A014284 -id:A014284 - cp's OEIS frontend

A175944 1 appears once, n-th prime p appears p times.

1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19

Offset: 1

Juri-Stepan Gerasimov, Oct 27 2010

a(A014284(n)) = a(A175965(n)) = A018252(n). [Reinhard Zumkeller, Mar 18 2011]

This is how A063905 would have been reckoned at the beginning of the 20th century, taking the primes as given in A008578 instead of the way they are given in A000040. [Alonso del Arte, Sep 09 2011]

Cf. A063905.

Cf. A005145.

a175944 n = a175944_list !! (n-1)
a175944_list =
   concat $ zipWith ($) (map replicate a018252_list) a018252_list
-- Reinhard Zumkeller, Mar 18 2011

Join[{1}, Flatten[Table[Prime[n], {n, 8}, {Prime[n]}]]] (* Alonso del Arte, Sep 08 2011 based on Robert G. Wilson v's program for A002024 *)
Join[{1},Flatten[Table[PadRight[{},n,n],{n,Prime[Range[10]]}]]] (* Harvey P. Dale, May 16 2019 *)

a(1)=1, a(n)=A063905(n-1) for n>1.

A089228 Numbers m such that 1 + Sum_{k=1..m} prime(k) is prime.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 19, 25, 29, 31, 49, 51, 57, 97, 99, 103, 109, 119, 123, 127, 163, 169, 179, 185, 195, 207, 209, 211, 213, 217, 221, 223, 233, 235, 239, 251, 261, 269, 273, 289, 295, 297, 303, 325, 329, 333, 347, 369, 371, 375, 409, 439, 449, 453, 455, 467

Offset: 1

Yalcin Aktar, Dec 10 2003

nonn

Also numbers n such that the sum of the first n "primes", as defined in A008578, is prime. Analogous to A013916. - Robert G. Wilson v, May 19 2015

Integers k such that A007504(k) + 1 is prime. - Michel Marcus, Aug 10 2023

			25 is a term: 1 + Sum_{k=1..25} prime(k) = 1061 is prime.

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

Cf. A007504, A008578, A013916, A014284.

a:=proc(n) if isprime(1+add(ithprime(k),k=1..n))=true then n else fi end: seq(a(n),n=1..600); # Emeric Deutsch, Jul 02 2005
# alternative
Primes:= select(isprime,[2,seq(2*i+1,i=1..10^5)]):
PS:= ListTools:-PartialSums(Primes):
select(t -> isprime(PS[t]+1), [$1..nops(PS)]); # Robert Israel, May 19 2015

Position[1 + Accumulate@ Prime@ Range@ 600, A013916%20*)%20(*%20_Robert%20G.%20Wilson%20v">?(PrimeQ@# &)] // Flatten (* after Harvey P. Dale from A013916 *) (* _Robert G. Wilson v, May 19 2015 *)

for(n=1,10^3,if(isprime(1+sum(i=1,n,prime(i))),print1(n,", "))) \\ Derek Orr, May 19 2015

Corrected and extended by Emeric Deutsch, Jul 02 2005

A175966 Complement of A175965(n), where A175965(n) = the lexicographically earliest sequence with first differences as increasing sequence of noncomposites A008578.

Original entry on oeis.org

3, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80

Offset: 1

Jaroslav Krizek, Oct 31 2010

nonn

Cf. A175965, A175967, A175968, A014284, A175969, A051349, A175970.

A175970 Complement of A051349(n), where A051349(n) = the lexicographically earliest sequence with first differences as increasing sequence of composites A002808(n).

Original entry on oeis.org

2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70

Offset: 1

Jaroslav Krizek, Oct 31 2010

nonn

Cf. A175965, A175966, A175967, A175968, A175969, A014284, A051349.

A175968 Complement of A175967(n), where A175967(n) = the lexicographically earliest sequence with first differences as increasing sequence of nonprimes A018252(n).

Original entry on oeis.org

3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Jaroslav Krizek, Oct 31 2010

nonn

Cf. A175965, A175966, A175967, A014284, A175969, A051349, A175970.

A036439 a(n) = a(n-1) + prime(n-1), with a(1)=2.

Original entry on oeis.org

2, 4, 7, 12, 19, 30, 43, 60, 79, 102, 131, 162, 199, 240, 283, 330, 383, 442, 503, 570, 641, 714, 793, 876, 965, 1062, 1163, 1266, 1373, 1482, 1595, 1722, 1853, 1990, 2129, 2278, 2429, 2586, 2749, 2916, 3089, 3268, 3449, 3640, 3833, 4030, 4229, 4440, 4663

Offset: 1

Pavel Bubak (pbub6070(AT)beta.ms.mff.cuni.cz)

Old name was "a(n) = 2 + the sum of the first n-1 prime numbers".

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A014284(n) + 1.

nxt[{n_,a_}]:={n+1,a+Prime[n]}; NestList[nxt,{1,2},50][[All,2]] (* Harvey P. Dale, Sep 21 2016 *)

a(n) = 2 + sum(i=1, n-1, prime(i)); \\ Michel Marcus, Aug 12 2013

More terms from James Sellers, Feb 06 2000

Offset changed and definition improved from Bruno Berselli, Jul 30 2013

A280055 Nachos sequence based on 1 plus primes (A008578).

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 3, 2, 3, 1, 2, 3, 2, 3, 4, 2, 1, 2, 3, 2, 3, 4, 2, 3, 4, 3, 4, 1, 2, 3, 2, 3, 4, 2, 3, 4, 3, 4, 2, 3, 1, 2, 3, 2, 3, 4, 2, 3, 4, 3, 4, 2, 3, 4, 3, 4, 5, 1, 2, 3, 2, 3, 4, 2, 3, 4, 3, 4, 2, 3, 4, 3, 4, 5, 3, 2, 1, 2, 3, 2, 3, 4, 2, 3, 4, 3, 4, 2, 3, 4, 3, 4, 5, 3, 2, 3, 4, 3

Offset: 1

N. J. A. Sloane, Jan 08 2017

nonn

Like A280053 but based on 1,2,3,5,7,11,... rather than squares. See that entry for further information.

Equivalently, greedily subtract terms of A014284 from n until reaching 0; a(n) = number of steps required.

			26 takes 4 phases to read 0:
subtract leaves
1   25
2   23
3   20
5   15
7   8
------
1   7
2   5
3   2
------
1   1
------
1   0
so a(26) = 4

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A280055, A014284.

For records see A280760.

A280055 := proc(n)
    local a,nres,i ;
    a := 0 ;
    nres := n;
    while nres > 0 do
        for i from 1 do
            if A014284(i) > nres then
                break;
            end if;
        end do:
        nres := nres-A014284(i-1) ;
        a := a+1 ;
    end do:
    a ;
end proc:
seq(A280055(n),n=1..80) ; # R. J. Mathar, Mar 05 2017

A345297 a(n) is the least k >= 0 such that A331835(k) = n.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 22, 23, 26, 27, 29, 30, 31, 43, 45, 46, 47, 54, 55, 58, 59, 61, 62, 63, 94, 95, 107, 109, 110, 111, 118, 119, 122, 123, 125, 126, 127, 187, 189, 190, 191, 222, 223, 235, 237, 238, 239, 246, 247, 250, 251, 253, 254, 255

Offset: 0

Rémy Sigrist, Jun 13 2021

Sequence A200947 gives the position of the last occurrence of a number in A331835.

			We have:
           n|  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18
  ----------+------------------------------------------------------------------
  A331835(n)|  0  1  2  3  3  4  5  6  5  6   7   8   8   9  10  11   7   8   9
So a(0) = 0,
   a(1) = 1,
   a(2) = 2,
   a(3) = 3,
   a(4) = 5,
   a(5) = 6,
   a(6) = 7,
   a(7) = 10,
   a(8) = 11,
   a(9) = 13,
   a(10) = 14,
   a(11) = 15.

Rémy Sigrist, Table of n, a(n) for n = 0..2000
Rémy Sigrist, C program for A345297

Cf. A014284, A200947, A331835.

C
```
See Links section.
```

from sympy import prime
def p(n): return prime(n) if n >= 1 else 1
def A331835(n): return sum(p(i)*int(b) for i, b in enumerate(bin(n)[:1:-1]))
def adict(klimit):
    adict = dict()
    for k in range(klimit+1):
        fk = A331835(k)
        if fk not in adict: adict[fk] = k
    n, alst = 0, []
    while n in adict: alst.append(adict[n]); n += 1
    return alst
print(adict(255)) # Michael S. Branicky, Jun 13 2021

a(A014284(n)) = 2^n - 1.

a(n) <= A200947(n).

A110996 Powers equal to (sum of first k primes) plus 1, for some k >= 0.

Original entry on oeis.org

1, 441, 970225, 1464100, 194379364, 1303400915339554201

Offset: 1

Walter Kehowski, Sep 30 2005

I have checked for powers out to the 250000th prime and the last element found is at the 6420th prime, 64067. It is interesting to note that the only powers so far are squares. Finding a higher power would be interesting.

			1 is a term (corresponding to k=0), since it is the empty sum plus 1. - _N. J. A. Sloane_, Dec 02 2015
441 is a term since sum(primes<=59) = 440 and 441 = 21^2.

Cf. A007504, A110979, A110997, A264858.

Intersection of A001597 and A014284.

with(numtheory); egcd := proc(n) local L; L:=map(proc(z) z[2] end, ifactors(n)[2]); igcd(op(L)) end: s := proc(n) option remember; local p; if n=1 then [1,2] else [n,s(n-1)[2]+ithprime(n)] fi end; t := proc(n) option remember; [n,s(n)[2]+1] fi end; PW:=[]; for z to 1 do for j from 1 to 250000 do if egcd(t(j)[2])>1 then PW:=[op(PW),t(j)] fi od od; PW;

lista(nn) = { print1(1, ", "); s = 1; for(k=1, nn, s += prime(k); if(ispower(s) || s==1, print1(s, ", ")););} \\ Altug Alkan, Nov 29 2015

New term 1 prepended by Altug Alkan, Nov 29 2015

a(6) from Jinyuan Wang, Aug 09 2023

A227547 a(n) = a(n-1) + prime(n-1), with a(1)=3.

Original entry on oeis.org

3, 5, 8, 13, 20, 31, 44, 61, 80, 103, 132, 163, 200, 241, 284, 331, 384, 443, 504, 571, 642, 715, 794, 877, 966, 1063, 1164, 1267, 1374, 1483, 1596, 1723, 1854, 1991, 2130, 2279, 2430, 2587, 2750, 2917, 3090, 3269, 3450, 3641, 3834, 4031, 4230, 4441, 4664

Offset: 1

Vincenzo Librandi, Jul 27 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007504, A014284, A036439.

[IsOne(n) select 3 else Self(n-1)+NthPrime(n-1): n in [1..50]]; // Bruno Berselli, Jul 30 2013

Mathematica
```
Accumulate[Join[{3}, Prime[Range[60]]]]
```

a(n) = A036439(n) + 1 = A014284(n) + 2 = A007504(n-1) + 3 (in this case, n>1).

cp's OEIS Frontend

A175944 1 appears once, n-th prime p appears p times.

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A089228 Numbers m such that 1 + Sum_{k=1..m} prime(k) is prime.

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A175966 Complement of A175965(n), where A175965(n) = the lexicographically earliest sequence with first differences as increasing sequence of noncomposites A008578.

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A175970 Complement of A051349(n), where A051349(n) = the lexicographically earliest sequence with first differences as increasing sequence of composites A002808(n).

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A175968 Complement of A175967(n), where A175967(n) = the lexicographically earliest sequence with first differences as increasing sequence of nonprimes A018252(n).

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A036439 a(n) = a(n-1) + prime(n-1), with a(1)=2.

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Mathematica

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Extensions

A280055 Nachos sequence based on 1 plus primes (A008578).

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Maple

A345297 a(n) is the least k >= 0 such that A331835(k) = n.

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A110996 Powers equal to (sum of first k primes) plus 1, for some k >= 0.

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