A061358 -id:A061358 - cp's OEIS frontend

A103273 Number of ways of writing prime(n)-1 in the form prime(i)+prime(j).

0, 0, 1, 1, 2, 1, 2, 2, 3, 2, 3, 4, 3, 4, 4, 3, 4, 6, 6, 5, 6, 7, 5, 4, 7, 6, 8, 6, 8, 7, 10, 7, 5, 8, 5, 12, 11, 10, 6, 6, 7, 14, 8, 11, 9, 13, 19, 11, 7, 12, 7, 9, 18, 9, 8, 9, 9, 19, 16, 14, 16, 8, 15, 12, 17, 10, 24, 19, 9, 16, 10, 10, 18, 18, 22, 10, 9, 21, 14, 20, 11, 30, 14, 19, 21, 13, 13

Offset: 1

Yasutoshi Kohmoto, Jan 27 2005

nonn

			11-1=3+7=5+5, so a(5)=2.

Table[Count[IntegerPartitions[p-1,{2}],?(AllTrue[#,PrimeQ]&)],{p,Prime[ Range[90]]}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Sep 09 2019 *)

a(n) = A061358(prime(n)-1). - David Wasserman, Oct 08 2005

More terms from David Wasserman, Oct 08 2005

A104886 Sequence A048138, the number of times that a positive integer occurs as the sum of proper divisors, if Goldbach partitions (two odd primes, which account for most of the values) are ignored.

Original entry on oeis.org

0, 1, 1, 0, 2, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 0, 1, 0, 2, 1, 2, 0, 1, 0, 1, 0, 1, 0, 1, 2, 2, 1, 1, 0, 1, 0, 1, 0, 3, 1, 2, 1, 2, 0, 2, 0, 1, 1, 2, 0, 0, 0, 2, 1, 1, 2, 1, 0, 1, 0, 2, 1, 3, 1, 1, 0, 1, 0, 1, 0, 1, 2, 3, 0, 3, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 0, 1, 0, 0, 2

Offset: 2

Nathan McNew (agreatnate(AT)yahoo.com), Mar 29 2005

nonn

			a(13)=1 because s(27)=1+3+9 and s(35)=1+5+7=13 however 35's factors 3 and 5 are a Goldbach partition, so 35 is not counted.

Cf. A048138, A001065, A061358.

a(n) = number of m such that the sum of the proper divisors of m is n, ignoring m if m is the product of two different odd primes.

A131834 Indices of records in A100949.

Original entry on oeis.org

6, 9, 11, 17, 38, 51, 62, 88, 93, 98, 122, 148, 152, 188, 222, 232, 248, 266, 272, 296, 308, 326, 388, 398, 458, 488, 500, 518, 572, 602, 686, 692, 708, 860, 912, 972, 992, 1068, 1112, 1128, 1146, 1152, 1270, 1272, 1340, 1356, 1422, 1536, 1542, 1578

Offset: 1

Jonathan Vos Post, Oct 04 2007

nonn

			a(15) = 222 because there are 22 partitions of n into a prime and a semiprime and that 22 is a record.
For n = 6, 9, 11, 17, 38, 51, 62, 88, 93, 98, 122, 148, 152, 188, 222, A100949(n) = 1, 2, 3, 5, 6, 8, 10, 11, 12, 13, 16, 17, 19, 21, 22.

Cf. A000040, A001358, A061358, A100949.

nPar[n_] := Length@ Select[Prime@ Range[ PrimePi@ n], PrimeOmega[n - #] == 2 &]; r = 0; L = {}; n = 2; While[Length[L] < 50, p = nPar[++n]; If[p > r, r = p; AppendTo[L, n]]]; L (* Giovanni Resta, Jun 19 2016 *)
DeleteDuplicates[Table[{n,Count[Sort/@(PrimeOmega/@IntegerPartitions[n,{2}]),{1,2}]},{n,1600}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]]//Rest (* Harvey P. Dale, Jun 14 2024 *)

Numbers n such that the number of partitions of n into a prime and a semiprime is a record.

Data corrected by Giovanni Resta, Jun 19 2016

A152165 Largest number which is not the sum of an n-almost prime and a prime.

Original entry on oeis.org

10, 300, 60060, 3573570, 446185740

Offset: 2

Jonathan Vos Post, Mar 30 2009

All the values are conjectural and untrustworthy. - N. J. A. Sloane, Oct 05 2009

For n=1 see A061358. - N. J. A. Sloane, Oct 05 2009

Cf. A130588, A146295, A146296, A146297, A100949.

A330210 Numbers that can be expressed as the sum of 2 prime numbers in a prime number of different ways.

Original entry on oeis.org

10, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 38, 40, 44, 48, 52, 54, 56, 62, 64, 68, 70, 74, 76, 78, 82, 86, 94, 96, 98, 104, 112, 124, 128, 130, 136, 140, 144, 148, 156, 158, 164, 168, 174, 176, 178, 186, 188, 192, 194, 198, 206, 208, 210, 216, 218, 222, 224

Offset: 1

Pietro Saia, Dec 05 2019

nonn

			24 can be expressed as the sum of 2 prime numbers in 3 different ways (5+19, 7+17, and 11+13), and 3 is prime.

Cf. A000040, A014091, A061358.

Select[Range[2, 224, 2], PrimeQ@ Length@ IntegerPartitions[#, {2}, Prime@ Range@ PrimePi@ #] &] (* Giovanni Resta, Dec 06 2019 *)

import math
from sympy import isprime
def main(n):
    x = {}
    a = 1
    b = 1
    for i in range(2, n):
        x[i] = []
        while a < i:
            if a + b == i:
                x[i].append(str(a) + "+" + str(b))
            b += 1
            if b == i:
                a += 1
                b = 1
        a = 1
        b = 1
    for i in x:
        x[i] = x[i][0:math.ceil(len(x[i])/2)]
    x[2] = ["1+1"]
    newdict = {}
    for i in x:
        newdict[i] = []
        for j in x[i]:
            if isprime(int(j.split("+")[0])) and isprime(int(j.split("+")[1])):
                newdict[i].append(j)
    finaloutput = []
    for i in newdict:
        if isprime(len(newdict[i])):
            finaloutput.append(i)
    return finaloutput
def a(n):
    x = 0
    while len(main(x)) != n:
        x += 1
    return main(x)[-1]

A334134 Number of integer-sided triangles with perimeter n whose side lengths can be written as the sum of two primes in the same number of ways.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 1, 1, 0, 0, 1, 1, 2, 3, 3, 4, 5, 6, 6, 6, 5, 5, 4, 6, 4, 6, 5, 5, 6, 7, 6, 7, 6, 5, 6, 4, 4, 7, 5, 3, 6, 6, 7, 7, 9, 6, 8, 5, 6, 8, 7, 5, 6, 7, 5, 7, 5, 6, 4, 5, 3, 8, 4, 6, 6, 8, 7, 9, 9, 10, 7, 9, 8, 12, 6, 8, 7, 9, 6, 11, 6, 11, 9, 11, 6, 14

Offset: 1

Wesley Ivan Hurt, Apr 15 2020

			a(4) = 0; no triangles can be made.
a(7) = 2; The two triangles [1,3,3] and [2,2,3] both have perimeter 7, and in each case, the side lengths can be written as the sum of two primes in the same number of ways (0 ways).
a(12) = 1; The triangle [4,4,4] has perimeter 12 and all of its side lengths can be written as the sum of two primes in the same number of ways (1 way).
a(15) = 3; the triangles [4,4,7], [4,5,6] and [5,5,5] all have perimeter 15. In each triangle, all the side lengths can be written as the sum of two primes in the same number of ways (1 way).

Wikipedia, Integer Triangle

Cf. A005044, A061358.

Table[Sum[Sum[KroneckerDelta[Sum[(PrimePi[r] - PrimePi[r - 1]) (PrimePi[k - r] - PrimePi[k - r - 1]), {r, Floor[k/2]}], Sum[(PrimePi[s] - PrimePi[s - 1]) (PrimePi[i - s] - PrimePi[i - s - 1]), {s, Floor[i/2]}], Sum[(PrimePi[t] - PrimePi[t - 1]) (PrimePi[(n - i - k) - t] - PrimePi[(n - i - k) - t - 1]), {t, Floor[(n - i - k)/2]}]]*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * [c(i) = c(k) = c(n-i-k)], where [] is the Iverson bracket and c = A061358.

A339020 Largest value of (p*q mod n), for primes p and q, where p + q = n and p <= q (or 0 if no such primes exist).

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 7, 5, 5, 0, 11, 9, 7, 11, 7, 0, 11, 15, 11, 17, 19, 0, 23, 21, 17, 0, 19, 0, 29, 27, 23, 29, 25, 0, 35, 0, 27, 35, 39, 0, 41, 39, 39, 41, 37, 0, 47, 45, 41, 0, 43, 0, 47, 51, 55, 0, 53, 0, 59, 57, 53, 59, 55, 0, 65, 0, 59, 65, 69, 0, 71, 69, 65, 71, 71, 0, 53

Offset: 1

Wesley Ivan Hurt, Nov 22 2020

nonn

a(m) = 0 for m in A014092.

			a(14) = 7; There are two partitions of 14 into two primes, (3,11) and (7,7). Since (3*11 mod 14) = 5 and (7*7 mod 14) = 7, then 7 is the largest. Therefore, a(14) = 7.

Cf. A014092, A061358, A338768.

Table[If[n == 1, 0, Max[Table[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i] - PrimePi[n - i - 1]) Mod[i (n - i), n], {i, Floor[n/2]}]]], {n, 100}]
Table[Max[Mod[Times@@#,n]&/@Select[IntegerPartitions[n,{2}],AllTrue[#,PrimeQ]&]],{n,80}]/.(-\[Infinity]->0) (* Harvey P. Dale, Mar 24 2024 *)

A352787 Numbers with as many divisors as Goldbach partitions.

Original entry on oeis.org

34, 46, 58, 102, 116, 122, 138, 150, 154, 162, 172, 184, 190, 196, 212, 228, 264, 266, 296, 304, 332

Offset: 1

Wesley Ivan Hurt, Apr 02 2022

If it exists, a(22) > 7*10^5. - Ivan N. Ianakiev, Apr 11 2022

Numbers k such that A000005(k) = A061358(k). - Michel Marcus, Apr 12 2022

			122 is in the sequence since it has 4 divisors {1,2,61,122} and 4 Goldbach partitions (13,109), (19,103), (43,79), (61,61).

Cf. A000005, A061358.

Select[Range[332],DivisorSigma[0,#]==Length[Select[#-Prime[Range[PrimePi[#/2]]], PrimeQ]]&] (* Ivan N. Ianakiev, Apr 11 2022 *)

nbgp(n) = my(s); forprime(q=2, n\2, s+=isprime(n-q)); s; \\ A061358
isok(k) = numdiv(k) == nbgp(k); \\ Michel Marcus, Apr 12 2022

cp's OEIS Frontend

A103273 Number of ways of writing prime(n)-1 in the form prime(i)+prime(j).

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A104886 Sequence A048138, the number of times that a positive integer occurs as the sum of proper divisors, if Goldbach partitions (two odd primes, which account for most of the values) are ignored.

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A131834 Indices of records in A100949.

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A152165 Largest number which is not the sum of an n-almost prime and a prime.

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A330210 Numbers that can be expressed as the sum of 2 prime numbers in a prime number of different ways.

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A334134 Number of integer-sided triangles with perimeter n whose side lengths can be written as the sum of two primes in the same number of ways.

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A339020 Largest value of (p*q mod n), for primes p and q, where p + q = n and p <= q (or 0 if no such primes exist).

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A352787 Numbers with as many divisors as Goldbach partitions.

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