A180852 -id:A180852 - cp's OEIS frontend

A180851 Sum of increasing powers of divisors: a(n) = Sum_{i=1..q} d(i)^i where d(1) < d(2) < ... < d(q) are the divisors of n.

1, 5, 10, 69, 26, 1328, 50, 4165, 739, 10130, 122, 2994048, 170, 38764, 50760, 1052741, 290, 34072601, 362, 64100694, 194834, 235592, 530, 110111416192, 15651, 459178, 532180, 482430598, 842, 656271867808, 962, 1074794565, 1187262

Offset: 1

Jason Earls, Sep 21 2010

			For n=4, the divisors of 4 are [1, 2, 4] and summing them as increasing powers yields: 1^1+2^2+4^3 = 69.
For n=12, the divisors of 12 are [1, 2, 3, 4, 6, 12] and summing them as increasing powers yields: 1^1+2^2+3^3+4^4+6^5+12^6 = 2994048.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A027750.
Positions of primes: A180852.
Comparable sequences: A055225, A264786, A344459.

f:= proc(n) local D,k;
  D:=sort(convert(numtheory:-divisors(n),list));
  add(D[k]^k,k=1..nops(D))
end proc:
map(f, [$1..100]); # Robert Israel, Sep 11 2020

Total[Divisors[#]^Range[DivisorSigma[0,#]]]&/@Range[40] (* Harvey P. Dale, Aug 16 2011 *)

a(n) = my(d = divisors(n)); sum(k=1, #d, d[k]^k); \\ Michel Marcus, Jan 01 2016

Name made precise by Peter Munn, Sep 19 2024

A358199 a(n) is the least integer whose sum of the i-th powers of the proper divisors is a prime for 1 <= i <= n, or -1 if no such number exists.

Original entry on oeis.org

4, 4, 981, 8829, 8829, 122029105, 2282761881

Offset: 1

Jean-Marc Rebert, Nov 02 2022

			4 is a term since the strict divisors of 4 are {1, 2}, 1^1 + 2^1 = 3 and 1^2 + 2^2 = 5 are prime and no number < 4 has this property.

Cf. A001065 (s1), A180852, A357324, A057709, A000203, A001157, A001158.

Subsequence of A037020.

card(n)=my(c=1,s=0);s=sigma(n)-n;while(isprime(s),c++;s=sigma(n,c)-n^c);c--
a(n)=my(x=0);for(k=1,+oo,x=card(k);if(x>=n,return(k)))

from itertools import count
from math import prod
from sympy import isprime, factorint
def A358199(n):
    for m in count(2):
        f = factorint(m).items()
        if all(map(isprime,(prod((p**((e+1)*i)-1)//(p**i-1) for p,e in f) - m**i for i in range(1,n+1)))):
            return m # Chai Wah Wu, Nov 15 2022

A376222 Numbers k such that Sum_{i=1..q-1} d(i)^i is prime, where d(1)

Original entry on oeis.org

4, 21, 27, 39, 57, 77, 183, 189, 203, 205, 219, 237, 253, 371, 387, 391, 417, 489, 565, 611, 655, 667, 669, 675, 687, 749, 767, 799, 831, 849, 897, 921, 955, 1007, 1047, 1135, 1189, 1207, 1349, 1371, 1379, 1407, 1421, 1461, 1469, 1497, 1513, 1569, 1633, 1643, 1659

Offset: 1

Michel Lagneau, Sep 16 2024

nonn

The corresponding primes are in A376223.

			39 is a term because the 3 first divisors of 39 are {1,3,13} and 1^1 + 3^2 + 13^3 = 2207 is prime.
189 is a term since the 7 first divisors of 189 are {1, 3, 7, 9, 21, 27, 63} and 1^1+3^2+7^3+9^4+21^5+27^6+63^7 = 3939372150671 is prime.

Cf. A180852, A376223.

with(numtheory):nn:=1700:
for n from 1 to nn do:
d:=divisors(n):n0:=nops(d):s:=sum(‘d[k]^k’, ‘k’=1..n0-1):
   if isprime(s)
    then
     printf(`%d,`,n):
    else
   fi:
od:

Select[Range[1700],PrimeQ[Sum[Part[Divisors[#],i]^i,{i,DivisorSigma[0,#]-1}]] &] (* Stefano Spezia, Sep 16 2024 *)

isok(k) = my(d=divisors(k)); isprime(sum(j=1, #d-1, d[j]^j)); \\ Michel Marcus, Sep 16 2024

A376223 a(n) = Sum_{i=1..q-1} d(i)^i where d(i) are the q sorted divisors of A376222(n).

Original entry on oeis.org

5, 353, 739, 2207, 6869, 1381, 226991, 3939372150671, 24439, 68947, 389027, 493049, 12289, 148927, 35726471189, 12457, 2685629, 4330757, 1442923, 103993, 2248117, 24919, 11089577, 74820287157480518691978649, 12008999, 1225093, 205549, 104113, 21253943, 22665197

Offset: 1

Michel Lagneau, Sep 16 2024

nonn

By the definition of A376222 all terms are prime.

			a(4) = 2207 because A376222(4) = 39 and the proper divisors of 39 are {1,3,13} with 1^1 + 3^2 + 13^3 = 2207.

Cf. A180852, A376222.

with(numtheory):nn:=900:
for n from 1 to nn do:
d:=divisors(n):n0:=nops(d):p:=sum(‘d[k]^k’, ‘k’=1..n0-1):
   if isprime(p)
    then
     printf(`%d,`,p):
    else
   fi:
od:

cp's OEIS Frontend

A180851 Sum of increasing powers of divisors: a(n) = Sum_{i=1..q} d(i)^i where d(1) < d(2) < ... < d(q) are the divisors of n.

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A358199 a(n) is the least integer whose sum of the i-th powers of the proper divisors is a prime for 1 <= i <= n, or -1 if no such number exists.

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A376222 Numbers k such that Sum_{i=1..q-1} d(i)^i is prime, where d(1)

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A376223 a(n) = Sum_{i=1..q-1} d(i)^i where d(i) are the q sorted divisors of A376222(n).

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