A065330 -id:A065330 - cp's OEIS frontend

A248909 Completely multiplicative with a(p) = p if p = 6k+1 and a(p) = 1 otherwise.

1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 13, 7, 1, 1, 1, 1, 19, 1, 7, 1, 1, 1, 1, 13, 1, 7, 1, 1, 31, 1, 1, 1, 7, 1, 37, 19, 13, 1, 1, 7, 43, 1, 1, 1, 1, 1, 49, 1, 1, 13, 1, 1, 1, 7, 19, 1, 1, 1, 61, 31, 7, 1, 13, 1, 67, 1, 1, 7, 1, 1, 73, 37, 1, 19, 7, 13, 79, 1, 1

Offset: 1

Tom Edgar, Mar 06 2015

To compute a(n) replace primes not of the form 6k+1 in the prime factorization of n by 1.
The first place this sequence differs from A170824 is at n = 49.
For p prime, a(p) = p if p is a term in A002476 and a(p) = 1 if p = 2, p = 3 or p is a term in A007528.
a(n) is the largest term of A004611 that divides n. - Peter Munn, Mar 06 2021

			a(49) = 49 because 49 = 7^2 and 7 = 6*1 + 1.
a(15) = 1 because 15 = 3*5 and neither of these primes is of the form 6k+1.
a(62) = 31 because 62 = 31*2, 31 = 6*5 + 1, and 2 is not of the form 6k+1.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Index to divisibility sequences

Sequences used in a definition of this sequence: A002476, A004611, A007528, A020639, A028234, A032742.
Equivalent sequence for distinct prime factors: A170824.
Equivalent sequences for prime factors of other forms: A000265 (2k+1), A343430 (3k-1), A170818 (4k+1), A097706 (4k-1), A343431 (6k-1), A065330 (6k+/-1), A065331 (<= 3).

A248909 := proc(n)
    local a,pf;
    a := 1 ;
    for pf in ifactors(n)[2] do
        if modp(op(1,pf),6) = 1 then
            a := a*op(1,pf)^op(2,pf) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Mar 14 2015

f[p_, e_] := If[Mod[p, 6] == 1, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((f[i,1] - 1) % 6, f[i, 1] = 1);); factorback(f);} \\ Michel Marcus, Mar 11 2015

from sympy import factorint
def A248909(n):
    y = 1
    for p,e in factorint(n).items():
        y *= (1 if (p-1) % 6 else p)**e
    return y # Chai Wah Wu, Mar 15 2015

n=100; sixnplus1Primes=[x for x in primes_first_n(100) if (x-1)%6==0]
[prod([(x^(x in sixnplus1Primes))^y for x,y in factor(n)]) for n in [1..n]]

(define (A248909 n) (if (= 1 n) n (* (if (= 1 (modulo (A020639 n) 6)) (A020639 n) 1) (A248909 (A032742 n))))) ;; Antti Karttunen, Jul 09 2017

a(1) = 1; for n > 1, if A020639(n) = 1 (mod 6), a(n) = A020639(n) * a(A032742(n)), otherwise a(n) = a(A028234(n)). - Antti Karttunen, Jul 09 2017

a(n) = a(A065330(n)). - Peter Munn, Mar 06 2021

A385045 The sum of the unitary divisors of n that are 5-rough numbers (A007310).

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 8, 1, 1, 6, 12, 1, 14, 8, 6, 1, 18, 1, 20, 6, 8, 12, 24, 1, 26, 14, 1, 8, 30, 6, 32, 1, 12, 18, 48, 1, 38, 20, 14, 6, 42, 8, 44, 12, 6, 24, 48, 1, 50, 26, 18, 14, 54, 1, 72, 8, 20, 30, 60, 6, 62, 32, 8, 1, 84, 12, 68, 18, 24, 48, 72, 1, 74, 38

Offset: 1

Amiram Eldar, Jun 16 2025

First differs from A186099 at n = 25; a(25) = 26, while A186099(25) = 31.

The number of these divisors is A385044(n), and the largest of them is A065330(n).

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

The unitary analog of A186099.
Cf. A002117, A007310, A034448, A065330, A385044.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), this sequence (5-rough), A385046 (3-smooth), A385047 (power of 2), A385048 (cubefull), A385049 (biquadratefree).

f[p_, e_] := If[p <= 3, 1, p^e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] < 5, 1, f[i, 1]^f[i, 2] + 1));}

Multiplicative with a(p^e) = 1 if p <= 3, and p^e + 1 if p >= 5.
a(n) = A034448(n)/A385046(n).
a(n) <= A034448(n), with equality if and only if n is 5-rough.
a(n) <= A186099(n).
Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s-1)) * ((1-1/2^(s-1))/(1-1/2^(2*s-1))) * ((1-1/3^(s-1))/(1-1/3^(2*s-1))).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*Pi^2/(91*zeta(3)) = 0.270679... .

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

Original entry on oeis.org

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

Antti Karttunen, Dec 15 2015

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

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

Cf. A064989, A065330, A065331.

Cf. also A192232, A206296, A265399.

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 *)

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)); };

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

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

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

Keyword mult added by Antti Karttunen, Aug 04 2018

A336561 Numbers k at which point A336459(k) appears multiplicative, but A051027(k) does not.

Original entry on oeis.org

506, 1819, 2024, 2714, 3674, 3818, 4554, 5088, 5750, 5786, 6026, 6762, 6842, 7215, 7276, 9487, 9523, 10442, 11895, 12397, 12650, 13178, 13303, 14235, 14696, 15272, 15962, 16346, 16371, 18216, 18458, 19274, 19514, 19690, 19706, 20179, 20378, 21079, 21255, 21626, 22066, 22586, 22682, 23000, 23144, 23322, 24104, 24246

Offset: 1

Antti Karttunen, Jul 25 2020

nonn

			506 = 2*11*23 is a term as A336459(2)*A336459(11)*A336459(23) = 1*7*5 = 35 = A336459(506), while A051027(2)*A051027(11)*A051027(23) = 4*28*60 = 6720 <> A051027(506) = 2520. Note that 2520 = 2^3 * 3^2 * 5 * 7, thus A065330(2520) = 5*7 = 35.

Antti Karttunen, Table of n, a(n) for n = 1..11481
Index entries for sequences related to sigma(n)

Cf. A051027, A065330, A336459.
Cf. also A336549.
Subsequence of A336548, and probably also of A336560.

is_fun_mult_on_n(fun,n) = { my(f=factor(n)); prod(k=1,#f~,fun(f[k,1]^f[k,2]))==fun(n); };
A051027(n) = sigma(sigma(n));
A336546(n) = is_fun_mult_on_n(A051027,n);
A065330(n) = (n>>valuation(n, 2)/3^valuation(n, 3));
A336459(n) = A065330(A051027(n));
isA336561(n) = (A336546(n)A336459,n));

A343431 Part of n composed of prime factors of the form 6k-1.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 11, 1, 1, 1, 5, 1, 17, 1, 1, 5, 1, 11, 23, 1, 25, 1, 1, 1, 29, 5, 1, 1, 11, 17, 5, 1, 1, 1, 1, 5, 41, 1, 1, 11, 5, 23, 47, 1, 1, 25, 17, 1, 53, 1, 55, 1, 1, 29, 59, 5, 1, 1, 1, 1, 5, 11, 1, 17, 23, 5, 71, 1, 1, 1, 25, 1, 11, 1, 1, 5, 1, 41, 83, 1, 85, 1, 29, 11, 89, 5

Offset: 1

Peter Munn, Apr 15 2021

Completely multiplicative with a(p) = p if p is of the form 6k-1 and a(p) = 1 otherwise.

Largest term of A259548 that divides n.

Equivalent sequence for distinct prime factors: A170825.
Equivalent sequences for prime factors of other forms: A000265 (2k+1), A343430 (3k-1), A170818 (4k+1), A097706 (4k-1), A248909 (6k+1), A065330 (6k+/-1), A065331 (<= 3), A355582 (<= 5).
Range of terms: A259548.

f[p_, e_] := If[Mod[p, 6] == 5, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* after Amiram Eldar at A248909 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((f[i, 1] + 1) % 6, f[i, 1] = 1); ); factorback(f); } \\ after Michel Marcus at A248909

from math import prod
from sympy import factorint
def A343431(n): return prod(p**e for p, e in factorint(n).items() if not (p+1)%6) # Chai Wah Wu, Dec 26 2022

a(n) = n / A065331(n) / A248909(n) = A065330(n) / A248909(n).

A384042 The number of integers k from 1 to n such that gcd(n,k) is a 5-rough number (A007310).

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 7, 4, 6, 5, 11, 4, 13, 7, 10, 8, 17, 6, 19, 10, 14, 11, 23, 8, 25, 13, 18, 14, 29, 10, 31, 16, 22, 17, 35, 12, 37, 19, 26, 20, 41, 14, 43, 22, 30, 23, 47, 16, 49, 25, 34, 26, 53, 18, 55, 28, 38, 29, 59, 20, 61, 31, 42, 32, 65, 22, 67, 34, 46

Offset: 1

Amiram Eldar, May 18 2025

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

Cf. A000010, A003586, A007310, A065330, A065331, A372671.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), A384040 (cubefull), A384041 (exponentially odd), this sequence (5-rough).

f[p_, e_] := If[p < 5, (p-1)*p^(e-1), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] < 5, (f[i,1]-1)*f[i,1]^(f[i,2]-1), f[i,1]^f[i,2]));}

Multiplicative with a(p^e) = (p-1)*p^(e-1) if p <= 3 and p^e if p >= 5.
a(n) >= A000010(n), with equality if and only if n is 3-smooth (A003586).
a(n) = A000010(A065331(n)) * A065330(n).
a(n) = 2 * n * phi(n)/phi(6*n) = n * A000010(n) / A372671(n).
Dirichlet g.f.: zeta(s-1) * (1-1/2^s) * (1-1/3^s).
Sum_{k=1..n} a(k) ~ n^2 / 3.

A065332 3-smooth numbers in their natural position, gaps filled with 0.

Original entry on oeis.org

1, 2, 3, 4, 0, 6, 0, 8, 9, 0, 0, 12, 0, 0, 0, 16, 0, 18, 0, 0, 0, 0, 0, 24, 0, 0, 27, 0, 0, 0, 0, 32, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 72, 0, 0, 0, 0, 0, 0, 0, 0, 81, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 96, 0, 0

Offset: 1

Reinhard Zumkeller, Oct 29 2001

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

Cf. A003586, A065330, A065331, A065333, A065334, A065335.

smooth3Q[n_] := n == 2^IntegerExponent[n, 2]*3^IntegerExponent[n, 3];
a[n_] := n Boole[smooth3Q[n]];
Array[a, 100] (* Jean-François Alcover, Oct 17 2021 *)

a(n) = if(n >> valuation(n, 2) == 3^valuation(n, 3), n, 0); \\ Amiram Eldar, Sep 16 2023

a(n) = if A065330(n) = 1 then n else 0.
a(n) = A065333(n) * n.
If a(k) > 0 then a(k) = (2^A065334(k)) * (3^A065335(k)).
From Amiram Eldar, Sep 16 2023: (Start)
Multiplicative with a(p^e) = p^e if p <= 3, and 0 otherwise.
Dirichlet g.f.: 6^s / ((2^s-2)*(3^s-3)).
Sum_{k=1..n} a(k) ~ (n/(log(2)*log(3))) * (log(n) + log(6)/2 - 1). (End)

A106799 Number of prime factors of n apart from 2 or 3, counted with multiplicity.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, May 17 2005

nonn

Self-similar in every second and in every third term, i.e., a(n) = a(2n) = a(3n).
Logarithmic since a(b*c) = a(b) + a(c).
Coincidentally, a(n) = A101040(n+78) for 1 < n < 20.

			a(24) = 0 since 24 = 2*2*2*3.
a(25) = 2 since 25 = 5*5.
a(26) = 1 since 26 = 2*13.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001222, A007814, A007949, A027746, A065330, A087436, A101040, A169611.

a106799 = a001222 . a065330  -- Reinhard Zumkeller, Nov 19 2015

a[n_] := PrimeOmega[n] - IntegerExponent[n, 2] - IntegerExponent[n, 3]; Array[a, 100] (* Amiram Eldar, Jan 16 2022 *)

a(n) = bigomega(n) - valuation(n, 2) - valuation(n, 3); \\ Michel Marcus, Jan 16 2022

a(n) = A001222(n) - A007814(n) - A007949(n) = A087436(n) - A007949(n).

a(n) = A001222(A065330(n)). - Reinhard Zumkeller, May 19 2005

A349438 Dirichlet convolution of A000027 with A349348 (Dirichlet inverse of A252463), where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 2, 1, 3, 0, 4, -1, 2, 0, 2, 1, 4, -1, 2, -2, 2, 0, 4, -2, 10, 0, 9, -2, 6, -4, 2, 1, 4, 0, 4, -4, 6, 0, 2, -4, 4, -4, 2, -4, 6, 0, 4, -3, 14, -4, 4, -2, 6, -6, 8, -4, 2, 0, 6, -6, 2, 0, 6, 1, 4, -8, 6, -4, 4, -8, 4, -6, 2, 0, 10, -2, 8, -4, 6, -6, 27, 0, 4, -6, 8, 0, 6, -8, 6, -16, 4, -4, 2, 0, 4

Offset: 1

Antti Karttunen, Nov 18 2021

sign

Convolving this sequence with A348045 gives Euler phi, A000010.

It might first seem that A000265(a(p^k)) = p^(k-1) for all odd primes and all exponents k >= 1, but this does not hold for prime 37. However, with p=37, identity A065330(A349438(37^k)) = 37^(k-1) seems to hold for all exponents k >= 1. - Antti Karttunen, Nov 20 2021

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A000027, A064989, A252463, A349348, A349437 (Dirichlet inverse), A349439 (sum with it).

Cf. also A000010, A000265, A065330, A348045.

f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := If[EvenQ[n], n/2, Times @@ f @@@ FactorInteger[n]]; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; a[n_] := DivisorSum[n, # * sinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

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)};
A252463(n) = if(!(n%2),n/2,A064989(n));
memoA349348 = Map();
A349348(n) = if(1==n,1,my(v); if(mapisdefined(memoA349348,n,&v), v, v = -sumdiv(n,d,if(dA252463(n/d)*A349348(d),0)); mapput(memoA349348,n,v); (v)));
A349438(n) = sumdiv(n,d,d*A349348(n/d));

a(n) = Sum_{d|n} d * A349348(n/d).

A146892 For definition see comments lines.

Original entry on oeis.org

1, 6, 6, 72, 72, 72, 6, 72, 72, 5184, 6, 5184, 72, 5184, 31104, 5184, 5184, 5184, 2592, 5184, 432, 373248, 36, 373248, 31104, 26873856, 26873856, 26873856, 373248, 31104, 36, 31104, 2239488, 2239488, 1934917632, 26873856, 31104, 2239488

Offset: 0

Yasutoshi Kohmoto, Apr 17 2009

Let USigma denote the unitary sigma function, A034448.
As in A146891, let PF_p(n) denote the largest power of the prime p dividing n. PF_2 is A006519, and PF_3 is A038500. Furthermore define PF_1(n)=1.
Extension to multi-prime-indices is done by multiplying the corresponding functions: PF_{p,q,..}(n) = PF_p(n)*PF_q(n)*... An example of this is PF_{2,3} = A065331.
[How to compute c(m)]
Case of Base Primes = {2}{3}
c(0)=2^m, b(0)=2^m
c(n)=c(n-1)/PF_2[USigma[b(n-1)]]*PF_3[USigma[b(n-1)]]
b(n)=USigma[b(n-1)]/ PF_2,3[USigma[b(n-1)]]
IF b(k)=1 THEN END
a(m)=c(k)
Sequence gives a(m)
Factorization of term becomes 2^r*3^s.

Cf. A146891.

A146892 := proc(n) local b,a,k ;
   b := [2^n] ;
   while op(-1,b) <> 1 do
       b := [op(b), A065330(A034448(op(-1,b))) ] ;
   od:
   a := 2^n ;
   for k from 2 to nops(b) do
       a := a/ A006519(A034448(op(k-1,b))) *A038500(A034448(op(k-1,b))) ;
   od:
   a ;
end: # R. J. Mathar, Jun 24 2009

More terms from R. J. Mathar, Jun 24 2009

cp's OEIS Frontend

A248909 Completely multiplicative with a(p) = p if p = 6k+1 and a(p) = 1 otherwise.

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A385045 The sum of the unitary divisors of n that are 5-rough numbers (A007310).

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

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A336561 Numbers k at which point A336459(k) appears multiplicative, but A051027(k) does not.

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A343431 Part of n composed of prime factors of the form 6k-1.

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A384042 The number of integers k from 1 to n such that gcd(n,k) is a 5-rough number (A007310).

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A065332 3-smooth numbers in their natural position, gaps filled with 0.

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