A033844 -id:A033844 - cp's OEIS frontend

A336322 a(n) = A225546(A122111(n)).

1, 2, 3, 4, 6, 8, 5, 16, 9, 12, 10, 32, 15, 24, 18, 256, 30, 64, 7, 48, 27, 20, 14, 512, 36, 40, 81, 96, 21, 128, 42, 65536, 54, 60, 72, 1024, 35, 120, 45, 768, 70, 192, 105, 80, 162, 28, 210, 131072, 25, 144, 90, 160, 11, 4096, 108, 1536, 135, 56, 22, 2048, 33, 84, 243, 4294967296, 216, 384, 66, 240, 270, 288, 55, 262144, 110, 168, 324, 480, 50

Offset: 1

Antti Karttunen and Peter Munn, Jul 17 2020

nonn

A225546 and A122111 are both self-inverse permutations of the positive integers based on prime factorizations, and they share further common properties. For instance, they map the prime numbers to powers of 2: A225546 maps the k-th prime to 2^2^(k-1), whereas A122111 maps it to 2^k.

In composing these permutations, this sequence maps the list of prime numbers to the squarefree numbers, as listed in A019565; and the "normal" numbers (A055932), as listed in A057335, to ascending powers of 2.

Index entries for sequences that are permutations of the natural numbers

A225546 composed with A122111.
Sorted even bisection: A335738.
Sorted odd bisection (excluding 1): A335740.
Sequences used to express relationship between terms of this sequence: A001222, A003961, A253560, A331590, A350066.
Sequences of sequences (S_1, S_2, ... S_j) with the property a(S_i) = S_{i+1}, or essentially so: (A033844, A000040, A019565), (A057335, A000079, A001146), (A000244, A011764), (A001248, A334110), (A253563, A334866).
The inverse permutation, A336321, lists sequences where the property is weaker (between the sets of terms).

a(A033844(m)) = A000040(m+1). [Offset corrected Peter Munn, Feb 14 2022]
a(A000040(m)) = A019565(m).
a(A057335(m)) = 2^m.
For m >= 1, a(2^m) = A001146(m-1).
a(A253563(m)) = A334866(m).
From Peter Munn, Feb 14 2022: (Start)
a(A253560(n)) = a(n)^2.
For n >= 2, a(A003961(n)) = A331590(a(n), 2^2^(A001222(n)-1)).
a(A350066(n, k)) = A331590(a(n), a(k)).
(End)

A018249 a(n) = prime(2^n)-1.

Original entry on oeis.org

1, 2, 6, 18, 52, 130, 310, 718, 1618, 3670, 8160, 17862, 38872, 84016, 180502, 386092, 821640, 1742536, 3681130, 7754076, 16290046, 34136028, 71378568, 148948138, 310248240, 645155196, 1339484196, 2777105128, 5750079046, 11891268400, 24563311308, 50685770166, 104484802056

Offset: 0

N. J. A. Sloane

nonn

Amiram Eldar, Table of n, a(n) for n = 0..78
Paul Cooijmans, Numbers, Item 18, 1995.
Paul Cooijmans, Short Test For Genius, Item 29, 2002.

Cf. A000040, A033844, A051438, A051440, A051439.

Prime[2^Range[0,30]]-1 (* Harvey P. Dale, Sep 30 2016 *)

a(n) = prime(2^n) - 1; \\ Amiram Eldar, Jul 11 2025

a(n) = A033844(n) - 1.

a(n) = A051440(n) - 2. - Amiram Eldar, Jul 11 2025

More terms from Amiram Eldar, Jul 11 2025

A051440 a(n) = prime(2^n) + 1.

Original entry on oeis.org

3, 4, 8, 20, 54, 132, 312, 720, 1620, 3672, 8162, 17864, 38874, 84018, 180504, 386094, 821642, 1742538, 3681132, 7754078, 16290048, 34136030, 71378570, 148948140, 310248242, 645155198, 1339484198, 2777105130, 5750079048, 11891268402, 24563311310, 50685770168, 104484802058

Offset: 0

N. J. A. Sloane

nonn

Amiram Eldar, Table of n, a(n) for n = 0..78

Cf. A000040, A018249, A033844, A051438, A051439.

a[n_] := Prime[2^n] + 1; Array[a, 33, 0] (* Amiram Eldar, Jul 11 2025 *)

a(n) = prime(2^n) + 1; \\ Amiram Eldar, Jul 11 2025

a(n) = A033844(n) + 1.

a(n) = A018249(n) + 2. - Amiram Eldar, Jul 11 2025

More terms from Amiram Eldar, Jul 11 2025

A065856 The (2^n)-th composite number.

Original entry on oeis.org

4, 6, 9, 15, 26, 48, 88, 168, 323, 627, 1225, 2406, 4736, 9351, 18504, 36655, 72730, 144450, 287147, 571208, 1136971, 2264215, 4510963, 8990492, 17923944, 35743996, 71298762, 142249762, 283859985, 566537515, 1130886504, 2257704401, 4507834166, 9001524190

Offset: 0

Labos Elemer, Nov 26 2001

nonn

a(n) = A002808(A000079(n)).

			composite[1] = composite[2^0] = 4, composite[2] = composite[2^1] = 6, composite[1024] = composite[2^10] = 1225, composite[1073741824] = composite[2^30] = 1130886504.

Chai Wah Wu, Table of n, a(n) for n = 0..70

Cf. A033844, A002808, A062298, A000720, A065855.

Composite[n_Integer] := Block[ {k = n + PrimePi[n] + 1 }, While[ k != n + PrimePi[k] + 1, k = n + PrimePi[k] + 1]; Return[ k ]]; Table[ Composite[2^n], {n, 0, 36} ]

a(n)-pi(a(n))-1 = 2^n.

More terms from Robert G. Wilson v, Nov 26 2001
Definition corrected by N. J. A. Sloane, Jun 07 2009
Further corrections from Reinhard Zumkeller, Jun 24 2009
a(32)-a(33) from Chai Wah Wu, Apr 16 2018

A074325 Difference between (1+2^n)-th and (2^n)-th primes. Also number of terms in blocks of A073798.

Original entry on oeis.org

1, 2, 4, 4, 6, 6, 2, 8, 2, 2, 6, 18, 18, 30, 8, 24, 6, 2, 18, 4, 26, 30, 34, 2, 10, 30, 10, 2, 30, 6, 70, 4, 12, 22, 6, 24, 26, 10, 88, 2, 50, 18, 6, 20, 14, 4, 12, 6, 2, 56, 4, 30, 42, 6, 70, 74, 14, 60, 170, 14, 44, 22, 52, 36, 96, 6, 86, 86, 72, 48, 42, 18, 20, 24, 10, 154, 54, 20, 12

Offset: 0

Labos Elemer, Aug 21 2002

			For n = 39: 2^39 = 549755813888, prime(549755813889) = 16149760533343, prime(549755813888) =  16149760533341, difference = 2 (just twin primes), so a(39) = 2.

Cf. A051439, A033844, A073798, A000079, A001223.

Table[Prime[1+2^n] - Prime[2^n], {n, 1, 39}]

a(n) = A051439(n) - A033844(n).

a(n) = A001223(A000079(n)). - Michel Marcus, May 10 2024

More terms from Michel ten Voorde Jun 13 2003
a(42) from Max Alekseyev, May 08 2009
Offset corrected by Max Alekseyev, Oct 24 2013
a(43)-a(57) from Chai Wah Wu, Aug 30 2019
a(58)-a(78) calculated using the data at A033844 and A051439 and added by Amiram Eldar, May 10 2024

A107655 a(n) is the smallest number m greater than 1 such that phi(m) = d(m)^n, where d(m) is number of positive divisors of m; if there is no such m, a(n)=1.

Original entry on oeis.org

3, 5, 85, 17, 1285, 4369, 559876, 257, 327685, 1114129, 1114521441417, 16843009, 160490068541289, 1925878801139721, 23110536763219977, 65537, 3327917287071744009, 39934999967815157769, 479219999336720898057, 5750639996603165650953, 69007679885506346588169, 828092158571811231498249, 9937105900443065378930697

Offset: 1

Farideh Firoozbakht, Jun 06 2005

nonn

For n=0,1,2,3, and 4, a(2^n) = A000215(n), the n-th Fermat prime.
Conjecture: A000005(a(n)) <= 12 for all n. [Max Alekseyev, May 07 2010]
This conjecture holds throughout the first 102 terms. - David A. Corneth, Jun 14 2020

			a(10) = 1114129 because phi(1114129) = d(1114129)^10 and 1114129 is the smallest number m greater than 1 that phi(m) = 1048576 = 4^10 = d(m)^10.

David A. Corneth, Table of n, a(n) for n = 1..100
Max Alekseyev, PARI scripts for various problems (see invphitau there).

Cf. A000005, A000215, A033844.

a(n)=res = oo; for(i=2, oo, if(i^n > res, return(res)); c=invphitau(i^n,i); if(#c>0, res=c[1])) \\ for invphitau, see Alekseyev link \\ David A. Corneth, Jun 14 2020

Terms a(11) onward from Max Alekseyev, May 07 2010

Terms a(20)-a(23), offset corrected by David A. Corneth, Jun 14 2020

A124012 Decimal expansion of Sum_{k>=1} 1/(k*prime(k)).

Original entry on oeis.org

8, 4, 8, 9, 6, 9, 0, 3, 4, 0, 4, 3

Offset: 0

Pierre CAMI, Nov 02 2006

From Robert Price, Jul 14 2010: (Start)
This series converges very slowly. I could not find any transform that converges faster, so I did this by brute force using 256 bits of precision.
After k=596765000000 terms (p(k)=17581469834441) the partial sum is 0.848 969 034 043 245 206 069 544 346 415 327 714...
The next two digits are either 29 or 30. (End)
The table in the Example section shows, for increasing values of j, the results of computing the partial sum s(j) = Sum_{k=1..j} 1/(k*prime(k)) and adding to it an approximate value for the tail (i.e., the sum for all the terms k > j). See the Links entry for an explanation of the method used in approximating the size of the tail of the summation beyond the j-th prime. - Jon E. Schoenfield, Jan 20 2019

			0.848969034043...
From _Jon E. Schoenfield_, Jan 14 2019: (Start)
We can obtain prime(2^d) for d = 0..57 from the b-file for A033844. Given the above result from _Robert Price_, and letting j_RP = 596765000000, the partial sum through
   prime(j_RP) = 17581469834441
is
   s(j_RP) = Sum_{k=1..j_RP} 1/(k*prime(k))
           = 0.848969034043245206069544346415327714...;
adding to this actual partial sum s(j_RP) the approximate tail value
   t(j_RP) =
         h'(prime(j_RP), prime(2^40))
       + (Sum_{d=41..57} h'(prime(2^(d-1)), prime(2^d)))
       + lim_{x->infinity} h(prime(2^57), x)
(see the Links entry for an explanation) gives the result 0.84896903404330021273712255895762255... (which seems likely to be correct to at least 20 significant digits).
The table below gives, for j = 2^16, 2^17, ..., 2^32, and j_RP, the actual partial sum s(j) and the sum s(j) + t(j) where t(j) is the approximate tail value beyond prime(j).
.
   j             s(j)                s(j) + t(j)
  ====  ======================  ======================
  2^16  0.84896790758922908159  0.84896903393397518971
  2^17  0.84896850050492294891  0.84896903400552099072
  2^18  0.84896878057566843770  0.84896903404214147367
  2^19  0.84896891330602605081  0.84896903404317536927
  2^20  0.84896897639243509768  0.84896903404350431035
  2^21  0.84896900645590169648  0.84896903404376063663
  2^22  0.84896902081581006534  0.84896903404343742139
  2^23  0.84896902768965496764  0.84896903404337393698
  2^24  0.84896903098637626311  0.84896903404331189996
  2^25  0.84896903257029535468  0.84896903404329806633
  2^26  0.84896903333252861584  0.84896903404330030271
  2^27  0.84896903369988697984  0.84896903404330084536
  2^28  0.84896903387717904236  0.84896903404330042023
  2^29  0.84896903396285181513  0.84896903404330024036
  2^30  0.84896903400430044877  0.84896903404330021861
  2^31  0.84896903402437548991  0.84896903404330021472
  2^32  0.84896903403410856545  0.84896903404330021655
  ...            ...                     ...
  j_RP  0.84896903404324520607  0.84896903404330021274
(End)

Jon E. Schoenfield, Notes on approximating the size of the summation's tail beyond the j-th prime
Eric Weisstein's World of Mathematics, Prime Number Theorem

Cf. A033286, A085548, A209329, A210473.

Offset and leading zero corrected by R. J. Mathar, Jan 31 2009
Four more terms (4,0,4,3) from Robert Price, Jul 14 2010
Title and example edited by M. F. Hasler, Jan 13 2015

A325093 Heinz numbers of integer partitions into distinct powers of 2.

Original entry on oeis.org

1, 2, 3, 6, 7, 14, 19, 21, 38, 42, 53, 57, 106, 114, 131, 133, 159, 262, 266, 311, 318, 371, 393, 399, 622, 719, 742, 786, 798, 917, 933, 1007, 1113, 1438, 1619, 1834, 1866, 2014, 2157, 2177, 2226, 2489, 2751, 3021, 3238, 3671, 4314, 4354, 4857, 4978, 5033

Offset: 1

Gus Wiseman, Mar 27 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are squarefree numbers whose prime indices are powers of 2. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    6: {1,2}
    7: {4}
   14: {1,4}
   19: {8}
   21: {2,4}
   38: {1,8}
   42: {1,2,4}
   53: {16}
   57: {2,8}
  106: {1,16}
  114: {1,2,8}
  131: {32}
  133: {4,8}
  159: {2,16}
  262: {1,32}
  266: {1,4,8}
  311: {64}

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

Cf. A000720, A001222, A018819, A033844, A056239, A102378, A112798, A318400, A325091, A325092.

P:= [seq(ithprime(2^i),i=0..20)]:f:= proc(S,N) option remember;
  if S = [] or S[1]>N then return {1} fi;
  procname(S[2..-1],N) union
    map(t -> S[1]*t, procname(S[2..-1], floor(N/S[1])))end proc:
sort(convert(f(P, P[20]),list));  # Robert Israel, Mar 28 2019

Select[Range[1000],SquareFreeQ[#]&&And@@IntegerQ/@Log[2,Cases[If[#==1,{},FactorInteger[#]],{p_,_}:>PrimePi[p]]]&]

isp2(q) = (q == 1) || (q == 2) || (ispower(q,,&p) && (p==2));
isok(n) = {if (issquarefree(n), my(f=factor(n)[,1]); for (k=1, #f, if (! isp2(primepi(f[k])), return (0));); return (1);); return (0);} \\ Michel Marcus, Mar 28 2019

A119773 (8^n)-th prime.

Original entry on oeis.org

2, 19, 311, 3671, 38873, 386093, 3681131, 34136029, 310248241, 2777105129, 24563311309, 215187847711, 1870358526653, 16149760533341, 138666449011757, 1184895616861903, 10082409897709157, 85476377250109733, 722285281729443799, 6085631874569939777

Offset: 0

Jim Snow (jsnow(AT)mitre.org), Jun 22 2006

Amiram Eldar, Table of n, a(n) for n = 0..26
Andrew Booker, The Nth Prime Page.

Cf. A033844, A119772, A119777, A119778, A119779.

a[n_] := Prime[8^n]; Array[a, 16, 0] (* Amiram Eldar, Jul 18 2025 *)

a(n)=prime(8^n) \\ Charles R Greathouse IV, Nov 02 2014

a(n) = A033844(3*n). - Amiram Eldar, Jul 18 2025

a(14)-a(19) from Charles R Greathouse IV, Nov 02 2014

A119777 (16^n)-th prime.

Original entry on oeis.org

2, 53, 1619, 38873, 821641, 16290047, 310248241, 5750079047, 104484802057, 1870358526653, 33089240375501, 579863159340527, 10082409897709157, 174160587542317721, 2991614170035124397, 51140670371058101123, 870566678511500413493, 14764978793012287880219

Offset: 0

Jim Snow (jsnow(AT)mitre.org), Jun 22 2006

Amiram Eldar, Table of n, a(n) for n = 0..19
Andrew Booker, The Nth Prime Page.

Cf. A033844, A119772, A119773, A119778, A119779.

a[n_] := Prime[16^n]; Array[a, 12, 0] (* Amiram Eldar, Jul 18 2025 *)

a(n)=prime(16^n) \\ Charles R Greathouse IV, Nov 02 2014

a(n) = A033844(4*n) = A119772(2*n). - Amiram Eldar, Jul 18 2025

a(10)-a(14) from Charles R Greathouse IV, Nov 02 2014

a(15)-a(17) from Amiram Eldar, Jul 18 2025

cp's OEIS Frontend

A336322 a(n) = A225546(A122111(n)).

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A018249 a(n) = prime(2^n)-1.

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A051440 a(n) = prime(2^n) + 1.

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A065856 The (2^n)-th composite number.

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A074325 Difference between (1+2^n)-th and (2^n)-th primes. Also number of terms in blocks of A073798.

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A107655 a(n) is the smallest number m greater than 1 such that phi(m) = d(m)^n, where d(m) is number of positive divisors of m; if there is no such m, a(n)=1.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A124012 Decimal expansion of Sum_{k>=1} 1/(k*prime(k)).

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Comments

Examples

Links

Crossrefs

Extensions

A325093 Heinz numbers of integer partitions into distinct powers of 2.

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