A000043 -id:A000043 - cp's OEIS frontend

A330818 Numbers of the form 2^(2*p+1), where p is a Mersenne exponent, A000043.

32, 128, 2048, 32768, 134217728, 34359738368, 549755813888, 9223372036854775808, 10633823966279326983230456482242756608, 766247770432944429179173513575154591809369561091801088

Offset: 1

Walter Kehowski, Jan 01 2020

nonn

Also the first factor of A330817, 2^(2*p+1)*M_p^2. The second factor of A330817 is A133049.

			a(1) = 2^(2*2+1) = 32. Since M_2=3, the number 2^5*3^2 has power-spectral basis {225,64}.

Walter Kehowski, Table of n, a(n) for n = 1..12

Cf. A000043, A000668, A132794, A133049, A330817, A330819, A330820.

A330818:=[]:
for www to 1 do
for i from 1 to 31 do
  #ithprime(31)=127
  p:=ithprime(i);
  q:=2^p-1;
  if isprime(q) then x:=2^(2*p+1); A330818:=[op(A330818),x]; fi;
od;
od;
A330818;

2^(2 * MersennePrimeExponent[Range[10]] + 1) (* Amiram Eldar, Jan 03 2020 *)

a(n) = 2^(2*A000043(n)+1).

A139116 a(n) = p*(p-1)/2, where p is A000043(n).

Original entry on oeis.org

1, 3, 10, 21, 78, 136, 171, 465, 1830, 3916, 5671, 8001, 135460, 183921, 817281, 2425503, 2600340, 5172936, 9041878, 9779253, 46933516, 49406770, 62860078, 198732016, 235455850, 269317236, 989969256, 3718884403, 6105401253, 8718403176, 23347552095, 286402257541

Offset: 1

Omar E. Pol, May 10 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Harvey P. Dale)

Cf. A000043, A000217, A036689, A139306, A139307, A139116, A139223, A139224, A139225, A139226.

(#(#-1))/2&/@MersennePrimeExponent[Range[47]] (* Harvey P. Dale, Aug 13 2021 *)

a(n) = A000043(n)*(A000043(n)-1)/2.

a(24)-a(32) from Harvey P. Dale, Aug 13 2021

A180221 Numbers that can be written as sum of one or more distinct elements of A000043. Numbers k for which sigma(A180162(k))=2^k, k>=2.

Original entry on oeis.org

2, 3, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Walter Kehowski, Aug 16 2010

nonn

The distinct values of log_2(sigma(m)), where m > 1 is a term of A046528. - Amiram Eldar, Jun 02 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1855 from Walter A. Kehowski)

Cf. A000043, A180162, A046528, A180169. Complement of A078426.

n = 10; p = MersennePrimeExponent[Range[n]]; Rest[-1 + Position[CoefficientList[Series[Product[(1 + x^p[[k]]), {k, 1, n}], {x, 0, p[[-1]]}], x], ?(# > 0 &)] // Flatten] (* _Amiram Eldar, Jun 02 2020 *)

A242025 Primes of the form R = 2^k-1+(2^k-2)/(2^(p-k)-1), where p are Mersenne prime exponents listed in A000043.

Original entry on oeis.org

5, 17, 29, 41, 2729, 8737, 65537, 74897, 174761, 715827881, 153722867280912929, 302379100949042568368129

Offset: 1

M. F. Hasler, Aug 17 2014

Related to the search for large primitive weird numbers: Kravitz has shown that 2^(k-1)*Q*R is a primitive weird number when Q > 2^k and R = (2^k*Q-Q-1)/(Q+1-2^k) both are prime, cf. subset A258882 of A002975. Here we consider such primes for the special case of Mersenne primes Q = 2^p-1, p in A000043. For such Q one has R = 2^k-1+(2^k-2)/(2^(p-k)-1), which must be an integer and prime number.
See A242998 for the number of exponents k leading to primes R, for given Q = A000668(n) = 2^p-1, p = A000043(n). But there is no one-to-one correspondence since the primes R are here listed according to their size (cf. example). The pairs (k,p) are given in A242999 and A243003.
Kravitz used his formula in 1976 to find the 53-digit PWN corresponding to a(11), cf. examples. In 2013, students of CWU used the same idea to find the next term in the series, corresponding to a(12), see examples. They found still larger PWN of the same form with other primes Q, see A320875. This renewed the interest in weird numbers and motivated several recent papers, cf. A002975. - M. F. Hasler, Nov 10 2018

			For given p = A000043(n), the following k yield a prime R and an associated (primitive) weird number W = 2^(k-1)*(2^p-1)*R in A258882 c A002975 c A006037:
For p = 2, no k yields a prime R = 2^k-1+(2^k-2)/(2^(p-k)-1).
For p = 3, k = 2 yields R = 5 and the (smallest) weird number W = 70 = A006037(1).
For p = 5, k = 4 yields R = 29 = a(3) and W = 7192 = A258882(3).
For p = 7, k = 4 yields R = 17 = a(2) and W = 17272 = A258882(7),
       and k = 5 yields R = 41 = a(4) and W = 83312 = A258882(9).
For p = 13, k = 11 yields R = 2729 = a(5) and W = 22889716736 = A258882(288)
For p = 17, k = 13 yields R = 8737 = a(6) and W = 4690605371392 = A258882(1203).
For p = 19, k = 16 yields R = 74897 = a(8), W = 1286718208049152 = A258882(7154),
        and k = 17 yields R = 174761 = a(9), W = 6004730783793152 = A258882(11466).
For p = 31, k = 16 yields R = 65537 = a(7) (smaller than both R's for p = 19),
        and k = 29 yields R = 715827881 = a(10).
For p = 61, only k = 57 yields a prime R = 153722867280912929 = a(11).
For p = 89, only k = 78 yields a prime R = 302379100949042568368129 = a(12).
For p = 107 through p = 86243, no k yields a prime R.
For p = 107 through p = 3021377, no k yields a prime R. - _Robert Price_, Sep 04 2019

M. F. Hasler, Table of n, a(n) for n = 1..12
CWU press release, CWU Math Students Break World Record for Largest Weird Number, Dec. 4, 2013.
S. Kravitz, A search for large weird numbers. J. Recreational Math. 9(1976), 82-85 (1977). Zbl 0365.10003

Cf. A258882 (PWN of the form 2^k*p*q).
Cf. A242993, A242998, A242999 and A243003: related to PWN with a prime factor R as defined here, with Q = A000668(n) = 2^A000043(n)-1.
Cf. A320875 for a more general pattern leading to more and larger PWN.

A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,
   1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,
   21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433,
   1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,
   24036583, 25964951, 30402457, 32582657, 37156667, 42643801,
   43112609};
lst = {};
For[i = 1, i <= Length[A000043], i++,
  p = A000043[[i]];
  For[k = 1, k < p, k++,
   r = 2^k - 1 + (2^k - 2)/(2^(p - k) - 1);
   If[! IntegerQ[r], Continue[]];
   If[PrimeQ[r], AppendTo[lst, r]]]];
Union[lst] (* Robert Price, Sep 04 2019 *)

A242999 Mersenne prime exponents p in A000043 such that R=2^k-1+(2^k-2)/(2^(p-k)-1) is prime for some k < p, listed with multiplicity (number of k's), see A243003 for the k-values.

Original entry on oeis.org

3, 5, 7, 7, 13, 17, 19, 19, 31, 31, 61, 89

Offset: 1

M. F. Hasler, Aug 17 2014

Related to the search for large primitive weird numbers: Kravitz has shown that 2^(k-1)*Q*R is a primitive weird number (cf A002975) when Q > 2^k and R = (2^k*Q-Q-1)/(Q+1-2^k) both are prime. In the special case of Mersenne primes Q = 2^p-1, p = A000043(n), considered here, one has R = 2^k-1+(2^k-2)/(2^(p-k)-1).
This sequence lists the p-values. See sequence A243003 for the k-values and A242998(n) for the number of possible k-values for a given p = A000043(n), i.e., the number of times this p appears here.
The next term, a(13), is larger than 80000 (if it exists).

			For given p = A000043(n), the following k's yield a prime R:
p : k's
2 : -
3 : 2
5 : 4
7 : 4, 5
13 : 11
17 : 13
19 : 16, 17
31 : 16, 29
61 : 57
89 : 78
107 through 86243 : none.
107 through 3021377: none. - _Robert Price_, Sep 04 2019
Thus the pairs (p,k) are (3,2), (5,4), (7,4), (7,5), (13,11), ... and the present sequence lists the first component of these pairs, sequence A243003 lists the second component.

S. Kravitz, A search for large weird numbers. J. Recreational Math. 9(1976), 82-85 (1977). Zbl 0365.10003

A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,
   1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,
   21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433,
   1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,
   24036583, 25964951, 30402457, 32582657, 37156667, 42643801,
   43112609};
lst = {};
For[i = 1, i <= 10, i++,
  p = A000043[[i]];
  For[k = 1, k < p, k++,
   r = 2^k - 1 + (2^k - 2)/(2^(p - k) - 1);
   If[! IntegerQ[r], Continue[]];
   If[PrimeQ[r], AppendTo[lst, p]]]];
lst (* Robert Price, Sep 04 2019 *)

forprime(p=1,,ispseudoprime(2^p-1)||next;for(k=p\2+1,p-1,(k-1)%(p-k)==0 && isprime(2^k-1+(2^k-2)/(2^(p-k)-1))&&print1(p","))) \\ M. F. Hasler, Jul 19 2016

A243003 Pairs (p,k) such that p is in A000043 and R=2^k-1+(2^k-2)/(2^(p-k)-1) is prime: this sequence lists the k-values, see A242999 for the p-values. (Ordered by p, then k.)

Original entry on oeis.org

2, 4, 4, 5, 11, 13, 16, 17, 16, 29, 57, 78

Offset: 1

M. F. Hasler, Aug 17 2014

Related to the search for large primitive weird numbers: Kravitz has shown that 2^(k-1)*Q*R is a primitive weird number (cf A002975) when Q > 2^k and R = (2^k*Q-Q-1)/(Q+1-2^k) both are prime. Here we consider the special case where Q = 2^p-1 is a Mersenne prime, p = A000043(n). For such Q one has R = 2^k-1+(2^k-2)/(2^(p-k)-1). [First R formula corrected by Jens Kruse Andersen, Aug 18 2014]
This sequence lists the k-values, see sequence A242999 for the p-values and A242998(n) for the number of possible k-values for given p = A000043(n) resp. Q = A000668(n).
This sequence can also be considered as a table whose n-th row holds the possible k-values for the n-th Mersenne prime Q = A000668(n); sequence A242998 gives the row lengths of the table (which are mostly 0).

			For given p=A000043(n), the following k's yield a prime R:
p : k's
2 : -
3 : 2
5 : 4
7 : 4, 5
13 : 11
17 : 13
19 : 16, 17
31 : 16, 29
61 : 57
89 : 78
107 through 86243 : none.
107 through 3021377: none. - _Robert Price_, Sep 04 2019
Thus the pairs (p,k) are (3,2), (5,4), (7,4), (7,5), (13,11), ... and the present sequence lists the second component of these pairs, the first components are listed in A242999.

S. Kravitz, A search for large weird numbers. J. Recreational Math. 9(1976), 82-85 (1977). Zbl 0365.10003

Cf. A258882 (PWN of the form 2^k*p*q).

Cf. A242993 (least k), A242998 (number of solution for given p in A000043), A242999 (values of p), A242025 list of all primes R.

A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,
   1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,
   21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433,
   1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,
   24036583, 25964951, 30402457, 32582657, 37156667, 42643801,
   43112609};
lst = {};
For[i = 1, i <= 10, i++,
  p = A000043[[i]];
  For[k = 1, k < p, k++,
   r = 2^k - 1 + (2^k - 2)/(2^(p - k) - 1);
   If[! IntegerQ[r], Continue[]];
   If[PrimeQ[r], AppendTo[lst, k]]]];
lst (* Robert Price, Sep 04 2019 *)

forprime(p=1,, ispseudoprime(2^p-1)||next; for(k=p\2+1, p-1, (k-1)%(p-k)==0 && isprime(2^k-1+(2^k-2)/(2^(p-k)-1))&&print1(k", "))) \\ M. F. Hasler, Jul 19 2016

One must have p/2 < k < p and (p-k) | (k-1).

A354532 Numbers k that are not Mersenne exponents (A000043) such that 2*(2^k-1) is in A354525.

Original entry on oeis.org

1, 9, 67, 137, 727

Offset: 1

Jianing Song, Aug 16 2022

2^a(n) - 1 is a semiprime for n = 2,3,4,5.

Conjecture: all terms beyond a(2) = 9 are primes.

			k = 9: 2^9 - 1 = 7*73 (not a prime), and we have 2*(2^9-1) + 7 = 7^3 is 7-smooth and 2*(2^9-1) + 73 = 3*5*73 is 73-smooth, so 9 is a term.
k = 67: 2^67 - 1 = 193707721*761838257287 (not a prime), and we have 2*(2^67-1) + 193707721 = 3*5^2*16033*1267117*193707721 is 193707721-smooth and 2*(2^67-1) + 761838257287 = 3*5011*25771*761838257287 is 761838257287-smooth, so 67 is a term.
k = 137: 2^137 - 1 = 32032215596496435569*5439042183600204290159 (not a prime), and we have 2*(2^137-1) + 32032215596496435569 = 379*28702069570449626861*32032215596496435569 is 32032215596496435569-smooth and 2*(2^137-1) + 5439042183600204290159 = 9007*7112738002996877*5439042183600204290159 is 5439042183600204290159-smooth, so 137 is a term.

Cf. A000043, A354525, A354531, A354534, A354537.

gpf(n) = vecmax(factor(n)[, 1]);
ispsmooth(n,p,{lim=1<<256}) = if(n<=lim, n==1 || gpf(n)<=p, my(N=n/p^valuation(n,p)); forprime(q=2, p, N=N/q^valuation(N,q); if((N<=lim && isprime(N)) || N==1, return(N<=p))); 0); \\ check if n is p-smooth, using brute force if n is too large
isA354532(n,{lim=256},{p_lim=1<<32}) = {
  my(N=2^n-1);
  if(isprime(N), return(0));
  if(n>lim, forprime(p=3, p_lim, if(N%p==0 && !ispsmooth(2*N+p,p), return(0)))); \\ first check if there is a prime factor p <= p_lim of 2^n-1 such that 2*(2^n-1)+p is not p-smooth (for large n)
  my(d=divisors(n));
  for(i=1, #d, my(f=factor(2^d[i]-1)[, 1]); for(j=1, #f, if(!ispsmooth(2*N+f[j],f[j],1<

By definition, equals A354531 \ A000043.

a(5) from Jinyuan Wang, Jan 21 2025

A063883 Number of ways writing n as a sum of different Mersenne prime exponents (terms of A000043).

Original entry on oeis.org

0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 3, 2, 4, 2, 4, 3, 3, 4, 2, 4, 2, 4, 3, 3, 4, 3, 4, 4, 4, 5, 4, 5, 4, 4, 5, 3, 5, 3, 4, 4, 3, 5, 3, 5, 4, 4, 5, 4, 5, 4, 4, 5, 3, 5, 4, 3, 6, 2, 6, 3, 5, 5, 3, 6, 3, 5, 4, 4, 4, 4, 4, 4, 4, 5, 3, 6, 3, 5, 5, 4, 6, 3, 7, 3, 6, 5, 5, 6, 5, 6, 5, 6, 6, 5, 6, 6

Offset: 1

Labos Elemer, Aug 28 2001

This sequence appears to be growing. However, for 704338 < n < 756839, a(n) is 0. See A078426 for the n such that a(n) = 0. - T. D. Noe, Oct 12 2006

Numbers k such that sigma(k) = 2^n. - Juri-Stepan Gerasimov, Mar 08 2017

			n = 50 = 2 + 5 + 7 + 17 + 19 = 2 + 17 + 31 = 19 + 31, so a(50) = 3. The first numbers for which the number of these Mersenne-exponent partitions is k = 0, 1, 2, 3, 4, 5, 6, 7, 8 are 1, 2, 5, 20, 22, 39, 66, 92, 107, respectively.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000043, A046528, A048947, A054784, A054973, A063889.

Numbers k such that a(k) = m: A078426 (m = 0), A283160 (m = 1).

N:= 500: # to get the first N terms
G:= mul(1+x^i,i=select(t -> numtheory:-mersenne(t)::integer, [$1..N])):
S:= series(G,x,N+1):
seq(coeff(S,x,n),n=1..N); # Robert Israel, Sep 22 2016

exponents[n_] := Reap[For[k = 1, k <= n, k++, If[PrimeQ[2^k-1], Sow[k]]]][[2, 1]]; r[n_] := Module[{ee, x, xx}, ee = exponents[n]; xx = Array[x, Length[ee]]; Reduce[And @@ (0 <= # <= 1 & /@ xx) && xx.ee == n, xx, Integers]]; a[n_] := Which[rn = r[n]; Head[rn] === Or, Length[rn],  Head[rn] === And, 1, Head[rn] === Equal, 1, rn === False, 0, True, Print["error ", rn]]; a[1] = 0; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Feb 05 2014 *)
With[{e = MersennePrimeExponent[Range[10]]}, Rest@ CoefficientList[Product[1 + x^e[[i]], {i, 1, Length[e]}], x]] (* Amiram Eldar, Dec 23 2024 *)

first(lim)=my(M=[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667], x='x); if(lim>M[#M], error("Need more Mersenne exponents to compute further")); M=select(p->p<=lim, M); Vec(prod(i=1, #M, 1+x^M[i], O(x^(lim\1+1))+1)) \\ Charles R Greathouse IV, Mar 08 2017

a(n) = sum(k=1, 2^n+1, sigma(k)==2^n); \\ Michel Marcus, Mar 07 2017

a(n) = A054973(2^n). - Michel Marcus, Mar 08 2017

A139115 a(n) = p*(p - 1), where p is A000043(n).

Original entry on oeis.org

2, 6, 20, 42, 156, 272, 342, 930, 3660, 7832, 11342, 16002, 270920, 367842, 1634562, 4851006, 5200680, 10345872, 18083756, 19558506, 93867032, 98813540, 125720156, 397464032, 470911700, 538634472, 1979938512, 7437768806, 12210802506

Offset: 1

Omar E. Pol, May 10 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Harvey P. Dale)

Cf. A000043, A036689, A139306, A139307, A139116, A139223, A139224, A139225, A139226.

#(#-1)&/@MersennePrimeExponent[Range[30]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 15 2020 *)

a(n) = A000043(n)*(A000043(n) - 1).

More terms from Vincenzo Librandi, May 11 2010

A139479 Numbers k such that 24*k+7 is a term of A000043.

Original entry on oeis.org

0, 1, 5, 25, 53, 184, 4604, 1001524

Offset: 1

Artur Jasinski, Apr 22 2008

Cf. A000043, A000668, A124477.

Select[(MersennePrimeExponent[Range[48]] - 7) / 24, IntegerQ] (* Amiram Eldar, Oct 17 2024 *)

cp's OEIS Frontend

A330818 Numbers of the form 2^(2*p+1), where p is a Mersenne exponent, A000043.

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A139116 a(n) = p*(p-1)/2, where p is A000043(n).

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A180221 Numbers that can be written as sum of one or more distinct elements of A000043. Numbers k for which sigma(A180162(k))=2^k, k>=2.

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A242025 Primes of the form R = 2^k-1+(2^k-2)/(2^(p-k)-1), where p are Mersenne prime exponents listed in A000043.

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A242999 Mersenne prime exponents p in A000043 such that R=2^k-1+(2^k-2)/(2^(p-k)-1) is prime for some k < p, listed with multiplicity (number of k's), see A243003 for the k-values.

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A243003 Pairs (p,k) such that p is in A000043 and R=2^k-1+(2^k-2)/(2^(p-k)-1) is prime: this sequence lists the k-values, see A242999 for the p-values. (Ordered by p, then k.)

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A354532 Numbers k that are not Mersenne exponents (A000043) such that 2*(2^k-1) is in A354525.

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A063883 Number of ways writing n as a sum of different Mersenne prime exponents (terms of A000043).

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