A062700 -id:A062700 - cp's OEIS frontend

A085379 Greatest prime as sum of distinct divisors of n.

3, 3, 7, 5, 11, 7, 13, 13, 17, 11, 23, 13, 23, 23, 31, 17, 37, 19, 41, 31, 23, 23, 59, 31, 41, 37, 53, 29, 71, 31, 61, 47, 53, 47, 89, 37, 59, 53, 89, 41, 89, 43, 83, 73, 71, 47, 113, 7, 83, 71, 97, 53, 113, 71, 113, 79, 89, 59, 167, 61, 31, 103, 127, 83, 139, 67

Offset: 2

Reinhard Zumkeller, Jun 26 2003

nonn

			The divisors of n = 50 are {1,2,5,10,25,50}, the sums of distinct divisors that are prime: 2, 3 = 2+1, 5, 7 = 5+2, 11 = 10+1, 13 = 10+2+1, 17 = 10+5+2, 31 = 25+5+1, 37 = 25+10+2, 41 = 25+10+5+1, 43 = 25+10+5+2+1, 53 = 50+2+1, 61 = 50+10+1, 67 = 50+10+5+2 and 83 = 50+25+5+2+1. Therefore a(50) = 83 < 89 = A070801(50) and A085381(3) = 50.

Amiram Eldar, Table of n, a(n) for n = 2..10001
Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A000203, A023194, A062700, A070801, A085380, A085381.

a[n_] := Module[{d = Divisors[n], c, x}, c = Rest@CoefficientList[Series[Product[1 + x^d[[i]], {i, 1, Length[d]}], {x, 0, Total[d]}], x]; Max[Select[Position[c, ?(# > 0 &)] // Flatten, PrimeQ]]]; Array[a, 100, 2] (* _Amiram Eldar, Mar 01 2024 *)

a(n) <= A070801(n) <= A000203(n).
a(A085380(n)) = A070801(A085380(n)).
a(A085381(n)) < A070801(A085381(n)).
a(A023194(n)) = A000203(A023194(n)) = A062700(n).

A229266 Primes of the form sigma(k) + tau(k) + phi(k), where sigma(k) = A000203(k), tau(k) = A000005(k) and phi(k) = A000010(k).

Original entry on oeis.org

3, 23, 557, 1289, 2447, 3779, 9209, 10331, 11351, 18367, 14051, 34351, 42953, 67883, 95717, 96587, 134807, 164249, 193057, 310553, 253159, 321397, 383723, 548213, 657311, 499151, 630023, 516251, 732181, 713927, 927013, 932431, 784627, 906473, 855331, 1121987

Offset: 1

Paolo P. Lava, Sep 18 2013

nonn

			The third term of A229265 is 200 and sigma(200) +  tau(200) + phi(200) = 465 + 12 + 80 = 557 is prime.

Cf. A000005, A000010, A000203, A009087, A023194, A038344, A055813, A062700, A064205, A115919, A141242, A229264, A229265, A065061, A229268.

with(numtheory); P:=proc(q) local a, n; for n from 1 to q do a:=sigma(n)+tau(n)+phi(n);
if isprime(a) then print(a); fi; od; end: P(10^6);

Select[Table[DivisorSigma[0,n]+DivisorSigma[1,n]+EulerPhi[n],{n,10^6}],PrimeQ] (* Harvey P. Dale, Oct 03 2023 *)

A187822 Smallest k such that the partial sums of the divisors of k (taken in increasing order) contain exactly n primes.

Original entry on oeis.org

1, 2, 4, 16, 64, 140, 440, 700, 1650, 2304, 5180, 3960, 3900, 14400, 15600, 43560, 39600, 57600, 56700, 81900, 25200, 112896, 100100, 177840, 198000, 411840, 222768, 226800, 637560, 752400, 556920, 907200, 409500, 565488, 1306800, 1984500, 1884960

Offset: 0

Michel Lagneau, Dec 27 2012

nonn

It appears that a(n) is even for n > 0 and nonsquarefree for n > 1. We also conjecture that there is an infinite subsequence of squares 1, 4, 16, 64, 2304, 14400, 57600, 112896, ....
The corresponding triangle in which row n gives the n primes starts with:
k =   1 -> no prime
k =   2 -> 3;
k =   4 -> 3, 7;
k =  16 -> 3, 7, 31;
k =  64 -> 3, 7, 31, 127;
k = 140 -> 3, 7, 19, 29, 43;
k = 440 -> 3, 7, 41, 61, 83, 167; ...

			a(4) = 64 because the partial sums of the divisors {1, 2, 4, 8, 16, 32, 64} that generate 4 prime numbers are:
1 + 2 = 3;
1 + 2 + 4 = 7;
1 + 2 + 4 + 8 + 16  = 31;
1 + 2 + 4 + 8 + 16 + 32 + 64 = 127.

Amiram Eldar, Table of n, a(n) for n = 0..126

Cf. A023194, A062700, A000203.

read("transforms") :
A187822 := proc(n)
    local k,ps,pct ;
    if n = 0 then
        return 1;
    end if;
    for k from 1 do
        ps := sort(convert(numtheory[divisors](k),list)) ;
        ps := PSUM(ps) ;
        pct := 0 ;
        for p in ps do
            if isprime(p) then
                pct := pct+1 ;
            end if;
        end do:
        if pct = n then
            return k ;
        end if;
    end do:
end proc: # R. J. Mathar, Jan 18 2013

a[n_] := Catch[ For[k = 1, True, k++, cnt = Count[ Accumulate[ Divisors[k]], ?PrimeQ]; If[cnt == n, Print[{n, k}]; Throw[k]]]]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover, Dec 27 2012 *)

A187822(n)={n<1||for(k=1,9e9,numdiv(k)M. F. Hasler, Dec 29 2012

A229265 Numbers k such that sigma(k) + tau(k) + phi(k) is a prime, where sigma(k) = A000203(k), tau(k) = A000005(k) and phi(k) = A000010(k).

Original entry on oeis.org

1, 8, 200, 512, 968, 1458, 3200, 4232, 5618, 5832, 6962, 10368, 16928, 26912, 36992, 40328, 53792, 61952, 84050, 101250, 110450, 140450, 147968, 220448, 247808, 249218, 253472, 257762, 279752, 282752, 320000, 336200, 344450, 359552, 361250, 445568, 472392, 512072

Offset: 1

Paolo P. Lava, Sep 18 2013

nonn

			sigma(200) = 465, tau(200) = 12, phi(200) = 80 and 465 + 12 + 80 = 557 is prime.

Cf. A000005, A000010, A000203, A009087, A023194, A038344, A055813, A062700, A064205, A115919, A141242, A229264, A229266-A229268.

with(numtheory); P:=proc(q) local n; for n from 1 to q do
if isprime(sigma(n)+tau(n)+phi(n)) then print(n); fi; od; end: P(10^6);

A071166 a(n) = n - A006530(A000203(n)), difference between n and largest prime factor of the sum of its divisors.

Original entry on oeis.org

-1, 1, -3, 2, 3, 5, 3, -4, 7, 8, 5, 6, 11, 12, -15, 14, 5, 14, 13, 19, 19, 20, 19, -6, 19, 22, 21, 24, 27, 29, 25, 30, 31, 32, 23, 18, 33, 32, 35, 34, 39, 32, 37, 32, 43, 44, 17, 30, 19, 48, 45, 50, 49, 52, 51, 52, 53, 54, 53, 30, 59, 50, -63, 58, 63, 50, 61, 66, 67, 68, 59, 36, 55, 44, 69, 74, 71, 74, 49, 70, 75, 76, 77, 82, 75, 82, 83

Offset: 2

Labos Elemer, May 15 2002

sign

Terms are mostly positive. At cases when sigma(n) is prime the differences are negative. See A071167.

			n=12, divisors={1,2,3,4,6,12}, sigma(12)=28, its largest prime factor is 7, so a(12)=12-7=5.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A000203, A006530, A062700, A023194, A071167.

gpf:= n -> max(numtheory:-factorset(n)):
seq(n - gpf(numtheory:-sigma(n)), n=2..100); # Robert Israel, Feb 12 2017

ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] t=Table[w-ma[DivisorSigma[1, w]], {w, 2, 128}]

a(n)=n-factor(sigma(n))[1,1] \\ Charles R Greathouse IV, Feb 19 2013

A187825 Smallest k such that the partial sums of the divisors of k (in decreasing order) generate n primes.

Original entry on oeis.org

1, 3, 2, 140, 560, 2160, 2772, 2016, 16830, 5148, 20592, 10640, 69300, 31200, 156240, 177840, 288288, 143520, 927360, 1203840, 752400, 1242360, 2702700, 2948400, 3996720, 1884960, 5896800, 2692800, 1244880, 15800400, 4586400, 11060280, 15301440, 14414400

Offset: 0

Michel Lagneau, Dec 27 2012

nonn

It appears that a(n) is even for n > 0 and nonsquarefree for n > 2. The corresponding triangle of k in which row n gives the n primes starts:
k = 1 -> no prime
k = 3  -> 3;
k = 2  -> 2, 3;
k = 140 -> 293, 307, 317;
k = 560 -> 1373, 1451, 1481, 1487.

			a(3) = 140 because the partial sums of the divisors in decreasing order {140, 70, 35, 28, 20, 14, 10, 7, 5, 4, 2, 1} that generate 3 prime numbers are
140 + 70 + 35 + 28 + 20 = 293;
140 + 70 + 35 + 28 + 20 + 14 = 307;
140 + 70 + 35 + 28 + 20 + 14 + 10 = 317.

Amiram Eldar, Table of n, a(n) for n = 0..77

Cf. A023194, A062700, A000203, A187822.

with(numtheory):for n from 0 to 40
do:ii:=0:for k from 1 to 4000000 while(ii=0) do:s:=0:x:=divisors(k):n1:=nops(x):it:=0:for a from n1 by -1 to 1 do:s:=s+x[a]:if type(s,prime)=true then it:=it+1:else fi:od: if it = n then ii:=1: printf ( "%d %d \n",n,k):else fi:od:od:

a[n_] := Catch[ For[k = 1, True, k++, cnt = Count[ Accumulate[ Divisors[k] // Reverse], ?PrimeQ]; If[cnt == n, Print[{n, k}]; Throw[k]]]]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover, Dec 27 2012 *)

a(19)-a(33) by Jean-François Alcover, Dec 28 2012

A248963 Prime powers p^m for which sigma(p^2m) is not prime.

Original entry on oeis.org

1, 7, 9, 11, 13, 16, 19, 23, 25, 29, 31, 32, 37, 43, 47, 53, 61, 67, 73, 79, 81, 83, 97, 103, 107, 109, 113, 121, 127, 128, 137, 139, 149, 151, 157, 163, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, 257, 263, 269, 271, 277, 281, 283, 307, 311, 313, 317, 331

Offset: 1

M. F. Hasler, Oct 18 2014

nonn

sigma(x) cannot be prime unless x is a square of a prime power, x = p^2m, cf. A055638 and A023194. This sequence lists the complement: prime powers whose square does not have a prime sum of divisors.

Although generally 1 is not considered a prime power, it seemed logical for various good reasons to include the initial term a(1)=1.

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

Cf. A000961, A055638, A023194, A023195, A062700.

for(n=1,999,isprimepower(n)||next;isprime(sigma(n^2))||print1((n)","))

A248963 = A000961 \ A055638, i.e., the complement of A055638 in A000961.

A292446 Numbers k such that sigma((k + 1) / 2) is a prime q.

Original entry on oeis.org

3, 7, 17, 31, 49, 127, 577, 1457, 3361, 4801, 6961, 8191, 10081, 15841, 20401, 31249, 34321, 55777, 57121, 59857, 131071, 167041, 171697, 293377, 524287, 559681, 916657, 982801, 1062881, 1104097, 1158241, 1195057, 1367857, 1407841, 1414561, 1468897, 1659841

Offset: 1

Jaroslav Krizek, Sep 16 2017

Corresponding values of primes q are in A062700.
Prime terms are in A292447.
Mersenne primes p = 2^k - 1 (A000668) are terms: sigma((p + 1) / 2) = sigma((2^k - 1 + 1) / 2) = sigma(2^(k - 1)) = 2^k - 1.
This sequence also has terms of the form p^(q-1) where p and q are odd primes, i.e., A002315(1)^2 = 7^2 and A002315(3)^2 = 239^2. Terms that are not squarefree are 49, 55777, 57121, 167041, 2789521, 50060017, ... - Altug Alkan, Oct 02 2017

			49 is a term because sigma((49 + 1) / 2) = sigma(25) = 31 (prime).

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

Cf. A000203, A000668, A023194, A023195, A062700, A292447, A292448.

[n: n in [1..10^8] | IsOdd(n) and IsPrime(SumOfDivisors((n+1) div 2))];

Select[Range[1,166*10^4,2],PrimeQ[DivisorSigma[1,(#+1)/2]]&] (* Harvey P. Dale, Jun 22 2022 *)

isok(n) = (n%2) && isprime(sigma((n+1)/2)); \\ Michel Marcus, Sep 16 2017

a(n) = 2*A023194(n) - 1.

cp's OEIS Frontend

A085379 Greatest prime as sum of distinct divisors of n.

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A229266 Primes of the form sigma(k) + tau(k) + phi(k), where sigma(k) = A000203(k), tau(k) = A000005(k) and phi(k) = A000010(k).

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A187822 Smallest k such that the partial sums of the divisors of k (taken in increasing order) contain exactly n primes.

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A229265 Numbers k such that sigma(k) + tau(k) + phi(k) is a prime, where sigma(k) = A000203(k), tau(k) = A000005(k) and phi(k) = A000010(k).

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A071166 a(n) = n - A006530(A000203(n)), difference between n and largest prime factor of the sum of its divisors.

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A187825 Smallest k such that the partial sums of the divisors of k (in decreasing order) generate n primes.

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Maple

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Extensions

A248963 Prime powers p^m for which sigma(p^2m) is not prime.

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PARI

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A292446 Numbers k such that sigma((k + 1) / 2) is a prime q.

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