A023195 -id:A023195 - cp's OEIS frontend

A285017 Primes of the form 1 + n + n^2 + n^3 + ... + n^k, n > 1, k > 1 where n is not prime.

43, 73, 157, 211, 241, 421, 463, 601, 757, 1123, 1483, 2551, 2971, 3307, 3907, 4423, 4831, 5701, 6007, 6163, 6481, 8191, 9901, 11131, 12211, 12433, 13807, 14281, 19183, 20023, 20593, 21757, 22621, 22651, 23563, 24181, 26083, 26407, 27061, 28393, 31153, 35533

Offset: 1

Bernard Schott, Apr 08 2017

nonn

These numbers are Brazilian primes belonging to A085104.
Brazilian primes with n prime are A023195, except 3 which is not Brazilian.
A085104 = This sequence Union { A023195 \ number 3 }.
k + 1 is necessarily prime, but that's not sufficient: 1 + 10 + 100 = 111.
Most of these terms come from A185632 which are prime numbers 111_n with n no prime, the first other term is 22621 = 11111_12, the next one is 245411 = 11111_22.
Number of terms < 10^k: 0, 2, 9, 23, 64, 171, 477, 1310, 3573, 10098, ..., . - Robert G. Wilson v, Apr 15 2017

			157 = 12^2 + 12 + 1 = 111_12 is prime and 12 is composite.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1310 from Robert G. Wilson v)
Bernard Schott, Les nombres brésiliens, Reprinted from Quadrature, no. 76, April-June 2010, pages 30-38, included here with permission from the editors of Quadrature.

Cf. A002383, A023195, A053183, A053696, A085104, A088548, A088550, A185632, A190527, A194257.

N:= 40000: # to get all terms <= N
res:= NULL:
for k from 2 to ilog2(N) do if isprime(k) then
  for n from 2 do
    p:= (n^(k+1)-1)/(n-1);
    if p > N then break fi;
    if isprime(p) and not isprime(n) then res:= res, p fi
od fi od:
res:= {res}:
sort(convert(res,list)); # Robert Israel, Apr 14 2017

mx = 36000; g[n_] := Select[Drop[Accumulate@Table[n^ex, {ex, 0, Log[n, mx]}], 2], PrimeQ]; k = 1; lst = {}; While[k < Sqrt@mx, If[CompositeQ@k, AppendTo[lst, g@k]; lst = Sort@Flatten@lst]; k++]; lst (* Robert G. Wilson v, Apr 15 2017 *)

isok(n) = {if (isprime(n), forcomposite(b=2, n, d = digits(n, b); if ((#d > 2) && (vecmin(d) == 1) && (vecmax(d)== 1), return(1)););); return(0);} \\ Michel Marcus, Apr 09 2017

A285017_vec(n)={my(h=vector(n,i,1),y,c,z=4,L:list);L=List();forprime(x=3,,forcomposite(m=z,x-1,y=digits(x,m);if((y==h[1..#y])&&2<#y,listput(L,x);z=m;if(c++==n,return(Vec(L))))))} \\ R. J. Cano, Apr 18 2017

A152677 Subsequence of odd terms in A000203 (sum-of-divisors function sigma), in the order in which they occur and with repetitions.

Original entry on oeis.org

1, 3, 7, 15, 13, 31, 39, 31, 63, 91, 57, 93, 127, 195, 121, 171, 217, 133, 255, 403, 363, 183, 399, 465, 403, 399, 511, 819, 307, 847, 549, 381, 855, 961, 741, 1209, 931, 1023, 553, 1651, 921, 781, 1815, 1281, 1143, 1093, 1767, 1953, 871, 2223, 2821, 993, 1995

Offset: 1

Omar E. Pol, Dec 10 2008

Equivalently: subsequence of A000203 (sigma) with indices equal to a square or twice a square (A028982).

See A060657 for the set of odd values in the range of the sigma function, i.e., the list of odd values in ordered by increasing size and without repetitions.

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

Cf. A000203 (sigma = sum-of-divisors function), A152678 (even terms in A000203), A028982 (squares and twice the squares).
See A062700 and A023195 for the subsequence resp. subset of primes; A023194 for the indices of A000203 which yield these primes.
Cf. A002117.

[d:k in [1..1000]|IsOdd(d) where d is DivisorSigma(1,k)]; // Marius A. Burtea, Jan 09 2020

Select[DivisorSigma[1, Range[1000]], OddQ[#] &] (* Giovanni Resta, Jan 08 2020 *)
With[{max = 1000}, DivisorSigma[1, Union[Range[Sqrt[max]]^2, 2*Range[Sqrt[max/2]]^2]]] (* Amiram Eldar, Nov 28 2023 *)

A152677_upto(lim)=apply(sigma,vecsort(concat(vector(sqrtint(lim\1), i, i^2), vector(sqrtint(lim\2), i, 2*i^2)))) \\ Gives [a(n) = sigma(k) with k = A028982(n) <= lim]. - Charles R Greathouse IV, Feb 15 2013, corrected by M. F. Hasler, Jan 08 2020

a(n) = A000203(A028982(n)). - R. J. Mathar, Dec 12 2008

Sum_{k=1..n} a(k) ~ c * n^3, where c = (16-10*sqrt(2))*zeta(3)/Pi^2 = 0.226276... . - Amiram Eldar, Nov 28 2023

Extended by R. J. Mathar, Dec 12 2008

Edited and definition reworded by M. F. Hasler, Jan 08 2020

A330832 Numbers of the form p*q, where p is prime and q=(p^k-1)/(p-1) is also prime for some integer k>1.

Original entry on oeis.org

6, 14, 39, 62, 155, 254, 3279, 5219, 16382, 19607, 70643, 97655, 208919, 262142, 363023, 402233, 712979, 1040603, 1048574, 1508597, 2265383, 2391483, 4685519, 5207819, 6728903, 21243689, 25239899, 56328959, 61035155, 67977559, 150508643, 310747739, 344964203

Offset: 1

Walter Kehowski, Jan 08 2020

Also numbers with power-spectral basis {q,p^k}. The equation q=(p^k-1)/(p-1) is equivalent to the decomposition of the identity q + p^k = pq + 1 in Z/pqZ, and it is now easily verified that {q,p^k} is the spectral basis of p*q, consisting of primes and powers.

The numbers p^(r^e)*q, where p, q, r are primes, and q=(p^(r^e)-1)/(p^(r^(e-1))-1), e>0, have power-spectral basis {q,p^(r^e)}. However, the primes q for e>1 are usually quite large, while e=1 is accessible. For example, the table in A003424 has 4738 entries with all primes q<10^12, but only 8 have y>1.

			a(5) = 5*(5^3-1)/(5-1) = 5*31 = 155. The number 155 has spectral basis {31,125}.

Walter Kehowski, Table of n, a(n) for n = 1..4731

Cf. A003424, A023194, A023195, A085104, A330833, A330834, A330835.

a(n) = A330833(n) * A330835(n).

A330833 a(n) = first prime factor p of the term A330832(n) = p*q.

Original entry on oeis.org

2, 2, 3, 2, 5, 2, 3, 17, 2, 7, 41, 5, 59, 2, 71, 13, 89, 101, 2, 17, 131, 3, 167, 173, 23, 29, 293, 383, 5, 13, 43, 677, 701, 743, 17, 761, 773, 827, 839, 857, 911, 1091, 1097, 5, 1163, 1181, 1193, 1217, 73, 1373, 1427, 79, 1487, 1559, 1583, 83, 2, 1709, 1811, 1847, 1931, 1973, 2129, 2273, 2309, 2339, 2411, 2663, 2729, 2789, 2957

Offset: 1

Walter Kehowski, Jan 08 2020

			a(5) = 5 and, since A330834(5) = 3, then A330835(5) = (5^3-1)/(5-1) = 31 is prime.

Walter Kehowski, Table of n, a(n) for n = 1..4731

Cf. A003424, A023194, A023195, A085104, A330832, A330834, A330835.

A330834 The exponents k of A330832, that is, if A330832(n)=p*q, where p is prime and q=(p^k-1)/(p-1) is prime, then a(n)=k.

Original entry on oeis.org

2, 3, 3, 5, 3, 7, 7, 3, 13, 5, 3, 7, 3, 17, 3, 5, 3, 3, 19, 5, 3, 13, 3, 3, 5, 5, 3, 3, 11, 7, 5, 3, 3, 3, 7, 3, 3, 3, 3, 3, 3, 3, 3, 13, 3, 3, 3, 3, 5, 3, 3, 5, 3, 3, 3, 5, 31, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 7, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Walter Kehowski, Jan 08 2020

nonn

			a(5) = 3, and, since A330833(5)=5, then A330835(5)=(5^3-1)/(5-1) = 31 is prime.

Walter Kehowski, Table of n, a(n) for n = 1..4731
G. Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly 108(4), April 2001.

Cf. A003424, A023194, A023195, A085104, A330832, A330833, A330835.

A330835 Primes q appearing in A330832: that is, if A330832(n)=p*q, where p is prime and q=(p^k-1)/(p-1) is prime, then a(n)=q.

Original entry on oeis.org

3, 7, 13, 31, 31, 127, 1093, 307, 8191, 2801, 1723, 19531, 3541, 131071, 5113, 30941, 8011, 10303, 524287, 88741, 17293, 797161, 28057, 30103, 292561, 732541, 86143, 147073, 12207031, 5229043, 3500201, 459007, 492103, 552793, 25646167, 579883, 598303, 684757

Offset: 1

Walter Kehowski, Jan 08 2020

nonn

The terms in the b-file are the same as those of A003424 with y=1, but with an ordering based on that of A330832. The ordering allows the inclusion of the only duplicate 2^5-1=31 and (5^3-1)/(5-1)=31.

			a(5)=31 since A330833(5)=5, A330834(5)=3, and (5^3-1)/(5-1) = 31 is prime.

Walter Kehowski, Table of n, a(n) for n = 1..4731
G. Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly 108(4), April 2001.

Cf. A003424, A023194, A023195, A085104, A330832, A330833, A330834.

a(n) = (A330833(n) ^ A330834(n) - 1) / (A330833(n) - 1).

A368651 Numbers k such that 2^sigma(k) - k is a prime.

Original entry on oeis.org

3, 5, 17, 49, 53, 185, 503, 1301, 1689, 1797, 5929, 14747, 20433, 29903, 42137, 64763

Offset: 1

J.W.L. (Jan) Eerland, Jan 02 2024

If it exists, a(17) > 120000. - Michael S. Branicky, Aug 19 2024

			5 is in the sequence because 2^sigma(5)-5 = 2^6-5 = 59 is prime.

Cf. A000043, A000203, A000668, A023194, A023195, A253850, A253851.

[n: n in[1..10000] | IsPrime((2^SumOfDivisors(n)) - n)];

a[n_] := Select[Range@ n, PrimeQ[2^DivisorSigma[1, #] - #] &]; a[20000]
DeleteCases[ParallelTable[If[PrimeQ[2^DivisorSigma[1,k]-k],k,n],{k,1,10^4}],n]

a(16) from J.W.L. (Jan) Eerland, Jan 25 2024

A127727 Primes of the form p^e - p^(e-1) + p^(e-2) - ... + (-1)^e, where p is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 43, 61, 157, 521, 547, 683, 2731, 4423, 6163, 13421, 19183, 22651, 26407, 37057, 43691, 113233, 121453, 143263, 174763, 208393, 292141, 375157, 398581, 412807, 527803, 590593, 843643, 981091, 1041421, 1193557, 1246573

Offset: 1

T. D. Noe, Jan 25 2007

These primes are important in studying k-imperfect numbers (A127724), see Iannucci-link. Except for the cases p^e = 3 and 8, which yield primes 2 and 5, e is an even number such that e+1 is prime. In fact, except for those two cases, all the primes are of the form (1+p^q)/(1+p), where q is an odd prime; that is, repunit primes with negative prime base.

			From _David A. Corneth_, Oct 28 2017: (Start)
For (p, e) = (3, 1) we have the prime 3^1 - 3^0 = 2.
For (p, e) = (2, 3) we have the prime 2^3 - 2^2 + 2^1 - 2^0 = 5.
The examples above are the cases mentioned in the comments not of the form (1+p^q)/(1+p). A prime of that form is below;
For (p, e) = (2, 4) we have the prime 2^4 - 2^3 + 2^2 - 2^1 + 2^0 = 11 = (1+p^(e+1)) / (1+p) = 33/3. (End)

David A. Corneth, Table of n, a(n) for n = 1..33914 (terms < 10^14) (the first 4799 terms < 10^12 from T. D. Noe)
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), Article 00.2.7.
Douglas E. Iannucci, On a variation of perfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 6 (2006), #A41.

Cf. A023195, A065507, A084741, A206369.

upto(n) = {my(res = List([2,5])); forprime(p = 2, sqrtnint(n, 2), forprime(q = 3, logint(n * (1+p), p), r = (1+p^q)/(1+p); if(isprime(r), listput(res, r)))); listsort(res, 1); res} \\ David A. Corneth, Oct 28 2017

A253849 Numbers k such that 2^sigma(k) - 1 is a prime.

Original entry on oeis.org

2, 4, 9, 16, 25, 64

Offset: 1

Jaroslav Krizek, Jan 16 2015

Also numbers n such that sigma(n) is in A000043, i.e., p = 2^sigma(n) - 1 is a Mersenne prime (A000668). The sequence of corresponding primes p reads: 7, 127, 8191, 2147483647, 2147483647, 170141183460469231731687303715884105727, ..., see A253851.
Subsequence of A023194 (numbers n such that sigma(n) is a prime), see there for an explanation why all terms except the first one are squares.
The sequence of values of sigma(a(n)) is 3, 7, 13, 31, 31, 127, ... and each term of this sequence must be a prime from the sequence of Mersenne exponents (A000043). See A253850.
Sequence differs from A023194 because A023194(7) = 289 but if a(7) exists, it must be a number n such that sigma(n) > A000043(43) = 30402457.
a(n) must be an even power of a prime. If it is the square of an odd prime, then this prime must be in A053182. If a(n) is an even power of 2, a(n)=2^(2k), then sigma(a(n))=2^(2k+1)-1. Thus, 2k+1 must be a double Mersenne prime exponent, i.e., such that the corresponding Mersenne prime is again a Mersenne exponent, cf. A103901. Only 4 such primes are known, and a(6)=2^6 (k=3) corresponds to the largest known prime of this type, 2^(2k+1)-1 = 127. - M. F. Hasler, Jan 21 2015

			4 is in the sequence because 2^sigma(4)-1 = 2^7-1 = 127 is prime.

Cf. A000043, A000203, A000668, A023194, A023195, A253850, A253851.

[n: n in[1..10000] | IsPrime((2^SumOfDivisors(n)) - 1)];

a253849[n_] := Select[Range@ n, PrimeQ[2^DivisorSigma[1, #] - 1] &]; a253849[20000] (* Michael De Vlieger, Jan 19 2015 *)

A367460 Numbers k such that 3^sigma(k) - k is a prime.

Original entry on oeis.org

1, 10, 52, 400, 2480, 7202, 28222

Offset: 1

J.W.L. (Jan) Eerland, Jan 26 2024

a(8) > 67569.

			10 is in the sequence because 3^sigma(10) - 10 = 3^18 - 10 = 387420479 is prime.

Cf. A000043, A000203, A000668, A023194, A023195, A253850, A253851, A368651.

[n: n in[1..10000] | IsPrime((3^SumOfDivisors(n)) - n)]

a[n_] := Select[Range@ n, PrimeQ[3^DivisorSigma[1, #] - #] &]; a[20000]
DeleteCases[ParallelTable[If[PrimeQ[3^DivisorSigma[1,k]-k],k,n],{k,1,10^4}],n]

a(7) from Michael S. Branicky, Jan 27 2024

cp's OEIS Frontend

A285017 Primes of the form 1 + n + n^2 + n^3 + ... + n^k, n > 1, k > 1 where n is not prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

A152677 Subsequence of odd terms in A000203 (sum-of-divisors function sigma), in the order in which they occur and with repetitions.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A330832 Numbers of the form p*q, where p is prime and q=(p^k-1)/(p-1) is also prime for some integer k>1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A330833 a(n) = first prime factor p of the term A330832(n) = p*q.

Views

Author

Keywords

Examples

Links

Crossrefs

A330834 The exponents k of A330832, that is, if A330832(n)=p*q, where p is prime and q=(p^k-1)/(p-1) is prime, then a(n)=k.

Views

Author

Keywords

Examples

Links

Crossrefs

A330835 Primes q appearing in A330832: that is, if A330832(n)=p*q, where p is prime and q=(p^k-1)/(p-1) is prime, then a(n)=q.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A368651 Numbers k such that 2^sigma(k) - k is a prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

Extensions

A127727 Primes of the form p^e - p^(e-1) + p^(e-2) - ... + (-1)^e, where p is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI