A076793 -id:A076793 - cp's OEIS frontend

A080355 a(1)=1; thereafter, a(n+1) = a(n) + 2^(prime(n)-1).

1, 3, 7, 23, 87, 1111, 5207, 70743, 332887, 4527191, 272962647, 1346704471, 70066181207, 1169577808983, 5567624320087, 75936368497751, 4579535995868247, 292809912147579991, 1445731416754426967, 75232707711592633431, 1255824328429003936855, 5978190811298649150551

Offset: 1

N. J. A. Sloane, based on information supplied by Artur Jasinski, Mar 21 2003

Original name: a(1)=1; for n>1, a(n) = a(n-1) + 2^(j-1), where j = prime(n-1) is position of n-th 1 in A080339.
Or, take an initial segment of A080339, stopping at the n-th 1, reverse and interpret as a binary number. E.g., to get the 4th term: 11101 -> 10111 = 23, so a(4) = 23.
Indices of noncomposite terms in the sequence are 1, 2, 3, 4, 9, 310, 418, .... Next term (i.e., index of a prime), if it exists, is > 2000. See also post to SeqFan list by Tomasz Ordowski. - M. F. Hasler, Oct 30 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..475
Tomasz Ordowski, Primes in primes, SeqFan list, Oct 28 2018.

Cf. A076793.

[n le 1 select 1 else Self(n-1) + 2^(NthPrime(n-1)-1): n in [1..25]]; // Vincenzo Librandi, Oct 31 2018

a:=n->1+add(2^(ithprime(k)-1),k=1..n-1): seq(a(n),n=1..25); # Muniru A Asiru, Oct 31 2018

RecurrenceTable[{a[1]==1, a[n] == 2^(Prime[n-1] - 1) + a[n-1]}, a, {n, 25}] (* Vincenzo Librandi, Oct 31 2018 *)
nxt[{n_,a_}]:={n+1,a+2^(Prime[n]-1)}; NestList[nxt,{1,1},30][[All,2]] (* Harvey P. Dale, Aug 07 2019 *)

apply( A080355(n)=1+sum(i=1,n-1,2^(prime(i)-1)), [1..50]) \\ M. F. Hasler, Oct 30 2018

a(n) = 1 + Sum_{k=1..n-1} 2^(prime(k)-1).

a(n) = A076793(n-1) / 2 + 1. - Georg Fischer, Aug 12 2023

More terms from Vladeta Jovovic, Mar 26 2003

A135482 a(n) = (1/4)*Sum_{j=1..n} 2^prime(j).

Original entry on oeis.org

0, 1, 3, 11, 43, 555, 2603, 35371, 166443, 2263595, 136481323, 673352235, 35033090603, 584788904491, 2783812160043, 37968184248875, 2289767997934123, 146404956073789995, 722865708377213483, 37616353855796316715, 627912164214501968427, 2989095405649324575275

Offset: 0

Ctibor O. Zizka, Feb 07 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..450 (a(0) = 0 added by _M. F. Hasler_).

Cf. A000040, A000079, A076793, A080355.

Partial sums of A135620.

[&+[2^(NthPrime(k)-2): k in [1..n]]: n in [1..25]]; // Bruno Berselli, Sep 24 2015

A135482:= n-> add(2^ithprime(i)/4, i=1..n): seq(A135482(n), n=0..20); # Wesley Ivan Hurt, Feb 02 2014

Accumulate[Table[Floor[2^i/4],{i,Prime[Range[20]]}]] (* Harvey P. Dale, Dec 05 2013 *)

a(n) = sum(k=1, n, 2^prime(k))/4; \\ Michel Marcus, Oct 15 2016

a(n) = A076793(n)/4. - M. F. Hasler, Oct 30 2018

More terms from Harvey P. Dale, Dec 05 2013

a(0) = 0 prepended by M. F. Hasler, Oct 30 2018

A293390 Least m such that the exponents in expression for n as a sum of distinct powers of 2 are pairwise distinct mod m; a(0) = 0 by convention.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Oct 08 2017

The set of exponents in expression for n as a sum of distinct powers of 2 corresponds to the n-th row of A133457.
The sum of digits of n in base 2^a(n), say s, can be computed without carry in base 2; the Hamming weight of s equals the Hamming weight of n.
a(n) >= A000120(n) for any n > 0.
Apparently, a(n) = A000120(n) iff n = 0 or n belongs to A100290.
a(n) <= A070939(n) for any n >= 0.
For any sequence s of distinct nonnegative integers (s(n) being defined for n >= 0):
- let D_s be defined for any n > 0 by D_s(n) = a(Sum_{k=0..n-1} 2^s(k)),
- then D_s is the discriminator of s as introduced by Arnold, Benkoski, and McCabe in 1985,
- D_s(1) = 1,
- D_s(n) >= n for any n >= 1,
- D_s(n+1) >= D_s(n) for any n >= 1.

			For n=42:
- 42 = 2^5 + 2^3 + 2^1,
- 5 mod 1 = 3 mod 1,
- 5 mod 2 = 3 mod 2,
- 5 mod 3, 3 mod 3 and 1 mod 3 are all distinct,
- hence a(42) = 3.

Robert Israel, Table of n, a(n) for n = 0..10000
Sajed Haque, Jeffrey Shallit, Discriminators and k-Regular Sequences, arXiv:1605.00092 [cs.DM], 2016.

Cf. A000041, A000045, A000058, A000069, A000108, A000110, A000120, A000215, A001147, A001566, A001969, A002808, A003095, A005823, A016726, A062383, A070939, A076793, A100290, A133457, A173195, A270097, A270151, A270176, A272633, A272881, A272882, A273037, A273041, A273043, A273044, A273056, A273062, A273064, A273068, A273237, A273377

f:= proc(n) local L,D,k;
  L:= convert(n,base,2);
  L:= select(t -> L[t+1]=1, [$0..nops(L)-1]);
  if nops(L) = 1 then return 1 fi;
  D:= {seq(seq(L[j]-L[i],i=1..j-1),j=2..nops(L))};
  D:= `union`(seq(numtheory:-divisors(i),i=D));
  min({$2..max(D)+1} minus D)
end proc:
0, seq(f(i),i=1..100); # Robert Israel, Oct 08 2017

{0}~Join~Table[Function[r, SelectFirst[Range@ 10, Length@ Union@ Mod[r, #] == Length@ r &]][Join @@ Position[#, 1] - 1 &@ Reverse@ IntegerDigits[n, 2]], {n, 86}] (* Michael De Vlieger, Oct 08 2017 *)

a(n) = if (n, my (d=Vecrev(binary(n)), x = []); for (i=1, #d, if (d[i], x = concat(x, i-1))); for (m=1, oo, if (#Set(vector(#x, i, x[i]%m))==#x, return (m))), return (0))

a(2*n) = a(n) for any n >= 0.
a(2^k-1) = k for any k >= 0.
a(n) = 1 iff n = 2^k for some k >= 0.
a(n) = 2 iff n belongs to A173195.
a(Sum_{k=1..n} 2^(k^2)) = A016726(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000069(k)) = A062383(n) for any n >= 1.
a(Sum_{k=0..n} 2^(2^k)) = A270097(n) for any n >= 0.
a(Sum_{k=1..n} 2^A000045(k+1)) = A270151(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000041(k)) = A270176(n) for any n >= 1.
a(A076793(n)) = A272633(n) for any n >= 0.
a(Sum_{k=1..n} 2^A001969(k)) = A272881(n) for any n >= 1.
a(Sum_{k=1..n} 2^A005823(k)) = A272882(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000215(k-1)) = A273037(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000108(k)) = A273041(n) for any n >= 1.
a(Sum_{k=1..n} 2^A001566(k)) = A273043(n) for any n >= 1.
a(Sum_{k=1..n} 2^A003095(k)) = A273044(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000058(k-1)) = A273056(n) for any n >= 1.
a(Sum_{k=1..n} 2^A002808(k)) = A273062(n) for any n >= 1.
a(Sum_{k=1..n} 2^(k!)) = A273064(n) for any n >= 1.
a(Sum_{k=1..n} 2^(k^k)) = A273068(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000110(k)) = A273237(n) for any n >= 1.
a(Sum_{k=1..n} 2^A001147(k)) = A273377(n) for any n >= 1.

A076794 Numbers of the form Sum_{k=1..m} prime(r)^prime(k) for some values of m and r.

Original entry on oeis.org

4, 9, 12, 25, 36, 44, 49, 121, 150, 169, 172, 279, 289, 361, 392, 529, 841, 961, 1369, 1452, 1681, 1849, 2209, 2220, 2366, 2466, 2809, 3275, 3481, 3721, 4489, 5041, 5202, 5329, 6241, 6889, 7220, 7921, 9409, 10201, 10412, 10609, 11449, 11881, 12696, 12769

Offset: 1

Walter Carlini, Nov 17 2002

Union of sets S(r) := { Sum_{k=1..m} prime(r)^prime(k) | m = 1, 2, ... } sorted in increasing order.

Cf. A053810, A076793.

Corrected and extended by Arkadiusz Wesolowski, May 21 2013

A176496 a(n) = Sum_{k=1..n} 2^nonprime(k).

Original entry on oeis.org

2, 18, 82, 338, 850, 1874, 5970, 22354, 55122, 120658, 382802, 1431378, 3528530, 7722834, 24500050, 58054482, 125163346, 259381074, 527816530, 1601558354, 5896525650, 14486460242, 31666329426, 66026067794, 134745544530, 409623451474, 959379265362, 2058890893138

Offset: 1

David Lazar (lazar6(AT)illinois.edu), Apr 19 2010

			a(2) = 2 + 2^A018252(2) = 2 + 2^4 = 18 ; a(3) = 18 + 2^A018252(3) = 18 + 2^6 = 82.

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A076793, A018252.

With[{t=Select[Range[50],!PrimeQ[#]&]},Accumulate[2^t]] (* Harvey P. Dale, Feb 14 2023 *)

a(n) = my(s = 0, k = 1); while(n, if(!isprime(k), s += 2^k; n--); k++); return(s)

a(1) = 2, a(n) = a(n-1) + 2^A018252(n).

A320877 a(n) = 1 + Sum_{k=1..n} 2^prime(k).

Original entry on oeis.org

1, 5, 13, 45, 173, 2221, 10413, 141485, 665773, 9054381, 545925293, 2693408941, 140132362413, 2339155617965, 11135248640173, 151872736995501, 9159071991736493, 585619824295159981, 2891462833508853933, 150465415423185266861, 2511648656858007873709, 11956381622597298301101

Offset: 0

M. F. Hasler, Oct 30 2018

nonn

Indices of noncomposite terms are 0, 1, 2, 4, 5, 8, 43, 127, 251, ...

Cf. A076793.

Join[{1},Accumulate[2^Prime[Range[30]]]+1] (* Harvey P. Dale, Mar 26 2019 *)

apply( A320877(n)=1+sum(k=1,n,2^prime(k)), [0..30])

a(n) = A076793(n) + 1.

A333392 a(0) = 1; thereafter a(n) = 2^(prime(n)-1) + Sum_{k=1..n} 2^(prime(n)-prime(k)).

Original entry on oeis.org

1, 3, 7, 29, 117, 1873, 7493, 119889, 479557, 7672913, 491066433, 1964265733, 125713006913, 2011408110609, 8045632442437, 128730119078993, 8238727621055553, 527278567747555393, 2109114270990221573, 134983313343374180673, 2159733013493986890769, 8638932053975947563077

Offset: 0

Ilya Gutkovskiy, Mar 18 2020

nonn

			a(7) = 119889 (in base 10) = 11101010001010001 (in base 2).
                             ||| | |   | |   |
                             123 5 7  1113  17

Cf. A000040, A008578, A010051, A034785, A051006, A072762, A076793, A080339, A080355, A121240, A139104, A333393.

a[0] = 1; a[n_] := 2^(Prime[n] - 1) + Sum[2^(Prime[n] - Prime[k]), {k, 1, n}]; Table[a[n], {n, 0, 21}]

a(n) = if (n==0, 1, 2^(prime(n)-1) + sum(k=1, n, 2^(prime(n)-prime(k)))); \\ Michel Marcus, Mar 18 2020

a(n) = floor(c * 2^prime(n)) for n > 0, where c = 0.91468250985... = 1/2 + A051006.

A289898 a(n) = floor((2^prime(n+1))/Sum_{k=0|n,2^prime(k)}).

Original entry on oeis.org

2, 2, 2, 11, 3, 12, 3, 12, 59, 3, 51, 15, 3, 12, 59, 62, 3, 51, 15, 3, 50, 15, 60, 251, 15, 3, 12, 3, 12, 15179, 15, 60, 3, 816, 3, 51, 62, 15, 60, 62, 3, 816, 3, 12, 3, 3226, 4094, 15, 3, 12, 59, 3, 816, 63, 63, 63, 3, 51, 15, 3, 808, 16363, 15, 3, 12, 15183

Offset: 1

Joseph Wheat, Jul 14 2017

nonn

Cf. A034785, A076793.

Table[Floor[2^Prime[n + 1]/Sum[ 2^Prime[k], {k, n}]], {n, 66}] (* Michael De Vlieger, Jul 16 2017 *)

a(n) = 2^prime(n+1)\sum(k=1, n, 2^prime(k)); \\ Michel Marcus, Jul 16 2017

a(n) = floor(2^prime(n+1)/(Sum_{k=1..n} 2^prime(k))).

cp's OEIS Frontend

A080355 a(1)=1; thereafter, a(n+1) = a(n) + 2^(prime(n)-1).

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A135482 a(n) = (1/4)*Sum_{j=1..n} 2^prime(j).

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A293390 Least m such that the exponents in expression for n as a sum of distinct powers of 2 are pairwise distinct mod m; a(0) = 0 by convention.

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A076794 Numbers of the form Sum_{k=1..m} prime(r)^prime(k) for some values of m and r.

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A176496 a(n) = Sum_{k=1..n} 2^nonprime(k).

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A320877 a(n) = 1 + Sum_{k=1..n} 2^prime(k).

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Mathematica

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A333392 a(0) = 1; thereafter a(n) = 2^(prime(n)-1) + Sum_{k=1..n} 2^(prime(n)-prime(k)).

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A289898 a(n) = floor((2^prime(n+1))/Sum_{k=0|n,2^prime(k)}).