A327937 -id:A327937 - cp's OEIS frontend

A327936 Multiplicative with a(p^e) = p if e >= p, otherwise 1.

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2

Offset: 1

Antti Karttunen, Oct 01 2019

			For n = 12 = 2^2 * 3^1, only prime factor p = 2 satisfies p^p | 12, thus a(12) = 2.
For n = 108 = 2^2 * 3^3, both prime factors p = 2 and p = 3 satisfy p^p | 108, thus a(108) = 2*3 = 6.

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

Cf. A001221, A129251, A327937, A327938, A327939.

Differs from A129252 for the first time at n=108.

Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; IntegerQ@ p :> If[e >= p, p, 1]] &, 120] (* Michael De Vlieger, Oct 01 2019 *)

A327936(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]>=f[k,1])); factorback(f); };

Multiplicative with a(p^e) = p if e >= p, otherwise 1.
A001221(a(n)) = A129251(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + (p-1)/p^p) = 1.3443209052633459342... . - Amiram Eldar, Nov 07 2022

A327938 Multiplicative with a(p^e) = p^(e mod p).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 2, 9, 10, 11, 3, 13, 14, 15, 1, 17, 18, 19, 5, 21, 22, 23, 6, 25, 26, 1, 7, 29, 30, 31, 2, 33, 34, 35, 9, 37, 38, 39, 10, 41, 42, 43, 11, 45, 46, 47, 3, 49, 50, 51, 13, 53, 2, 55, 14, 57, 58, 59, 15, 61, 62, 63, 1, 65, 66, 67, 17, 69, 70, 71, 18, 73, 74, 75, 19, 77, 78, 79, 5, 3, 82, 83, 21, 85, 86, 87, 22

Offset: 1

Antti Karttunen, Oct 01 2019

All terms are in A048103.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A048103, A129251, A327936, A327937, A327939, A327965.

Differs from A065883 for the first time at n=27.

f[p_, e_] := p^Mod[e, p]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 07 2022 *)

A327938(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]%f[k,1])); factorback(f); };

Multiplicative with a(p^e) = p^(e mod p).
a(n) = n / A327939(n).
For all n, A129251(a(n)) = 0, A327936(a(n)) = 1.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1/(1+1/p^p)) = 0.38559042841678887219... . - Amiram Eldar, Nov 07 2022

A377515 The largest divisor of n that is a term in A276078.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 9, 10, 11, 6, 13, 14, 15, 2, 17, 18, 19, 10, 21, 22, 23, 6, 25, 26, 9, 14, 29, 30, 31, 2, 33, 34, 35, 18, 37, 38, 39, 10, 41, 42, 43, 22, 45, 46, 47, 6, 49, 50, 51, 26, 53, 18, 55, 14, 57, 58, 59, 30, 61, 62, 63, 2, 65, 66, 67, 34, 69, 70

Offset: 1

Amiram Eldar, Oct 30 2024

First differs from A327937 at n = 625 = 5^4: a(625) = 125, while A327937(625) = 625.

The number of these divisors is A377516(n), and their sum is A377517(n).

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

Cf. A000720, A276078, A327937, A377516, A377517, A377518.

f[p_, e_] := p^Min[PrimePi[p], e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^min(primepi(f[i,1]), f[i,2]));}

Multiplicative with a(p^e) = p^min(pi(p), e), where pi(n) = A000720(n).
a(n) = n if and only if n is in A276078.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (p^((pi(p)+1)*s) - p^(pi(p)+1) - p^(pi(p)*s) + p^pi(p))/p^((pi(p)+1)*s).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/(p^pi(p) * (p+1))) = 0.80906238421914194523... .

A378424 Product_{n>=1} (1+x^n)^a(n) = Sum_{k>=0} C(k)*x^k, where C(k) = A000108(k).

Original entry on oeis.org

1, 2, 3, 10, 25, 78, 245, 810, 2700, 9250, 32065, 112710, 400023, 1432858, 5170575, 18784170, 68635477, 252088416, 930138521, 3446167850, 12815663595, 47820447026, 178987624513, 671825132838, 2528212128750, 9536895064398, 36054433807398, 136583761444354, 518401146543811, 1971076361996550, 7506908923471953, 28634752211620266

Offset: 1

Thomas Scheuerle, Nov 26 2024

nonn

Conjecture: A327937(n) divides a(n).

Cf. A000108, A157161, A179277, A327937.

A179277(n) = if(n<=1, 1, sum(k=0,floor(n/2),A179277(k)*binomial(2*n-4*k, n-2*k)/(n-2*k+1)))
a(max_n) = {my(va,vb,vc); vc=va=vector(max_n);vb = vector(max_n,k,A179277(k)); for(k=1,max_n,vc[k]=k*vb[k]-sum(m=1,k-1,vc[m]*vb[k-m])); for(k=1,max_n,va[k]=1/k*sumdiv(k,m,moebius(k/m)*vc[m])); va;}

Inverse Euler transform of A179277.

cp's OEIS Frontend

A327936 Multiplicative with a(p^e) = p if e >= p, otherwise 1.

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A327938 Multiplicative with a(p^e) = p^(e mod p).

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A377515 The largest divisor of n that is a term in A276078.

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A378424 Product_{n>=1} (1+x^n)^a(n) = Sum_{k>=0} C(k)*x^k, where C(k) = A000108(k).

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