A105221 -id:A105221 - cp's OEIS frontend

A024934 Sum of remainders n mod p, over all primes p < n.

0, 0, 0, 1, 1, 3, 1, 4, 6, 7, 4, 8, 8, 13, 10, 8, 12, 18, 20, 27, 28, 26, 21, 29, 33, 37, 31, 37, 37, 46, 46, 56, 65, 62, 54, 53, 59, 70, 61, 57, 62, 74, 75, 88, 89, 95, 84, 98, 108, 116, 124, 119, 119, 134, 145, 145, 152, 146, 131, 147, 154, 171, 156, 164, 180, 180, 182, 200, 200, 193, 198, 217

Offset: 0

Clark Kimberling

nonn

			a(5) = 3. The remainder when 5 is divided by primes 2, 3 respectively is 1, 2, and their sum = 3.
10 = 2*5+0 = 3*3+1 = 5*2+0 = 7*1+3: a(10) = 0+1+0+3 = 4.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A004125, A067435, A067436, A013939, A101336.

a[n_] := Sum[Mod[n, Prime[i]], {i, PrimePi@ n}]; Array[a, 72, 0] (* Giovanni Resta, Jun 24 2016 *)
Table[Total[Mod[n,Prime[Range[PrimePi[n]]]]],{n,0,80}] (* Harvey P. Dale, Jul 02 2025 *)

a(n)=my(r=0);forprime(p=2,n,r+=n%p); r; \\ Joerg Arndt, Nov 05 2016

a(n) = n*A000720(n) - A024924(n). - Max Alekseyev, Feb 10 2012

a(n) = a(n-1) + A000720(n-1) - A105221(n). - Max Alekseyev, Nov 28 2017

Edited by Max Alekseyev, Jan 30 2012

a(0)=0 prepended by Max Alekseyev, Dec 10 2013

A003508 a(1) = 1; for n>1, a(n) = a(n-1) + 1 + sum of distinct prime factors of a(n-1) that are < a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 11, 12, 18, 24, 30, 41, 42, 55, 72, 78, 97, 98, 108, 114, 139, 140, 155, 192, 198, 215, 264, 281, 282, 335, 408, 431, 432, 438, 517, 576, 582, 685, 828, 857, 858, 888, 931, 958, 1440, 1451, 1452, 1469, 1596, 1628, 1679, 1776, 1819, 1944

Offset: 1

N. J. A. Sloane

R. K. Guy reports, Apr 14 2005: In Math. Mag. 48 (1975) 301 one finds "C. W. Trigg, C. C. Oursler and R. Cormier & J. L. Selfridge have sent calculations on Problem 886 [Nov 1973] for which we had received only partial results [Jan 1975]. Cormier and Selfridge sent the following results: There appear to be five sequences beginning with integers less than 1000 which do not merge. These sequences were carried out to 10^8 or more." The five sequences are A003508, A105210-A105213.
This suggests that there may be infinitely many different (non-merging) sequences obtained by choosing different starting values.
All terms of these five sequences are distinct up to least 10^30. - T. D. Noe, Oct 19 2007

			a(6)=8, so a(7) = 8 + 1 + 2 = 11.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..2000
Doug Engel, Problem 886, Math. Mag., 48 (1975), 57-58.

Cf. A096460, A105221, A105233.

Cf. A027748, A105210, A105211, A105212, A105213.

a003508 n = a003508_list !! (n-1)
a003508_list = 1 : map
      (\x -> x + 1 + sum (takeWhile (< x) $ a027748_row x)) a003508_list
-- Reinhard Zumkeller, Jan 15 2015

a[1] = 1; a[n_] := a[n] = a[n - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[ a[n - 1]]], # < a[n - 1] &]; Table[ a[n], {n, 54}] (* Robert G. Wilson v, Apr 13 2005 *)
nxt[n_]:=n+1+Total[Select[Transpose[FactorInteger[n]][[1]],#Harvey P. Dale, Jul 19 2015 *)

More terms from Henry Bottomley, May 09 2000

A136021 Sum of the proper prime divisors of all numbers up to 10^n.

Original entry on oeis.org

0, 19, 1047, 64373, 4481640, 340900331, 27436000061, 2292176360707, 196818634871899, 17246903703574357, 1534951275195670059, 138293592048140425181, 12583738258227621100170, 1154435823206834353336284, 106638384745041347295504523

Offset: 0

Enoch Haga, Dec 10 2007

The sum of the distinct prime factors less than k for all 1 <= k <= 10^n, as tabulated for the individual k in A105221.

			a(1)=19 because 10^1=10 and the factors to be summed are 2 for 4, added to 2 and 3 for 6, added to 2 for 8, added to 3 for 9, added to 2 and 5 for 10.

Cf. A136022, A136023, A136024, A136025.

A105221 := proc(n) local a,pfs,i ; a :=0 ; pfs := ifactors(n)[2] ; for i in pfs do if op(1,i) <> 1 and op(1,i) <> n then a := a+op(1,i) ; fi ; od: RETURN(a) ; end: A136021 := proc(n) add(A105221(i),i=2..10^n) ; end: for n from 1 do print(n,A136021(n)) ; od: # R. J. Mathar, Dec 12 2007

f[n_] := Plus @@ (First@# & /@ FactorInteger@ n); k = 2; s = 0; lst = {}; Do[While[k < 10^n + 1, If[ ! PrimeQ@k, s = s + f@k]; k++ ]; AppendTo[ lst, s]; Print[{n, s}], {n, 8}] (* Robert G. Wilson v, Aug 06 2010 *)

10 'distinct prime factors of composites <=10^n 20 S=0:N=N+1:Z=N\2 30 'print N; 40 for F=1 to Z:Q=N/F: if Q<>int(Q) then 60 50 S=S+F: if F=prmdiv(F) and F>1 then C=C+1:G=G+F 60 next F 70 'print C,G 80 if N=10^1 or N=10^2 or N=10^3 or N=10^4 or N=10^5 or N=10^6 or N=10^7 then print G:stop 90 C=0 100 goto 20

a(n) = Sum_{k=1..10^n} A105221(k). - R. J. Mathar, Dec 12 2007

a(n) = Sum_{prime p<10^n} p*floor((10^n-p)/p) = A006880(n)*10^n - A024934(10^n) - A046731(n). - Max Alekseyev, Jan 30 2012

One more term from R. J. Mathar, Dec 12 2007
Edited by R. J. Mathar, Apr 17 2009
a(7) & a(8) from Robert G. Wilson v, Aug 06 2010
a(9)-a(11) from Max Alekseyev, Jan 30 2012
a(12)-a(14) from Hiroaki Yamanouchi, Jun 29 2014

A048055 Numbers k such that (sum of the nonprime proper divisors of k) - (sum of prime divisors of k) = k.

Original entry on oeis.org

532, 945, 2624, 5704, 6536, 229648, 497696, 652970, 685088, 997408, 1481504, 11177984, 32869504, 52813084, 132612224, 224841856, 2140668416, 2404135424, 2550700288, 6469054976, 9367192064, 19266023936, 23414463358, 31381324288, 45812547584, 55620289024

Offset: 1

Naohiro Nomoto

From Peter Luschny, Dec 14 2009: (Start)
A term of this sequence is a Zumkeller number (A083207) since the set of its divisors can be partitioned into two disjoint parts so that the sums of the two parts are equal.
1 + sigma*(k) = sigma'(k) + k
sigma*(k) := Sum_{1 < d < k, d|k, d not prime}, (A060278),
sigma'(k) := Sum_{1 < d < k, d|k, d prime}, (A105221). (End)

			532 = 1 - 2 + 4 - 7 + 14 - 19 + 28 + 38 + 76 + 133 + 266.

Donovan Johnson, Table of n, a(n) for n = 1..34 (terms <= 10^12)
Donovan Johnson, 82 terms > 10^12.
Peter Luschny, Zumkeller Numbers.

Cf. A083207, A105221, A060278, A000203, A027751, A010051, A002808.

import Data.List (partition)
a048055 n = a048055_list !! (n-1)
a048055_list = [x | x <- a002808_list,
               let (us,vs) = partition ((== 1) . a010051) $ a027751_row x,
               sum us + x == sum vs]
-- Reinhard Zumkeller, Apr 05 2013

with(numtheory): A048055 := proc(n) local k;
if sigma(n)=2*(n+add(k,k=select(isprime,divisors(n))))
then n else NULL fi end: seq(A048055(i),i=1..7000);
# Peter Luschny, Dec 14 2009

zummableQ[n_] := DivisorSigma[1, n] == 2*(n + Total[Select[Divisors[n], PrimeQ]]); n = 2; A048055 = {}; While[n < 10^6, If[zummableQ[n], Print[n]; AppendTo[A048055, n]]; n++]; A048055 (* Jean-François Alcover, Dec 07 2011, after Peter Luschny *)

from sympy import divisors, primefactors
A048055 = []
for n in range(1,10**4):
    s = sum(divisors(n))
    if not s % 2 and 2*n <= s and (s-2*n)/2 == sum(primefactors(n)):
        A048055.append(n) # Chai Wah Wu, Aug 20 2014

a(15)-a(19) from Donovan Johnson, Dec 07 2008
a(20)-a(24) from Donovan Johnson, Jul 06 2010
a(25)-a(26) from Donovan Johnson, Feb 09 2012

A136025 Sum of distinct proper prime divisors of odd integers below 10^n.

Original entry on oeis.org

3, 373, 24307, 1691682, 127867801, 10233538789, 850896280551, 72812857079241, 6363727756215813, 565232434009370012, 50843507342073211151, 4620323131256374760046, 423405369424475640435621, 39074878176445767411791424

Offset: 1

Enoch Haga, Dec 12 2007

Through 10^5 about 37.5% of total sums for all integers N comprise sums of odd N and the remaining 62.5% of even N.

			a(0)=3 because the only odd N <=10^1-1 having a prime factor is 9 and its factor is 3 and sum is 3.

Cf. A136021, A136024, A143851.

A105221 := proc(n) local a,ifs,p; ifs := ifactors(n)[2] ; a := 0 ; for p in ifs do if op(1,p) <> 1 and op(1,p) <> n then a := a+op(1,p) ; fi ; od: RETURN(a) ; end: A136025 := proc(n) local a,k ; a := 0 ; for k from 5 to 10^n-1 by 2 do a := a+A105221(k) ; od: RETURN(a) ; end: for n from 1 do print(A136025(n)); od: # R. J. Mathar, Jan 29 2008

a(n) = sum_{k=1,2,...,A093143(n)} A105221(2k-1). - R. J. Mathar, Jan 29 2008

a(n) = sum_{prime p, 3<=p<10^n} p*floor((10^n-p)/(2p)). - Max Alekseyev, Jan 30 2012

a(6) from R. J. Mathar, Jan 29 2008
a(7)-a(11) from Max Alekseyev, Jan 30 2012
a(12)-a(14) from Hiroaki Yamanouchi, Jul 06 2014

A206773 Sum of nonprime proper divisors (or nonprime aliquot parts) of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 11, 1, 1, 1, 13, 1, 16, 1, 15, 1, 1, 1, 31, 1, 1, 10, 19, 1, 32, 1, 29, 1, 1, 1, 50, 1, 1, 1, 43, 1, 42, 1, 27, 25, 1, 1, 71, 1, 36, 1, 31, 1, 61, 1, 55, 1, 1, 1, 98, 1, 1, 31, 61, 1, 62, 1, 39, 1, 60, 1, 118, 1, 1, 41, 43, 1

Offset: 1

Michel Lagneau, Jan 10 2013

nonn

Sum of nonprime divisors of n that are less than n.
a(n) = 1 if n is prime or semiprime.
Up to 3*10^12, a(n) = n only for n = 42, 1316, and 131080256. In general, if p = 2^k-1 and q = 4^k-2*2^k-1 are two primes, then n = 2^(k-1)*p*q satisfies a(n) = n. This happens for k= 2, 3, 7, and 19, which give the aforementioned values and 3777871569031248714137. This property makes these values terms of A225028. - Giovanni Resta, May 03 2016

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A001065, A105221, A023890, A187793, A001358, A000469, A225028.

with(numtheory):for n from 1  to 100 do:x:=factorset(n):n1:=nops(x):s:=sum('x[i] ', 'i'=1..n1): s1:=sigma(n)-s-n: if type(n,prime)=true then printf(`%d, `,1) else printf(`%d, `,s1):fi:od:

Table[Plus@@Select[Divisors[n],#Giovanni Resta, May 03 2016 *)

a(n) = A001065(n) - A105221(n)

A137324 a(n) = Sum_{prime p < n} gcd(n,p).

Original entry on oeis.org

1, 3, 2, 6, 3, 5, 6, 9, 4, 8, 5, 13, 12, 7, 6, 10, 7, 13, 16, 19, 8, 12, 13, 22, 11, 16, 9, 17, 10, 12, 23, 28, 21, 14, 11, 31, 26, 17, 12, 22, 13, 25, 20, 37, 14, 18, 21, 20, 33, 28, 15, 19, 30, 23, 36, 45, 16, 24, 17, 49, 26, 19, 34, 31, 18, 36, 43, 30, 19, 23, 20, 58, 27, 40, 37

Offset: 3

Max Sills, Apr 06 2008

			a(10) = 9 because gcd(10,2) = 2, gcd(10,3) = 1, gcd(10,5) = 5, gcd(10,7) = 1; 2 + 1 + 5 + 1 = 9.
The underlying irregular table of gcd(n,2), gcd(n,3), gcd(n,5), gcd(n,7), etc., for which a(n) provides row sums, is obtained by deleting columns from A050873(n,k) and looks as follows for n=3,4,5,...:
  1
  2 1
  1 1
  2 3 1
  1 1 1
  2 1 1 1
  1 3 1 1
  2 1 5 1
  1 1 1 1
  2 3 1 1 1
  1 1 1 1 1
  2 1 1 7 1 1
  1 3 5 1 1 1
  2 1 1 1 1 1
  1 1 1 1 1 1
  2 3 1 1 1 1 1
  1 1 1 1 1 1 1
  2 1 5 1 1 1 1 1

Cf. A000720, A006579, A048865, A105221.

[&+[Gcd(n,p):p in PrimesInInterval(1,n-1)]:n in [3..77]]; // Marius A. Burtea, Aug 07 2019

A137324 := proc(n) local a,i; a :=0 ; for i from 1 to numtheory[pi](n-1) do a := a+gcd(n,ithprime(i)) ; od: a; end: seq(A137324(n),n=3..80) ; # R. J. Mathar, Apr 09 2008

Table[Plus @@ GCD[n, Select[Range[n - 1], PrimeQ[ # ] &]], {n, 3, 70}] (* Stefan Steinerberger, Apr 09 2008 *)

a(n) = sum(k=1, n-1, gcd(n,k)*isprime(k)); \\ Michel Marcus, Nov 07 2014

from math import gcd
from sympy import primerange
def a(n): return sum(gcd(n, p) for p in primerange(1, n))
print([a(n) for n in range(3, 78)]) # Michael S. Branicky, Nov 21 2021

a(p) = A000720(p) - 1 for prime p. - R. J. Mathar, Apr 09 2008

a(n) = A048865(n) + A105221(n). - Wesley Ivan Hurt, Nov 21 2021

Corrected and extended by R. J. Mathar and Stefan Steinerberger, Apr 09 2008

A140599 Grand total of prime factors in composite runs between two primes: first composite.

Original entry on oeis.org

4, 6, 12, 14, 18, 24, 54, 62, 68, 72, 80, 90, 108, 110, 158, 192, 234, 258, 264, 318, 338, 350, 354, 380, 420, 432, 458, 462, 570, 588, 602, 632, 662, 858, 978, 992, 1050, 1088, 1094, 1152, 1172, 1290, 1302, 1304, 1400, 1428, 1434, 1482, 1532, 1722, 1824, 1862

Offset: 1

Enoch Haga, May 18 2008

			In the composite run between primes 23 and 29 there are five composites. The sum of prime factors in all 5 composites is 37 (3+2+5+13+2+3+7+2). The first composite in this run is 24, a(6) and the last is 28.

Cf. A140600.

Compute composite runs between two primes. If the grand total of prime factors in each run is also prime, add the firat composite to the sequence.

{A008864(j): sum_{k=A008864(j)..A006093(j+1)} A105221(k) in A000040}. - R. J. Mathar, May 19 2008

A343394 Sum of indices of n's distinct prime factors below n.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 1, 2, 4, 0, 3, 0, 5, 5, 1, 0, 3, 0, 4, 6, 6, 0, 3, 3, 7, 2, 5, 0, 6, 0, 1, 7, 8, 7, 3, 0, 9, 8, 4, 0, 7, 0, 6, 5, 10, 0, 3, 4, 4, 9, 7, 0, 3, 8, 5, 10, 11, 0, 6, 0, 12, 6, 1, 9, 8, 0, 8, 11, 8, 0, 3, 0, 13, 5, 9, 9, 9, 0, 4, 2, 14, 0, 7, 10, 15, 12, 6, 0, 6

Offset: 1

Ilya Gutkovskiy, Apr 13 2021

nonn

			a(7) = a(prime(4)) = 0.
a(21) = a(3 * 7) = a(prime(2) * prime(4)) = 2 + 4 = 6.

Index entries for sequences computed from indices in prime factorization

Cf. A066328, A087624, A105221.

nmax = 90; CoefficientList[Series[Sum[k x^(2 Prime[k])/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := If[PrimeQ[n], 0, Plus @@ (PrimePi[#[[1]]] & /@ FactorInteger[n])]; Table[a[n], {n, 90}]

G.f.: Sum_{k>=1} k * x^(2*prime(k)) / (1 - x^prime(k)).

a(n) = 0 if n is prime, A066328(n) otherwise.

A352167 a(n) is the sum of the prime factors of n (with multiplicity) that are less than n.

Original entry on oeis.org

0, 0, 0, 4, 0, 5, 0, 6, 6, 7, 0, 7, 0, 9, 8, 8, 0, 8, 0, 9, 10, 13, 0, 9, 10, 15, 9, 11, 0, 10, 0, 10, 14, 19, 12, 10, 0, 21, 16, 11, 0, 12, 0, 15, 11, 25, 0, 11, 14, 12, 20, 17, 0, 11, 16, 13, 22, 31, 0, 12, 0, 33, 13, 12, 18, 16, 0, 21, 26, 14, 0, 12, 0, 39, 13, 23, 18, 18, 0, 13

Offset: 1

Ilya Gutkovskiy, Mar 06 2022

nonn

Eric Weisstein's World of Mathematics, Sum of Prime Factors

Cf. A001414, A061397, A105221, A144494.

a[n_] := If[n == 1 || PrimeQ[n], 0, Plus @@ Times @@@ FactorInteger@n]; Table[a[n], {n, 80}]

a(n) = if (isprime(n), 0, my(f=factor(n)); sum(k=1, #f~, f[k,1]*f[k,2])); \\ Michel Marcus, Mar 07 2022

a(n) = 0 if n is prime, A001414(n) otherwise.

a(n) = A001414(n) - A061397(n).

cp's OEIS Frontend

A024934 Sum of remainders n mod p, over all primes p < n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A003508 a(1) = 1; for n>1, a(n) = a(n-1) + 1 + sum of distinct prime factors of a(n-1) that are < a(n-1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A136021 Sum of the proper prime divisors of all numbers up to 10^n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

UBASIC

Formula

Extensions

A048055 Numbers k such that (sum of the nonprime proper divisors of k) - (sum of prime divisors of k) = k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

Extensions

A136025 Sum of distinct proper prime divisors of odd integers below 10^n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A206773 Sum of nonprime proper divisors (or nonprime aliquot parts) of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A137324 a(n) = Sum_{prime p < n} gcd(n,p).

Views

Author

Keywords

Examples