 1, 4, 6, 7, 8, 11, 10, 2, 1, 13, 8, 17, 12, 30, 14, 19, 4, 9, 22, 8, 20, 23, 1, 29, 1, 42, 26, 3, 1, 31, 34, 66, 24, 37, 15, 41, 28, 108, 38, 43, 32, 25, 18, 78, 44, 47, 4, 105, 40, 102, 46, 53, 8, 59, 58, 180, 1, 165, 21, 61, 52, 114, 6

Offset: 1

Rémy Sigrist, May 18 2020

This sequence has similarities with A038554; here we consider prime exponents, there binary digits.

Index entries for sequences related to XOR-triangles

Cf. A006530, A038554, A100778.

a(n) = { my (v=1, p=2, e=valuation(n,p)); n/=p^e; forprime (q=p+1, oo, if (n==1, return (v), my (f=valuation(n,q)); n/=q^f; v*=p^bitxor(e,f); [p,e]=[q,f])) }

a(n) = 1 iff n belongs to A100778.
a(n^2) = a(n)^2.
a(n^k) = a(n)^k for any k >= 0 and any squarefree number n.
a(prime(n+1)) = prime(n).
A006530(a(n)) < A006530(n) for any n > 1.

A343534 Quotients of successive terms of A343458.

Original entry on oeis.org

2, 2, 3, 2, 2, 5, 2, 3, 2, 2, 7, 3, 2, 2, 5, 2, 3, 2, 11, 2, 3, 2, 2, 5, 13, 2, 7, 3, 2, 2, 2, 3, 17, 2, 5, 2, 3, 2, 2, 11, 2, 7, 19, 3, 2, 5, 2, 3, 2, 2, 23, 2, 3, 2, 5, 13, 2, 7, 2, 3, 2, 29, 2, 11, 3, 2, 5, 2, 2, 3, 2, 31, 17, 2, 7, 3, 2, 5, 2, 2, 3, 2, 37

Offset: 1

Hal M. Switkay, Apr 18 2021

nonn

The entries in this sequence are computed as follows. a(n) is that prime number p such that A100778(n+1) is a power of p#.

Hal M. Switkay, Table of n, a(n) for n = 1..352

Cf. A100778, A343458.

A384180 Irregular triangle read by rows where row n lists the Heinz numbers of all uniform (equal multiplicities) and normal (covering an initial interval) multisets of length n.

Original entry on oeis.org

2, 4, 6, 8, 30, 16, 36, 210, 32, 2310, 64, 216, 900, 30030, 128, 510510, 256, 1296, 44100, 9699690, 512, 27000, 223092870, 1024, 7776, 5336100, 6469693230, 2048, 200560490130, 4096, 46656, 810000, 9261000, 901800900, 7420738134810, 8192, 304250263527210

Offset: 1

Gus Wiseman, May 25 2025

A permutation of A100778 (powers of primorials).
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
An integer partition is uniform iff all parts appear with the same multiplicity, and normal iff it covers an initial interval of positive integers.

			The uniform normal multisets of length 6 are: {1,1,1,1,1,1}, {1,1,1,2,2,2}, {1,1,2,2,3,3}, {1,2,3,4,5,6}, so row 6 is: 64, 216, 900, 30030.
Triangle begins:
    2
    4       6
    8      30
   16      36    210
   32    2310
   64     216    900    30030
  128  510510
  256    1296  44100  9699690

Row lengths are A000005.
Final term in each row is A002110.
The union is A100778.
Reversing rows gives A322792.
For just normal multisets we have A324939.
A047966 counts uniform partitions.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A381431 is the section-sum transform.
Cf. A000009, A001694, A056239, A106529, A112798, A116540, A122111, A258280, A325326, A325337.

Table[Table[Times@@Prime/@Range[d]^(n/d),{d,Divisors[n]}],{n,10}]

cp's OEIS Frontend

A363455 The number of distinct primorial numbers (A002110) larger than 1 in the representation of A025487(n) as a product of primorial numbers.

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A343458 Distinct values of the least common multiple of initial segments of numbers of least prime signature (A025487).

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A334990 a(1) = 1 and for any n > 1 with prime factorization Product_{k = 1..m} prime(k)^e_k (where prime(k) denotes the k-th prime number and e_m > 0), a(n) = Product_{k = 1..m-1} prime(k)^(e_k XOR e_{k+1}) (where XOR denotes the bitwise XOR operator).

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A343534 Quotients of successive terms of A343458.

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A384180 Irregular triangle read by rows where row n lists the Heinz numbers of all uniform (equal multiplicities) and normal (covering an initial interval) multisets of length n.

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Author

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Examples

Crossrefs

Programs

Mathematica