A063956 -id:A063956 - cp's OEIS frontend

A063958 Sum of the non-unitary prime divisors of n: sum of those prime divisors for which the exponent in the prime factorization exceeds 1.

0, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, 2, 0, 0, 0, 2, 0, 3, 0, 2, 0, 0, 0, 2, 5, 0, 3, 2, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, 2, 7, 5, 0, 2, 0, 3, 0, 2, 0, 0, 0, 2, 0, 0, 3, 2, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 5, 2, 0, 0, 0, 2, 3, 0, 0, 2, 0, 0, 0, 2, 0, 3, 0, 2, 0, 0, 0, 2, 0, 7, 3, 7, 0, 0, 0, 2, 0

Offset: 1

Labos Elemer, Sep 04 2001

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from Harry J. Smith)

Cf. A034444, A048105, A056169, A056170, A063956.

Cf. A007947 (rad), A008472 (sopf).

a:= proc(n) option remember; add(`if`(i[2]>1, i[1], 0), i=ifactors(n)[2]) end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 24 2018

Array[Total@ Select[FactorInteger@ #, Last@ # > 1 &][[All, 1]] &, 105] (* Michael De Vlieger, Dec 06 2018 *)

{ for (n=1, 1000, f=factor(n)~; a=0; for (i=1, length(f), if (f[2, i]>1, a+=f[1, i])); write("b063958.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 04 2009

G.f.: Sum_{k>=1} prime(k) * x^(prime(k)^2) / (1 - x^(prime(k)^2)). - Ilya Gutkovskiy, Apr 06 2020
a(n) = sopf(rad(n/rad(n))). - Wesley Ivan Hurt, Nov 21 2021
a(n) = Sum_{p^2|n} p. - Wesley Ivan Hurt, Feb 21 2022
From Amiram Eldar, Jul 24 2024: (Start)
a(n) = A008472(n) - A063956(n).
Additive with a(p^e) = p if e >= 2, and 0 otherwise. (End)

A063955 Sum of the unitary prime divisors of n!.

Original entry on oeis.org

0, 2, 5, 3, 8, 5, 12, 12, 12, 7, 18, 18, 31, 24, 24, 24, 41, 41, 60, 60, 60, 49, 72, 72, 72, 59, 59, 59, 88, 88, 119, 119, 119, 102, 102, 102, 139, 120, 120, 120, 161, 161, 204, 204, 204, 181, 228, 228, 228, 228, 228, 228, 281, 281, 281, 281, 281, 252, 311, 311

Offset: 1

Labos Elemer, Sep 04 2001

			Prime divisors of 20! which have exponent 1 (i.e., unitary prime divisors) are {11, 13, 17, 19}, so a(20) = 11 + 13 + 17 + 19= 60. (The sum of all its prime divisors (unitary and non-unitary) is A034387(20).)

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. A010051, A034387, A034444, A056169, A056170, A056171, A056172, A063956.

a:= n-> add(`if`(i[2]=1, i[1], 0), i=ifactors(n!)[2]):
seq(a(n), n=1..60);  # Alois P. Heinz, Jun 24 2018

a[n_] := Select[FactorInteger[n!], #[[2]] == 1&][[All, 1]] // Total;
Array[a, 60] (* Jean-François Alcover, Jan 01 2022 *)

a(n) = my(f=factor(n!)~); sum(i=1, length(f), if (f[2, i]==1, f[1, i])); \\ Harry J. Smith, Sep 04 2009

a(n) = Sum_{k=floor(n/2)+1..n} k*c(k), where c is the prime characteristic (A010051). - Wesley Ivan Hurt, Dec 23 2023

a(n) = A063956(n!). - Amiram Eldar, Jul 24 2024

A242152 Numbers n such that the sum of their unitary prime divisors divides sigma(n).

Original entry on oeis.org

15, 24, 28, 35, 40, 42, 54, 60, 66, 95, 96, 110, 114, 117, 119, 120, 132, 135, 140, 143, 147, 168, 195, 198, 209, 224, 240, 250, 252, 258, 280, 287, 290, 315, 319, 322, 323, 360, 375, 377, 380, 384, 408, 460, 468, 470, 476, 480, 486, 496, 506, 507, 510, 520

Offset: 1

Paolo P. Lava, May 05 2014

			Divisors of 315 are 1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105, 315. Its unitary prime divisors are 5 and 7. Finally, sigma(315) = 624 and 624 / (5 + 7) = 52.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paolo P. Lava)

Cf. A000203, A063956.

with(numtheory): P:=proc(q) local a,b,k,n; for n from 1 to q do a:=divisors(n); b:=0;
for k from 1 to nops(a) do if isprime(a[k]) then if gcd(a[k],n/a[k])=1 then b:=b+a[k]; fi; fi; od;
if b>0 then if type(sigma(n)/b,integer) then print(n); fi; fi; od; end: P(10^10);

unitaryPrimeSum[1]=0; unitaryPrimeSum[n_] := Total[(f = FactorInteger[n])[[;;,1]] * (Boole[# == 1]& /@ f[[;;,2]])]; Select[Range[500], (ups = unitaryPrimeSum[#]) > 0 && Divisible[DivisorSigma[1, #], ups] &] (* Amiram Eldar, Nov 26 2019 *)

isok(n) = (v = sumdiv(n, d, d*isprime(d)*(gcd(d, n/d)==1))) && ! (sigma(n) % v); \\ Michel Marcus, May 05 2014

A286875 If n = Product (p_j^k_j) then a(n) = Sum (k_j >= 2, p_j^k_j).

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 8, 9, 0, 0, 4, 0, 0, 0, 16, 0, 9, 0, 4, 0, 0, 0, 8, 25, 0, 27, 4, 0, 0, 0, 32, 0, 0, 0, 13, 0, 0, 0, 8, 0, 0, 0, 4, 9, 0, 0, 16, 49, 25, 0, 4, 0, 27, 0, 8, 0, 0, 0, 4, 0, 0, 9, 64, 0, 0, 0, 4, 0, 0, 0, 17, 0, 0, 25, 4, 0, 0, 0, 16, 81, 0, 0, 4, 0, 0, 0, 8, 0, 9, 0, 4, 0, 0, 0, 32, 0, 49, 9, 29, 0, 0, 0, 8, 0, 0, 0, 31

Offset: 1

Ilya Gutkovskiy, Aug 02 2017

Sum of unitary, proper prime power divisors of n.

			a(360) = a(2^3*3^2*5) = 2^3 + 3^2 = 17.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors.

Cf. A005117, A008475, A222416, A023888, A023889, A034448, A063956, A077610, A092261, A246547, A284117.

Table[DivisorSum[n, # &, CoprimeQ[#, n/#] && PrimePowerQ[#] && !PrimeQ[#] &], {n, 108}]
f[p_, e_] := If[e == 1, 0, p^e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

A286875(n) = { my(f=factor(n)); for (i=1, #f~, if(f[i, 2] < 2, f[i, 1] = 0)); vecsum(vector(#f~,i,f[i,1]^f[i,2])); }; \\ Antti Karttunen, Oct 07 2017

from sympy import primefactors, isprime, gcd, divisors
def a(n): return sum(d for d in divisors(n) if gcd(d, n//d)==1 and len(primefactors(d))==1 and not isprime(d))
print([a(n) for n in range(1, 109)]) # Indranil Ghosh, Aug 02 2017

a(n) = Sum_{d|n, d = p^k, p prime, k >= 2, gcd(d, n/d) = 1} d.
a(A246547(k)) = A246547(k).
a(A005117(k)) = 0.
Additive with a(p^e) = p^e if e >= 2, and 0 otherwise. - Amiram Eldar, Jul 24 2024

A064143 Sum of unitary prime divisors of binomial(n, floor(n/2)).

Original entry on oeis.org

0, 2, 3, 5, 7, 5, 12, 14, 9, 7, 23, 21, 27, 27, 29, 31, 48, 46, 62, 60, 59, 59, 81, 79, 66, 66, 59, 59, 93, 93, 124, 126, 120, 118, 125, 123, 141, 141, 158, 156, 193, 193, 225, 225, 194, 194, 243, 241, 241, 241, 245, 245, 298, 298, 321, 321, 314, 314, 365, 365, 395

Offset: 1

Labos Elemer, Sep 11 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A001405, A008472, A034444, A056169, A063956.

a(n) = { my(f=factor(binomial(n, n\2))); sum(i=1, #f~, if (f[i,2]==1, f[i,1])) } \\ Harry J. Smith, Sep 09 2009

a(n) = A063956(A001405(n)).

A345373 Sum of the divisor complements of the unitary prime divisors of n.

Original entry on oeis.org

0, 1, 1, 0, 1, 5, 1, 0, 0, 7, 1, 4, 1, 9, 8, 0, 1, 9, 1, 4, 10, 13, 1, 8, 0, 15, 0, 4, 1, 31, 1, 0, 14, 19, 12, 0, 1, 21, 16, 8, 1, 41, 1, 4, 9, 25, 1, 16, 0, 25, 20, 4, 1, 27, 16, 8, 22, 31, 1, 32, 1, 33, 9, 0, 18, 61, 1, 4, 26, 59, 1, 0, 1, 39, 25, 4, 18, 71, 1, 16, 0, 43

Offset: 1

Wesley Ivan Hurt, Jun 16 2021

nonn

a(p) = 1 for p prime.

			a(30) = 30 * Sum_{p|30, p prime} floor(1/gcd(p,30/p))/p = 30 * (1/2 + 1/3 + 1/5) = 31.

Cf. A063956.

a(n) = n * Sum_{p|n, p prime} floor(1/gcd(p,n/p)) / p.

cp's OEIS Frontend

A063958 Sum of the non-unitary prime divisors of n: sum of those prime divisors for which the exponent in the prime factorization exceeds 1.

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A063955 Sum of the unitary prime divisors of n!.

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A242152 Numbers n such that the sum of their unitary prime divisors divides sigma(n).

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A286875 If n = Product (p_j^k_j) then a(n) = Sum (k_j >= 2, p_j^k_j).

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A064143 Sum of unitary prime divisors of binomial(n, floor(n/2)).

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A345373 Sum of the divisor complements of the unitary prime divisors of n.

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