A265398 Perform one x^2 -> x+1 reduction for the polynomial with nonnegative integer coefficients that is encoded in the prime factorization of n.

1, 2, 3, 4, 6, 6, 15, 8, 9, 12, 35, 12, 77, 30, 18, 16, 143, 18, 221, 24, 45, 70, 323, 24, 36, 154, 27, 60, 437, 36, 667, 32, 105, 286, 90, 36, 899, 442, 231, 48, 1147, 90, 1517, 140, 54, 646, 1763, 48, 225, 72, 429, 308, 2021, 54, 210, 120, 663, 874, 2491, 72, 3127, 1334, 135, 64, 462, 210, 3599, 572, 969, 180, 4087, 72

Offset: 1

Author

Antti Karttunen, Dec 15 2015

Keywords

• nonn
• mult

Comments

Completely multiplicative with a(2) = 2, a(3) = 3, a(prime(k)) = prime(k-1) * prime(k-2) for k > 2. - Andrew Howroyd & Antti Karttunen, Aug 04 2018

Links

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

Crossrefs

Cf. A064989, A065330, A065331.

Cf. also A192232, A206296, A265399.

Programs

Mathematica

a[n_] := a[n] = Module[{k, p, e}, Which[n<4, n, PrimeQ[n], k = PrimePi[n]; Prime[k-1] Prime[k-2], True, Product[{p, e} = pe; a[p]^e, {pe, FactorInteger[n]}]]];
a /@ Range[1, 72] (* Jean-François Alcover, Sep 20 2019 *)
f[p_, e_] := If[p < 5, p, NextPrime[p,-1]*NextPrime[p,-2]]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 01 2022 *)

PARI

A065330(n) = { while(0 == (n%2), n = n/2); while(0 == (n%3), n = n/3); n; }
A065331 = n -> n/A065330(n);
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A265398(n) = { my(a); if(1 == n, n, a = A064989(A065330(n)); A064989(a)*a*A065331(n)); };

PARI

r(p) = {my(q = precprime(p-1)); q*precprime(q-1)};
a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,1]<5, f[i,1], r(f[i,1]))^f[i,2])}; \\ Amiram Eldar, Dec 01 2022

Scheme

(definec (A265398 n) (if (= 1 n) n (* (A065331 n) (A064989 (A065330 n)) (A064989 (A064989 (A065330 n))))))

Formula

a(1) = 1; for n > 1, a(n) = A064989(A064989(A065330(n))) * A064989(A065330(n)) * A065331(n).

Sum_{k=1..n} a(k) = c * n^3, where c = (1/3) * Product_{p prime} (p^3-p^2)/(p^3-a(p)) = 0.093529982... . - Amiram Eldar, Dec 01 2022

Extensions

Keyword mult added by Antti Karttunen, Aug 04 2018