A008472 -id:A008472 - cp's OEIS frontend

A063996 Numbers k such that ud(k) = sopf(k)-1, where ud(k)=A034444(k) and sopf(k)=A008472(k).

3, 6, 9, 12, 18, 24, 27, 36, 48, 54, 72, 81, 96, 108, 144, 162, 192, 210, 216, 243, 288, 324, 384, 420, 432, 486, 576, 630, 648, 729, 768, 840, 864, 972, 1050, 1152, 1260, 1296, 1458, 1470, 1536, 1680, 1728, 1890, 1944, 2100, 2187, 2304, 2520, 2592, 2916

Offset: 1

Jason Earls, Sep 06 2001

Amiram Eldar, Table of n, a(n) for n = 1..10000 (term 1..1000 from Harry J. Smith)

Cf. A034444, A008472.

sopf(n,s,fac,i)=fac=factor(n); for(i=1,matsize(fac)[1], s=s+fac[i,1]); return(s); ud(n) = 2^omega(n); j=[]; for(n=1,7000, if(ud(n)==sopf(n)-1, j=concat(j,n))); j

sopf(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) } { n=0; for (m=1, 10^9, if (2^omega(m)==sopf(m) - 1, write("b063996.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 05 2009

A064019 Numbers k such that sopf(k) = sopf(k^2 - 1), where sopf(k) = A008472(k).

Original entry on oeis.org

1, 5, 51, 99, 155, 209, 2369, 2569, 2882, 5745, 15143, 21691, 34573, 36566, 40516, 41237, 65304, 82718, 101638, 112305, 185701, 238302, 247221, 254865, 291399, 439104, 445794, 483107, 532645, 538531, 570020, 690238, 698561, 772485, 805013

Offset: 1

Jason Earls, Sep 07 2001

nonn

Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 100 terms from Harry J. Smith)

sopf(n,s,fac,i)=fac=factor(n); for(i=1,matsize(fac)[1],s=s+fac[i,1]); return(s);
for(n=1,10^6, if(sopf(n)==sopf(n^2-1),print(n)))

sopf(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) }
{ n=0; for (m=1, 10^9, if (sopf(m)==sopf(m^2 - 1), write("b064019.txt", n++, " ", m); if (n==100, break)) ) } \\ Harry J. Smith, Sep 06 2009

a(21)-a(35) from Donovan Johnson, Jun 15 2009

A065132 Arithmetic mean of first n terms of A008472 is an integer.

Original entry on oeis.org

2, 13, 134, 167, 2239, 62268, 75255, 135681, 439867, 18139940, 23671044, 40892256, 312083956, 724031017, 1990127567, 2144843867, 2588619526, 7439533243, 15054156002, 54892225873, 69959798320, 79760490898, 282311798922

Offset: 1

Labos Elemer, Oct 15 2001

nonn

			Sum of first 13 terms of A008472 gives A024924(13) = 65 which is divisible by n = 13, so 13 is here: 0+2+3+2+5+5+7+2+3+7+11+5+13 = 65 = 13*5 = A024924(13).

Cf. A024924, A008472.

s=0; Do[s=s+sp[n]; If[IntegerQ[n/25000], Print[n]]; If[IntegerQ[s/n], Print[{n, s, s/n}]], {n, 2, 4000000}] where sp[n]=A008472(n).

Integers n that divide A024924(n)=A008472(1)+A008472(2)+...+A008472(n).
Also, integers n that divide A024934(n).
Prime terms are listed in A143851.

a(10)-a(19) from Donovan Johnson, Nov 22 2009

a(20)-a(23) from Donovan Johnson, Aug 31 2010

A075658 Numbers k such that the sum of prime divisors of k (A008472) is composite.

Original entry on oeis.org

14, 15, 21, 26, 28, 30, 33, 35, 38, 39, 42, 45, 46, 51, 52, 55, 56, 57, 60, 62, 63, 65, 66, 69, 70, 74, 75, 76, 77, 78, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 98, 99, 102, 104, 105, 106, 110, 111, 112, 114, 115, 117, 119, 120, 122, 123, 124, 126, 129, 130, 132, 133

Offset: 1

Floor van Lamoen, Sep 23 2002 and Oct 02 2002

nonn

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

Cf. A008472.

f:=func; [k:k in [2..150]| not IsPrime(f(k))]; // Marius A. Burtea, Nov 14 2019

A075658 := proc(n) local i,j,t1,t; t := NULL; for i from 2 to n do t1 := 0; for j from 1 to i do if i mod ithprime(j) = 0 then t1 := t1+ithprime(j); fi; od; if not(isprime(t1)) then t := t,i; fi; od; t; end;

Select[Range[133], CompositeQ[Plus @@ FactorInteger[#][[;;,1]]] &] (* Amiram Eldar, Nov 14 2019 *)

Offset corrected by Amiram Eldar, Nov 14 2019

A161606 a(n) = gcd(A008472(n), A001222(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 5, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 5, 1, 1, 1, 3, 2, 3, 1, 1, 1, 1, 1, 4, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3

Offset: 1

Leroy Quet, Jun 14 2009

nonn

			28 has a prime-factorization of: 2^2 * 7^1. The sum of the distinct primes dividing 28 is 2+7 = 9. The sum of the exponents in the prime-factorization of 28 is 2+1 = 3. a(28) therefore equals gcd(9,3) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001222, A008472.

A008472 := proc(n) if n = 1 then 0 ; else add(p, p= numtheory[factorset](n)) ; end if ; end proc:
A161606 := proc(n) igcd(A008472(n),numtheory[bigomega](n)) ; end proc:
seq(A161606(n),n=2..80) ; # R. J. Mathar, Jul 08 2011

Table[GCD[DivisorSum[n, # &, PrimeQ], PrimeOmega@ n], {n, 105}] (* Michael De Vlieger, Jul 20 2017 *)

from sympy import primefactors, gcd
def a001222(n): return 0 if n==1 else a001222(n//primefactors(n)[-1]) + 1
def a(n): return gcd(sum(primefactors(n)), a001222(n))
print([a(n) for n in range(1, 151)]) # Indranil Ghosh, Jul 20 2017

(define (A161606 n) (gcd (A001222 n) (A008472 n))) ;; Antti Karttunen, Jul 20 2017

Term a(1)=0 prepended and more terms computed by Antti Karttunen, Jul 20 2017

A190722 Primes p such that A008472(p-1) = A008472(p+1) and is a prime.

Original entry on oeis.org

3, 45751, 149351, 171529, 223099, 434237, 678077, 706841, 1996297, 3993037, 6340457, 7199113, 7419761, 9000317, 13129271, 15052777, 17193217, 18436879, 18749881, 18998519, 23353469, 23689423, 33746663, 40985411, 41437751, 43547797, 51198097, 53773651, 56825687, 60207809, 62190113, 79778899, 81708353, 83019421

Offset: 1

Robert G. Wilson v, May 17 2011

nonn

A008472 is the sum of the distinct primes dividing n.

			For p = 45751, p-1 = 2*3*5^3*61; 2+3+5+61=71 and p+1 = 2^3*7*19*43; 2+7+19+43 = 71.

Amiram Eldar, Table of n, a(n) for n = 1..223 (calculated from the b-file at A203182)

Cf. A190680, A008472, A086711.

Subsequence of A203182.

[p:p in PrimesInInterval(3,10^8)|(&+PrimeDivisors(p-1) eq &+PrimeDivisors(p+1)) and IsPrime(&+PrimeDivisors(p-1))]; // Marius A. Burtea, Nov 14 2019

fQ[n_] := Block[{pn = Plus @@ (First@# & /@ FactorInteger[n - 1]), pp = Plus@@ (First@# & /@ FactorInteger[n + 1])}, pn == pp && PrimeQ[pn]];
p = 2; lst = {}; While[p < 10^8, If[fQ@p, AppendTo[lst, p]; Print@p]; p =
NextPrime@p]; lst
pQ[n_]:=Module[{p1=Total[FactorInteger[n-1][[All,1]]],p2=Total[ FactorInteger[ n+1][[All,1]]]},p1==p2&&PrimeQ[p1]]; Select[ Prime[ Range[5*10^6]],pQ] (* Harvey P. Dale, Jun 18 2017 *)

A203182 Primes p such that A008472(p-1) = A008472(p+1), where A008472 = sum of distinct primes dividing n.

Original entry on oeis.org

3, 18913, 24733, 29633, 32429, 42719, 45751, 46103, 61409, 117991, 149351, 171529, 174019, 176017, 223099, 294893, 326369, 363691, 421727, 423503, 434237, 472631, 658579, 678077, 686423, 706841, 735901, 770059, 771629, 906949, 936827, 937571, 1073447, 1256029

Offset: 1

Michel Lagneau, Dec 30 2011

nonn

Conjecture: the sequence is infinite.

			18913 is in the sequence because:
sum of the distinct prime divisors of 18912 = 2+3+197 = 202;
sum of the distinct prime divisors of 18914 = 2+7+193 = 202.

Donovan Johnson, Table of n, a(n) for n = 1..2000

Cf. A008472, A190680, A086711.

with(numtheory):for n from 1 to 100000 do:p:=ithprime(n):p1:=p-1: p2:=p+1:t1:=ifactors(p1)[2]; t11 := sum(t1[i][1], i=1..nops(t1)):t2:=ifactors(p2)[2]; t22 := sum(t2[i][1], i=1..nops(t2)):if t11=t22 then printf(`%d, `,p):else fi:od:

Select[Prime[Range[100000]],Total[Transpose[FactorInteger[#-1]][[1]]] == Total[Transpose[FactorInteger[#+1]][[1]]]&] (* Harvey P. Dale, Sep 22 2013 *)

A275665 Numbers n such that n and sopf(n) are relatively prime, where sopf(n) (A008472) is the sum of the distinct primes dividing n.

Original entry on oeis.org

1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104, 106, 108, 111, 112, 115, 116, 117, 118, 119, 122, 123, 124, 129, 133, 134, 135, 136, 141, 142, 143, 144, 145, 146, 147, 148, 152, 153, 155, 158, 159, 160, 161, 162, 164, 165

Offset: 1

Charles R Greathouse IV, Aug 04 2016

nonn

Hall shows that the density of this sequence is 6/Pi^2, so a(n) ~ (Pi^2/6)n.

Differs from A267114, from A030231, and from A007774 (shifted by one index) first at n=93. - R. J. Mathar, Aug 22 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
R. B. Hall, On the probability that n and f(n) are relatively prime, Acta Arithmetica 17 (1970), pp. 169-183.

Cf. A008472, A082300, A267114, A030231, A007774.

Select[Range@ 165, CoprimeQ[#, Total@ FactorInteger[#][[All, 1]]] &] (* Michael De Vlieger, Aug 06 2016 *)

sopf(n)=vecsum(factor(n)[,1])
is(n)=gcd(sopf(n),n)==1

A336445 Integers m such that m/sopf(m) is a prime number where sopf(m) is A008472(m), the sum of the distinct primes dividing m.

Original entry on oeis.org

4, 9, 25, 30, 49, 70, 84, 105, 121, 169, 231, 234, 260, 286, 289, 361, 456, 529, 532, 627, 646, 805, 841, 897, 961, 1116, 1364, 1369, 1581, 1665, 1681, 1798, 1849, 1924, 2064, 2150, 2209, 2632, 2809, 2967, 3055, 3339, 3481, 3526, 3721, 4489, 4543, 4824, 5025, 5041

Offset: 1

Michel Marcus, Jul 22 2020

nonn

All squares of primes (A001248) are terms.

			4 is a term since sopf(4)=2 and 4/2 = 2 is a prime.
30 is a term since sopf(30)=10 and 30/10 = 3 is a prime.

Michel Marcus, Table of n, a(n) for n = 1..1049 (terms up to 10^7)

Cf. A008472 (sopf).
Subsequence of A071139.
A001248 is a subsequence.
Cf. A336296, A336297.

sopf(n)=vecsum(factor(n)[, 1]); \\ A008472
isokp(k) = my(q=k/sopf(k)); (denominator(q)==1) && isprime(q);

A362948 Numbers whose sum of (distinct) prime divisors (A008472) equals 5.

Original entry on oeis.org

5, 6, 12, 18, 24, 25, 36, 48, 54, 72, 96, 108, 125, 144, 162, 192, 216, 288, 324, 384, 432, 486, 576, 625, 648, 768, 864, 972, 1152, 1296, 1458, 1536, 1728, 1944, 2304, 2592, 2916, 3072, 3125, 3456, 3888, 4374, 4608, 5184, 5832, 6144, 6912, 7776, 8748, 9216, 10368, 11664, 12288

Offset: 1

M. F. Hasler, Jul 20 2023

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A008472 (sopf), A000351 (5^n), A033845 (2^m*3^n).

seq[max_] := Union[Join[5^Range[Floor[Log[5, max]]], Flatten@ Table[2^i*3^j, {i, 1, Log2[max]}, {j, 1, Log[3, max/2^i]}]]]; seq[13000] (* Amiram Eldar, Jul 27 2023 *)

select( {is_A362948(n)=vecsum(factor(n,0)[,1])==5}, [1..11^4]) \\ alternatively: [n | n<-[1..11^4], A008472(n)==5]

Union of A000351 = {5^k ; k > 0} and A033845 = {2^j*3^k, j,k > 0}.

Sum_{n>=1} 1/a(n) = 3/4. - Amiram Eldar, Jul 27 2023

cp's OEIS Frontend

A063996 Numbers k such that ud(k) = sopf(k)-1, where ud(k)=A034444(k) and sopf(k)=A008472(k).

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A064019 Numbers k such that sopf(k) = sopf(k^2 - 1), where sopf(k) = A008472(k).

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A065132 Arithmetic mean of first n terms of A008472 is an integer.

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A075658 Numbers k such that the sum of prime divisors of k (A008472) is composite.

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A161606 a(n) = gcd(A008472(n), A001222(n)).

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A190722 Primes p such that A008472(p-1) = A008472(p+1) and is a prime.

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A203182 Primes p such that A008472(p-1) = A008472(p+1), where A008472 = sum of distinct primes dividing n.

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A275665 Numbers n such that n and sopf(n) are relatively prime, where sopf(n) (A008472) is the sum of the distinct primes dividing n.

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