A076768 -id:A076768 - cp's OEIS frontend

A132399 Number of ordered ways of writing n = i + j, where i is 0 or a prime and j is a triangular number (A000217) >= 0.

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

Offset: 0

N. J. A. Sloane, Mar 23 2008

Based on a posting by Zhi-Wei Sun to the Number Theory Mailing List, Mar 23 2008, where he conjectures that a(n) > 0 except for n = 216.
Zhi-Wei Sun has offered a monetary reward for settling this conjecture.
No counterexample below 10^10. - D. S. McNeil
Note that A076768 contains 216 and the numbers n whose only representation has 0 instead of a prime; all other integers appear to be the sum of a prime and a triangular number. Except for n=216, there is no other n < 2*10^9 for which a(n)=0.
It is clear that a(t) > 0 for any triangular number t because we always have the representation t = t+0. Triangular numbers tend to have only a few representations. Hence by not plotting a(n) for triangular n, the plot (see link) more clearly shows how a(n) slowly increases as n increases. This is more evidence that 216 is the only exception.
216 is the only exception less than 10^12. Let p(n) be the least prime (or 0 if n is triangular) such that n = p(n) + t(n), where t(n) is a triangular number. For n < 10^12, the largest value of p(n) is only 2297990273, which occurs at n=882560134401. - T. D. Noe, Jan 23 2009

			0 = 0+0, so a(0) = 1,
3 = 3+0 = 2+1 = 0+3, so a(3) = 3.
8 = 7+1 = 5+3 = 2+6, so a(8) = 3.

T. D. Noe, Table of n, a(n) for n = 0..10000
T. D. Noe, Plot of A132399(n) for n to 10^6
Zhi-Wei Sun, Posing to Number Theory List (1)
Zhi-Wei Sun, Posting to Number Theory List (2)
Zhi-Wei Sun, Conjectures on sums of primes and triangular numbers, J. Combin. Number Theory 1 (2009) 65-76 and arXiv:0803.3737

Cf. A117054, A144590.
Cf. A065397 (primes p whose only representation as the sum of a prime and a triangular number is p+0), A090302 (largest prime p for each n).
Cf. A154752 (smallest prime p for each n). - T. D. Noe, Jan 19 2009

Corrected, edited and extended by T. D. Noe, Mar 26 2008

Edited by N. J. A. Sloane, Jan 15 2009

A138666 Numbers n such that every sum of consecutive positive numbers ending in n is not prime.

Original entry on oeis.org

1, 8, 14, 18, 20, 25, 26, 28, 32, 33, 35, 38, 39, 44, 46, 48, 50, 56, 58, 60, 62, 63, 65, 68, 72, 74, 77, 78, 80, 81, 85, 86, 88, 92, 93, 94, 95, 98, 102, 104, 105, 108, 110, 111, 116, 118, 119, 122, 123, 124, 125, 128, 130, 133, 134, 138, 140, 143, 144, 145, 146, 148

Offset: 1

T. D. Noe, Mar 26 2008

Also numbers n such that all terms in row n of A087401 are not prime. Also the index of the triangular numbers in A076768. See A087572 for the least prime, if it exists. David Wasserman points out (in A087572) that n is in this sequence if and only if n and 2n-1 are both not prime. This sequence is infinite because 2k^2 is a term for all k>1.

			8 is in this sequence because 8, 15=7+8, 21=6+7+8, 26=5+6+7+8, 30=4+5+6+7+8, 33=3+4+5+6+7+8, 35=2+3+4+5+6+7+8 and 36=1+2+3+4+5+6+7+8 are all composite.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A010051.

a138666 n = a138666_list !! (n-1)
a138666_list = map (head . tail) $
   filter (all (== 0) . map a010051 . tail) $ drop 2 a087401_tabl
-- Reinhard Zumkeller, Oct 03 2012

Select[Range[200], !PrimeQ[ # ] && !PrimeQ[2#-1] &]
Select[Range[150],AllTrue[Accumulate[Reverse[Range[#]]],!PrimeQ[#]&]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 18 2017 *)

from sympy import isprime
from itertools import accumulate
def ok(n): return all(not isprime(s) for s in accumulate(range(n, 0, -1)))
def aupto(nn): return [m for m in range(1, nn+1) if ok(m)]
print(aupto(148)) # Michael S. Branicky, Jan 08 2021

A100592 Least positive integer that can be represented as the sum of exactly two semiprimes in exactly n ways.

Original entry on oeis.org

1, 8, 18, 30, 43, 48, 60, 72, 91, 108, 132, 155, 120, 144, 192, 168, 216, 236, 227, 180, 320, 340, 240, 252, 348, 300, 324, 336, 488, 484, 456, 396, 614, 360, 524, 548, 706, 468, 536, 656, 628, 420, 624, 576, 612, 588, 540, 600, 648, 768, 732, 800, 832, 660

Offset: 0

Jonathan Vos Post, Nov 30 2004

A072931(a(n)) = n and A072931(m) < n for m < a(n). [From Reinhard Zumkeller, Jan 21 2010]

			a(0) = 1 because 1 is the smallest positive integer that cannot be represented as sum of two semiprimes (since 4 is the smallest semiprime). a(1) = 8 because 8 is the smallest such sum of two semiprimes: 4 + 4. Similarly a(2) = 18 because 18 = 14 + 4 = 9 + 9 where {4,9,14} are semiprimes and there is no third such sum for 18.

Reinhard Zumkeller and Zak Seidov, Table of n, a(n) for n = 0..1000 (Terms 0-250 from Reinhard Zumkeller)

Cf. A001358, A076768, A100570, A072966.

a(n) = min{i such that i = A001358(j) + A001358(k) in n ways}.

A090302 Begin with n and consider numbers obtained by successively subtracting 0, 1, 2, 3, ...; a(n) = largest prime that arises in the process, i.e., largest prime of the form n - T(r), where T(r) is the r-th triangular number; or 0 if no such number exists.

Original entry on oeis.org

0, 2, 3, 3, 5, 5, 7, 7, 3, 7, 11, 11, 13, 13, 5, 13, 17, 17, 19, 19, 11, 19, 23, 23, 19, 23, 17, 13, 29, 29, 31, 31, 23, 31, 29, 0, 37, 37, 29, 37, 41, 41, 43, 43, 17, 43, 47, 47, 43, 47, 41, 37, 53, 53, 19, 53, 47, 43, 59, 59, 61, 61, 53, 61, 59, 11, 67, 67, 59, 67, 71

Offset: 1

Amarnath Murthy, Nov 30 2003

nonn

a(p) = p if p is a prime.

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

Cf. A076768 (positions of 0's).

Cf. A132399.

Largest prime of the form n - r(r+1)/2.

More terms from Frank Ellermann, Dec 03 2003

A101182 Least positive integer that can be represented as the sum of a prime and a triangular number in exactly n ways.

Original entry on oeis.org

1, 2, 3, 8, 17, 83, 47, 89, 107, 212, 194, 347, 284, 674, 524, 464, 467, 947, 662, 1187, 1514, 1304, 1019, 1229, 1559, 2189, 1724, 2699, 2084, 3434, 2417, 4022, 3467, 3824, 3764, 3362, 5324, 5879, 5672, 7214, 5927, 6179, 6134, 7079, 6704, 7727, 10667

Offset: 0

Jonathan Vos Post and Ray Chandler, Dec 14 2004

			a(0) = 1 because 1 is the smallest positive integer that cannot be represented as sum of a prime and a triangular number (since 2 is the smallest prime).
a(1) = 2 = 2 + 0; a(2) = 3 = 3 + 0 = 2 + 1; a(3) = 8 = 2 + 6 = 5 + 3 = 7 + 1.

Donovan Johnson, Table of n, a(n) for n = 0..1000

Cf. A000040, A000217, A076768.

a(n) = min{i such that i = A000040(j) + A000217(k) in n ways}.

A111908 Numbers that are not the sum of a prime and a nonzero triangular number.

Original entry on oeis.org

1, 2, 7, 36, 61, 105, 171, 210, 211, 216, 325, 351, 406, 528, 561, 630, 741, 780, 990, 1081, 1176, 1275, 1596, 1711, 1830, 1953, 2016, 2145, 2346, 2628, 2775, 3003, 3081, 3240, 3321, 3655, 3741, 3916, 4278, 4371, 4465, 4560, 4851, 5253, 5460, 5565, 5886

Offset: 1

Stefan Steinerberger, Nov 25 2005

nonn

Can anybody prove or disprove a(n) = O(n^c) for some constant c?

Jonathan Vos Post has observed that every term in A076768 also occurs in this sequence.

			7 = 1+6 = 2+5 = 3+4; 7 is in the sequence because there is no sum where the first element is a prime and the second one a triangular number.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A064233, A000217, A076768, A020756, A072386, A014089, A014090, A000404, A001481, A002654, A100570, A101181.

lim=6000;plim=PrimePi[lim];tlim=Ceiling[Sqrt[2lim]];Complement[Range[lim],Union[Flatten[Table[Prime[i]+PolygonalNumber[j],{i,plim},{j,tlim}]]]] (* James C. McMahon, Jun 04 2024 *)

a(47) and offset corrected by Donovan Johnson, Feb 09 2013

A224362 Number of partitions of n into a prime and a triangular number.

Original entry on oeis.org

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

Offset: 0

Alex Ratushnyak, Apr 04 2013

nonn

Indices of zeros: 0 followed by A076768.

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A000040, A000217, A076768, A101182.

nn = 13; tri = Table[n*(n + 1)/2, {n, 0, nn}]; pr = Prime[Range[PrimePi[tri[[-1]]]]]; Table[Length[Intersection[pr, n - tri]], {n, 0, tri[[-1]]}] (* T. D. Noe, Apr 05 2013 *)

import math
primes = [2, 3]
def isprime(k):
  for p in primes:
    if k%p==0:  return 0
  primes.append(k)
  return 1
def rootTriangular(a):
    sr = 2**(int(math.log(a,2))+2)
    while a < sr*(sr+1)//2:
          sr>>=1
    b = sr>>1
    while b:
      s = sr+b
      if a >= s*(s+1)//2:
        sr = s
      b>>=1
    return sr
for i in range(1<<10):
    k = 0
    for p in primes:
      if i <= p:  continue
      r = rootTriangular(i - p)
      if r*(r+1)//2 == i-p:  k+=1
    if i>1:
      if i<=3:  k += 1
      else:     k += isprime(i)
    print(k, end=', ')

G.f.: (Sum_{i>=0} x^(i*(i+1)/2))*(Sum_{j>=1} x^prime(j)). - Ilya Gutkovskiy, Feb 07 2017

A100591 Least positive integer that can be represented as sum of semiprime and a triangular number in exactly n ways. Triangular numbers include t(0)=0 and (1)=1.

Original entry on oeis.org

1, 4, 7, 10, 35, 25, 49, 61, 121, 140, 211, 268, 224, 392, 472, 517, 565, 529, 707, 1006, 1039, 994, 1213, 989, 1274, 1717, 1769, 1822, 2047, 2272, 2419, 2573, 2642, 3029, 3149, 3152, 3848, 3359, 4199, 4019, 4307, 4847, 5027, 4877, 5492, 6077

Offset: 0

Jonathan Vos Post, Nov 30 2004

Computed by Ray Chandler.

			a(0) = 1 because 1 is the smallest positive integer that cannot be represented as sum of semiprime and a triangular number (since 4 is the smallest semiprime). a(1) = 4 because 4 is the smallest such sum, namely semiprime(1)=4 + t(0)=0. Similarly a(2) = 7 because 7 = 4 + 3 and 7 = 6 + 1, where 4 and 6 are semiprimes, 3 and 1 are triangular.

Cf. A000217, A001358, A076768, A100570.

a(n) = min(i such that i = A001358(j) + A000217(k) in n ways).

A100593 Greatest positive integer that can be represented as the sum of exactly two semiprimes in exactly n ways.

Original entry on oeis.org

33, 62, 105, 122, 135, 174, 285, 214, 294, 315, 318, 366, 525, 405, 394, 474, 498, 585, 495, 529, 765, 645, 735, 693, 945, 705, 761, 825, 1155, 1109, 901, 989, 1049, 1123, 1365, 1063, 1121, 1181, 1243, 1129, 1231, 1169, 1349, 1485, 1399, 1577

Offset: 0

Jonathan Vos Post, Nov 30 2004

nonn

			a(0) = 33 is only a conjecture, based on computer searches by Lior Manor and by Ray Chandler. a(1) = 62 because 62 = 58 + 4 is the only way to partition 62 into two semiprimes.

Cf. A001358, A076768, A100570, A072966.

a(n) = max{i such that i = A001358(j) + A001358(k) in n ways}.

A255904 Numbers that are neither triangular nor the sum of a positive triangular number and a prime.

Original entry on oeis.org

2, 7, 61, 211, 216

Offset: 1

Arkadiusz Wesolowski, Mar 10 2015

Probably finite.
First four terms are primes, and are in A065397.
If it exists, a(6) > 5*10^6. - Derek Orr, Mar 12 2015
If it exists, a(6) > 4*10^9. - Hiroaki Yamanouchi, Mar 16 2015

			a(2) = 7 is in the sequence because 7 is not a triangular number, and cannot be written as a sum of the primes < 7 {2, 3, 5} and one of the positive triangular numbers < 7 {1, 3, 6}.

Cf. A065397, A076768.

lst:=[]; r:=21; for n in [0..r*(r+1)/2] do if not IsSquare(8*n+1) then f:=0; k:=1; while k*(k+1)/2 lt n-1 do if IsPrime(n-Truncate(k*(k+1)/2)) then f:=1; break; end if; k+:=1; end while; if IsZero(f) then Append(~lst, n); end if; end if; end for; lst;

a(n)=k=1;while(n-(t=k*(k+1)/2)>=0,if(isprime(n-t)||n==t,return(k));k++)
n=1;while(n<10^5,if(!a(n),print1(n,", "));n++) \\ Derek Orr, Mar 12 2015

cp's OEIS Frontend

A132399 Number of ordered ways of writing n = i + j, where i is 0 or a prime and j is a triangular number (A000217) >= 0.

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A138666 Numbers n such that every sum of consecutive positive numbers ending in n is not prime.

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A100592 Least positive integer that can be represented as the sum of exactly two semiprimes in exactly n ways.

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A090302 Begin with n and consider numbers obtained by successively subtracting 0, 1, 2, 3, ...; a(n) = largest prime that arises in the process, i.e., largest prime of the form n - T(r), where T(r) is the r-th triangular number; or 0 if no such number exists.

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Extensions

A101182 Least positive integer that can be represented as the sum of a prime and a triangular number in exactly n ways.

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Formula

A111908 Numbers that are not the sum of a prime and a nonzero triangular number.

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Mathematica

Extensions

A224362 Number of partitions of n into a prime and a triangular number.

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Mathematica

Python

Formula

A100591 Least positive integer that can be represented as sum of semiprime and a triangular number in exactly n ways. Triangular numbers include t(0)=0 and (1)=1.

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Formula

A100593 Greatest positive integer that can be represented as the sum of exactly two semiprimes in exactly n ways.

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