A323156 -id:A323156 - cp's OEIS frontend

A323155 a(n) = Product_{d|n, d-1 is prime} (d-1)^(1+A286561(n,d-1)), where A286561(n,k) gives the k-valuation of n (for k > 1).

1, 1, 2, 3, 1, 20, 1, 21, 2, 1, 1, 3960, 1, 13, 2, 21, 1, 340, 1, 57, 2, 1, 1, 1275120, 1, 1, 2, 39, 1, 2900, 1, 651, 2, 1, 1, 201960, 1, 37, 2, 399, 1, 10660, 1, 129, 2, 1, 1, 119861280, 1, 1, 2, 3, 1, 18020, 1, 1911, 2, 1, 1, 643678200, 1, 61, 2, 651, 1, 20, 1, 201, 2, 13, 1, 4617209520, 1, 73, 2, 111, 1, 20, 1, 31521, 2, 1, 1, 175186440, 1, 1, 2, 903, 1

Offset: 1

Antti Karttunen, Jan 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A010051, A286561, A323156, A323157.

Cf. also A185633.

A323155(n) = { my(m=1); fordiv(n, d, if(isprime(d-1), m *= (d-1)^(1+valuation(n,d-1)))); (m); }; \\ Antti Karttunen, Jan 09 2019

a(n) = Product_{d|n, d>2} [(d-1)^(1+A286561(n,d-1))]^A010051(d-1).

A323157 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A000265(A323155(n)) for all n, except with f(1) = 0.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 2, 2, 2, 6, 2, 7, 2, 5, 2, 8, 2, 9, 2, 2, 2, 10, 2, 2, 2, 11, 2, 12, 2, 13, 2, 2, 2, 14, 2, 15, 2, 16, 2, 17, 2, 18, 2, 2, 2, 19, 2, 2, 2, 3, 2, 20, 2, 21, 2, 2, 2, 22, 2, 23, 2, 13, 2, 4, 2, 24, 2, 7, 2, 25, 2, 26, 2, 27, 2, 4, 2, 28, 2, 2, 2, 29, 2, 2, 2, 30, 2, 31, 2, 3, 2, 2, 2, 32, 2, 33, 2, 9, 2, 34, 2, 35, 2

Offset: 1

Antti Karttunen, Jan 09 2019

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  A323156(i) = A323156(j) => a(i) = a(j),
  a(i) = a(j) => A322977(i) = A322977(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000265, A305801, A322977, A323155, A323156.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A000265(n) = (n/2^valuation(n, 2));
A323155(n) = { my(m=1); fordiv(n, d, if(isprime(d-1), m *= (d-1)^(1+valuation(n,d-1)))); (m); };
A323157aux(n) = if(1==n,0,A000265(A323155(n)));
v323157 = rgs_transform(vector(up_to, n, A323157aux(n)));
A323157(n) = v323157[n];

A322702 a(n) is the product of primes p such that p+1 divides n.

Original entry on oeis.org

1, 1, 2, 3, 1, 10, 1, 21, 2, 1, 1, 330, 1, 13, 2, 21, 1, 170, 1, 57, 2, 1, 1, 53130, 1, 1, 2, 39, 1, 290, 1, 651, 2, 1, 1, 5610, 1, 37, 2, 399, 1, 5330, 1, 129, 2, 1, 1, 2497110, 1, 1, 2, 3, 1, 9010, 1, 273, 2, 1, 1, 10727970, 1, 61, 2, 651, 1, 10, 1, 201, 2

Offset: 1

Daniel Suteu, Dec 23 2018

nonn

In general, a(n) is the product of A072627(n) distinct prime factors, with a(n) = 1 iff A072627(n) = 0.

			For n=12, the divisors of 12 are {1, 2, 3, 4, 6, 12}. The prime numbers p, such that p+1 is a divisor of 12, are {2, 3, 5, 11}, therefore a(12) = 2 * 3 * 5 * 11 = 330.

Antti Karttunen, Table of n, a(n) for n = 1..10080

Cf. A072627, A027760, A322356, A323156.

a:= n-> mul(`if`(isprime(d-1), d-1, 1), d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 29 2018

Array[Apply[Times, Select[Divisors@ #, PrimeQ[# - 1] &] - 1 /. {} -> {1}] &, 69] (* Michael De Vlieger, Jan 07 2019 *)

a(n) = my(d=divisors(n)); prod(k=1, #d, if(isprime(d[k]-1), d[k]-1, 1));

a(n) = Product_{p prime, p+1 divides n} p.
a(n) = denominator of Sum_{p prime, p+1 divides n} 1/p.
a(n) = Product_{d|n, d-1 is prime} (d-1), where d runs over the divisors of n.
a(2*n + 1) = 2, iff n == 1 (mod 3), else a(2*n + 1) = 1.
A001221(a(n)) = A072627(n). - Antti Karttunen, Jan 12 2019

cp's OEIS Frontend

A323155 a(n) = Product_{d|n, d-1 is prime} (d-1)^(1+A286561(n,d-1)), where A286561(n,k) gives the k-valuation of n (for k > 1).

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A323157 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A000265(A323155(n)) for all n, except with f(1) = 0.

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A322702 a(n) is the product of primes p such that p+1 divides n.

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