A093893 -id:A093893 - cp's OEIS frontend

A093894 Composite members of A093893.

49, 87, 91, 121, 133, 169, 183, 213, 217, 247, 249, 259, 287, 301, 339, 343, 361, 403, 411, 427, 445, 469, 473, 481, 501, 511, 527, 529, 553, 559, 581, 589, 591, 633, 679, 699, 703, 713, 717, 721, 763, 789, 793, 817, 841, 843, 871, 889, 895, 949, 951, 961

Offset: 1

Amarnath Murthy, Apr 23 2004

nonn

Comment: Most terms of this sequence have four divisors. Some terms (the squares of primes) have three divisors; very few terms have more than four divisors (the first such term is 4753, with six). Conjecture: This sequence is infinite. - Adam M. Kalman (mocha(AT)clarityconnect.com), Nov 11 2004

			133 is a term, the divisors are 1,7,19,133 and no sum of two or more gives a prime.

Charles R. Greathouse IV, Sep 10, 2008, Table of n, a(n) for n = 1..10001

Cf. A093890, A093891, A093892, A093893.

For[a:=4, a<=2000, s =Divisors[a];n := 1;d := False; While[(n<=2^Length[s])\[And]( ["not" character]d), If[Length[NthSubset[n, s]]>=2, If[ !PrimeQ[Plus@@NthSubset[n, s]], n++, d:= True], n++ ]]; If[ ["not" character]d, Print[a]];a++;While[PrimeQ[a], a+=2]]; (* Adam M. Kalman, Nov 11 2004 *)

Corrected and extended by Adam M. Kalman (mocha(AT)clarityconnect.com), Nov 11 2004

A161510 Number of primes formed as the sum of distinct divisors of n, counted with repetition.

Original entry on oeis.org

0, 2, 1, 4, 1, 6, 1, 6, 2, 7, 1, 20, 1, 5, 4, 11, 1, 16, 1, 19, 5, 5, 1, 66, 2, 5, 4, 17, 1, 64, 1, 18, 4, 6, 6, 120, 1, 5, 5, 63, 1, 62, 1, 18, 11, 5, 1, 237, 1, 15, 3, 18, 1, 47, 6, 60, 5, 7, 1, 863, 1, 3, 20, 31, 6, 58, 1, 16, 3, 62, 1, 808, 1, 4, 13, 16, 4, 56, 1, 216, 5, 5, 1, 839, 5, 5

Offset: 1

T. D. Noe, Jun 17 2009

nonn

That is, if a number has d divisors, then we compute all 2^d sums of distinct divisors and count how many primes are formed. Sequence A093893 lists the n that produce no primes except for the primes that divide n. The Mathematica code works well for numbers up to about 221760, which has 168 divisors and creates a polynomial of degree 950976. The coefficients of the prime powers of that polynomial sum to 28719307224839120896278355000770621322645671888269, the number of primes formed by the divisors of 221760. Records appear to occur at n=10 and n in A002182, the highly composite numbers.

			a(4) = 4 because the divisors (1,2,4) produce 4 primes (2,1+2,1+4,1+2+4).

Felix Huber, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

with(NumberTheory):
A161510:=proc(n)
    local b,l,j;
    l:=[(Divisors(n))[]]:
    b:=proc(m,i)
        option remember;
        `if`(m=0,1,`if`(i<1,0,b(m,i-1)+`if`(l[i]>m,0,b(m-l[i],i-1))))
    end;
    add(b(ithprime(j),nops(l)),j=1..pi(sigma(n)));
end:
seq(A161510(n),n=1..77); # Felix Huber, Jul 24 2025

CountPrimes[n_] := Module[{d=Divisors[n],t,lim,x}, t=CoefficientList[Product[1+x^d[[i]], {i,Length[d]}], x]; lim=PrimePi[Length[t]-1]; Plus@@t[[1+Prime[Range[lim]]]]]; Table[CountPrimes[n], {n,100}]

a(n) = my(nb=0, d=divisors(n)); forsubset(#d, s, nb+=isprime(sum(i=1, #s, d[s[i]]))); nb; \\ Michel Marcus, Jul 24 2025

A160763 Least number having n divisors such that every sum of two or more divisors is composite.

Original entry on oeis.org

1, 3, 49, 87, 130321, 4753, 7212549413161, 285541, 7890946561, 834472284661, 174913992535407978606601, 19699251391, 23205949656945057666311162427422570380321

Offset: 1

Robert G. Wilson v, May 25 2009, May 29 2009

nonn

First term of A093893 to have n divisors.

a(2)=3, a(3)=7^2, a(4)=3*29, a(5)=19^4, a(6)=7^2*97, a(7)=139^6, a(8)=31*61*151, a(9)=211^2*421^2, a(10)=211^4*421, a(11)=211^10, a(12)=211^2*421*1051, a(13)=2311^12, 5.92*10^20

Proof that a(n) exists for all n: We will show that there is a prime p such that the sums of two or more divisors of p^(n-1) are all composite. Let Q be the product of the primes less than or equal to n. Let p be a prime of the form Qk+1. Observe that the divisors of p^(n-1), which are just powers of p, have the same form Qk+1 (but with different k, of course). Hence a sum of r of these powers will have the form Qk+r (for some k). Due to the way Q is constructed and r <= n, r and Q have a common factor, making Qk+r composite. Furthermore, by Dirichlet's theorem, we know there are an infinite number of primes p that will work for each n. [T. D. Noe, Jun 01 2009]

If a(14) < 2311^6*50821, then a(14) = p^6*q with primes p,q such that 139<=p<1000 and p^6 in A093893. - Hagen von Eitzen, Jun 03 2009

If a(14) < 2311^6*50821, then a(14) = p^6*q with p in {139,151,181,211,241} and q being prime. - Max Alekseyev, Sep 24 2015

Cf. A093893, A000005.

(* first do *) Needs["Combinatorica`"] (* then *) f[n_] := Block[{d = Divisors@n, k, mx = 1 + 2^DivisorSigma[0, n]}, k = 2 + Length@d; While[k < mx, If[ PrimeQ[Plus @@ NthSubset[k, d]], Break[]]; k++ ]; If[k == mx, Length@d, 0]]; t = Table[0, {20}]; k = 1; While[k < 2*10^7, a = f@k; If[a > 0 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k += 2]; t

Definition revised by N. J. A. Sloane, May 30 2009
Term a(9) corrected, a(10)-a(13) and more upper bounds added by Max Alekseyev and Hagen von Eitzen, May 30 2009
a(19) from M. F. Hasler, May 31 2009
Edited by Max Alekseyev, Sep 25 2009
a(1)=1 prepended by Max Alekseyev, Mar 31 2015

A160994 a(n) is the least prime number p such that every sum of two or more divisors of p^n is composite.

Original entry on oeis.org

3, 7, 7, 19, 19, 139, 151, 211, 211, 211, 421, 2311, 2311, 92401, 120121, 120121, 180181, 2312311

Offset: 1

Robert G. Wilson v, Jun 01 2009

Cf. A093893, A093894.

(* first do *) Needs["Combinatorica`"] (* then *) f[n_] := Block[{d = Divisors@n, k, mx}, k = 1 + Length@d; mx = 2^Length[d]; While[k < mx && !PrimeQ[Plus @@ NthSubset[k, d]], k++ ]; If[k == mx, Length@d, 0]];
a[n_] := a[n] = Module[{p = If[n == 1, 2, a[n-1]]}, While[f[p^n] == 0, p = NextPrime[p]]; p]; Array[a, 13] (* second part of the program added by Amiram Eldar, Jul 30 2024 *)

Offset changed to 1 and name corrected by Amiram Eldar, Jul 30 2024

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A093894 Composite members of A093893.

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A161510 Number of primes formed as the sum of distinct divisors of n, counted with repetition.

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A160763 Least number having n divisors such that every sum of two or more divisors is composite.

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A160994 a(n) is the least prime number p such that every sum of two or more divisors of p^n is composite.

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