A087112 -id:A087112 - cp's OEIS frontend

A369979 Three-dimensional array giving all products of three (not necessarily distinct) odd primes.

27, 45, 75, 125, 63, 105, 175, 147, 245, 343, 99, 165, 275, 231, 385, 539, 363, 605, 847, 1331, 117, 195, 325, 273, 455, 637, 429, 715, 1001, 1573, 507, 845, 1183, 1859, 2197, 153, 255, 425, 357, 595, 833, 561, 935, 1309, 2057, 663, 1105, 1547, 2431, 2873, 867, 1445, 2023, 3179, 3757, 4913, 171, 285, 475, 399, 665, 931

Offset: 1

Antti Karttunen, Mar 09 2024

For n > 20, a(n) < A370138(n).

			Table T(x,y,z) = A065091(x) * A065091(y) * A065091(z), x >= y >= z >= 1, is read by lexicographical ordering of weakly decreasing triplets (x,y,z):
(1, 1, 1) -> 3*3*3 = 27;
(2, 1, 1) -> 5*3*3 = 45, (2, 2, 1) -> 5*5*3 = 75, (2, 2, 2) -> 5*5*5 = 125;
(3, 1, 1) -> 7*3*3 = 63, (3, 2, 1) -> 7*5*3 = 105, (3, 2, 2) -> 7*5*5 = 175, (3, 3, 1) -> 7*7*3 = 147, (3, 3, 2) -> 7*7*5 = 245, (3, 3, 3) -> 7*7*7 = 343.

Antti Karttunen, Table of n, a(n) for n = 1..15180

Cf. A000292, A046316 (same sequence sorted into ascending order), A065091, A276086, A370137, A370138.

Cf. also A087112.

Table[Prime[i]*Prime[j]*Prime[k], {i, 2, 8}, {j, 2, i}, {k, 2, j}] // Flatten (* Michael De Vlieger, Mar 09 2024 *)

up_to = 15180;
A369979list(up_to) = { my(v = vector(up_to), i=0); for(x=1,oo, for(y=1,x, for(z=1,y, i++; if(i > up_to, return(v)); v[i] = prime(1+x)*prime(1+y)*prime(1+z)))); (v); };
v369979 = A369979list(up_to);
A369979(n) = v369979[n];

For n > 1, a(n) = A276086(A370137(n)).

A370129 Triangle read by rows: T(n,k) = A003415(A002110(n)+A002110(k)), 0 <= k <= n; arithmetic derivatives of the sums of two primorial numbers.

Original entry on oeis.org

1, 1, 4, 1, 12, 16, 1, 80, 60, 92, 1, 216, 540, 608, 704, 1, 3740, 3100, 4548, 6324, 8164, 568, 60080, 40060, 56292, 116208, 61768, 110752, 33975, 1021040, 1041768, 794468, 2415104, 1091004, 1357128, 1942844, 28300, 9789116, 29099520, 19722884, 18576860, 35347200, 35779644, 26575580, 37935056, 704080, 335024060

Offset: 0

Antti Karttunen, Feb 29 2024

Apart from those positions (A014545) at the left edge where a(n) = 1, a(n) <= A087112(1+n) only at n=2, 4 and 5, i.e., never after the third row.

			Triangle begins as:
      1;
      1,       4;
      1,      12,       16;
      1,      80,       60,       92;
      1,     216,      540,      608,      704;
      1,    3740,     3100,     4548,     6324,     8164;
    568,   60080,    40060,    56292,   116208,    61768,   110752;
  33975, 1021040,  1041768,   794468,  2415104,  1091004,  1357128,  1942844;
  28300, 9789116, 29099520, 19722884, 18576860, 35347200, 35779644, 26575580, 37935056;

Cf. A002110, A003415, A024451, A370121, A370135, A370136.
Cf. A014545 (positions of 1's at the left edge), A087112.
Cf. also A024451 (arithmetic derivatives of primorials).

A002110(n) = prod(i=1,n,prime(i));
A370121(n) = { my(c = (sqrtint(8*n + 1) - 1) \ 2); (A002110(c) + A002110(n - binomial(c + 1, 2))); };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A370129(n) = A003415(A370121(n));

a(n) = A003415(A370121(n)).

For n, k >= 1, T(n,k) = A002110(k)*A370136(n,k) + A024451(k)*A370135(n,k).

A077553 Triangle in which the n-th row contains n distinct composite numbers with the least product and has least number of prime divisors. No member of a row is a multiple of another member of the row.

Original entry on oeis.org

4, 4, 6, 4, 6, 9, 4, 6, 9, 10, 4, 6, 9, 10, 15, 4, 6, 9, 10, 15, 25, 4, 6, 9, 10, 14, 15, 21, 4, 6, 9, 10, 14, 15, 21, 25, 4, 6, 9, 10, 14, 15, 21, 25, 35, 4, 6, 9, 10, 14, 15, 21, 25, 35, 49, 4, 6, 9, 10, 14, 15, 21, 22, 25, 33, 35, 4, 6, 9, 10, 14, 15, 21, 22, 25, 33, 35, 49, 4, 6, 9, 10

Offset: 1

Amarnath Murthy, Nov 10 2002

If there are two sets of distinct composite numbers satisfying the above condition then the set with lesser product is chosen irrespective of the number of prime divisors. Perhaps the ambiguity may not arise. E.g., row 6 is 4,6,9,10,15,25. This row cannot be extended to get the next row without bringing in another prime because every number divisible by 2,3 or 5 will be a multiple of one of the previous terms. Hence in row 7, prime 7 has to be brought in and then we get a new set of numbers: 4,6,9,10,14,15,21.

			4;
4,6;
4,6,9;
4,6,9,10;
4,6,9,10,15;
4,6,9,10,15,25;
4,6,9,10,14,15,21;

Sean A. Irvine, Table of n, a(n) for n = 1..5050 (rows 1..100 flattened)

Cf. A001358, A005843, A077554, A077555, A087112.

More terms from Ray Chandler, Aug 21 2003

Offset corrected by Sean A. Irvine, May 28 2025

A077554 Final terms of rows of A077553.

Original entry on oeis.org

4, 6, 9, 10, 15, 25, 21, 25, 35, 49, 35, 49, 55, 77, 121, 65, 77, 91, 121, 143, 169, 119, 121, 143, 169, 187, 221, 289, 169, 187, 209, 221, 247, 289, 323, 361, 247, 253, 289, 299, 323, 361, 391, 437, 529, 323, 361, 377, 391, 437, 493, 529, 551, 667, 841, 437

Offset: 0

Amarnath Murthy, Nov 10 2002

nonn

If there are two sets of distinct composite numbers satisfying the above condition then the set with lesser product is chosen irrespective of the number of prime divisors. Perhaps the ambiguity may not arise. E.g., row 6 is 4,6,9,10,15,25. This row cannot be extended to get the next row without bringing in another prime because every number divisible by 2,3 or 5 will be a multiple of one of the previous terms. Hence in row 7, prime 7 has to be brought in and then we get a new set of numbers: 4,6,9,10,14,15,21.

Cf. A002024.

Cf. A001358, A005843, A077553, A077555, A087112.

More terms from Ray Chandler, Aug 24 2003

A143215 a(n) = prime(n) * Sum_{i=1..n} prime(i).

Original entry on oeis.org

4, 15, 50, 119, 308, 533, 986, 1463, 2300, 3741, 4960, 7289, 9758, 12083, 15416, 20193, 25960, 30561, 38056, 45369, 51976, 62489, 72542, 85707, 102820, 117261, 130192, 146697, 161320, 180009, 218440, 242481, 272356, 295653, 339124, 366477

Offset: 1

Gary W. Adamson, Jul 30 2008

nonn

Row sums of triangle A087112.

Sum of semiprimes (A001358) with greater prime factor prime(n). - Gus Wiseman, Dec 06 2020

			The series begins (4, 15, 50, 119, 308,...) since the primes = (2, 3, 5, 7, 11,...) and partial sum of primes = (2, 5, 10, 17, 28,...).
a(5) = 308 = 11 * 28.
a(4) = 119 = sum of row 4 terms of triangle A087112: (14 + 21 + 35 + 49).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Row sums of A087112.
The squarefree version is A339194, row sums of A339116.
Semiprimes grouped by weight are A338904, with row sums A024697.
Squarefree semiprimes grouped by weight are A338905, with row sums A025129.
Squarefree numbers grouped by greatest prime factor are A339195, with row sums A339360.
A001358 lists semiprimes.
A006881 lists squarefree semiprimes.
A332765 is the greatest semiprime of weight n.
A338898/A338912/A338913 give the prime indices of semiprimes.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
Cf. A000040, A001222, A001748, A007504, A014342, A098350, A100484, A168472, A319613, A339003, A339114/A339115.

a143215 n = a000040 n * a007504 n  -- Reinhard Zumkeller, Nov 25 2012

A143215:= func< n | NthPrime(n)*(&+[NthPrime(j): j in [1..n]]) >;
[A143215(n): n in [1..50]]; // G. C. Greubel, Aug 27 2024

A143215:=n->ithprime(n)*sum(ithprime(i), i=1..n); seq(A143215(n), n=1..50); # Wesley Ivan Hurt, Mar 26 2014

Table[Prime[n]*Sum[Prime[i], {i, n}], {n, 50}] (* Wesley Ivan Hurt, Mar 26 2014 *)

a(n) = prime(n)*vecsum(primes(n)); \\ Michel Marcus, Jun 15 2024

def A143215(n): return nth_prime(n)*sum(nth_prime(j) for j in range(1,n+1))
[A143215(n) for n in range(1,51)] # G. C. Greubel, Aug 27 2024

a(n) = A000040(n) * A007504(n).

More terms from Vladimir Joseph Stephan Orlovsky, Sep 21 2009

A077555 Product of terms of n-th row of A077553.

Original entry on oeis.org

4, 24, 216, 2160, 32400, 810000, 9525600, 238140000, 8334900000, 408410100000, 6051137400000, 296505732600000, 16307815293000000, 1255701777561000000, 151939915084881000000, 1074848105961630000000, 82763304159045510000000

Offset: 0

Amarnath Murthy, Nov 10 2002

nonn

If there are two sets of distinct composite numbers satisfying the above condition then the set with lesser product is chosen irrespective of the number of prime divisors. Perhaps the ambiguity may not arise. E.g., row 6 is 4,6,9,10,15,25. This row cannot be extended to get the next row without bringing in another prime because every number divisible by 2,3 or 5 will be a multiple of one of the previous terms. Hence in row 7, prime 7 has to be brought in and then we get a new set of numbers: 4,6,9,10,14,15,21.

Cf. A001358, A005843, A077553, A077554, A087112.

More terms from Ray Chandler, Aug 21 2003

A219603 a(n) = prime(n) * prime(2*n-1).

Original entry on oeis.org

4, 15, 55, 119, 253, 403, 697, 893, 1357, 1943, 2263, 3071, 3977, 4429, 5123, 6731, 8083, 9089, 10519, 11857, 13067, 15089, 16351, 18779, 22019, 23533, 24823, 27499, 29321, 31301, 35941, 40217, 42881, 46009, 51703, 53303, 57619, 61777, 64963, 69373, 75001

Offset: 1

Reinhard Zumkeller, Nov 25 2012

nonn

A020639(a(n)) = A000040(n); A006530(a(n)) = A031368(n);

a(n) is central term of row 2n-1 of semiprimes triangle (A087112).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Subsequence of A001358.

Cf. A006094.

Haskell
```
a219603 n = a000040 n * a031368 n
```

a(n)=prime(n) * prime(2*n-1) \\ Charles R Greathouse IV, Feb 07 2017

a(n) = A000040(n) * A031368(n).

A370134 Triangle read by rows: T(n,k) = A002110(n) + A002110(k), 1 <= k <= n; sums of two primorials > 1, not necessarily distinct.

Original entry on oeis.org

4, 8, 12, 32, 36, 60, 212, 216, 240, 420, 2312, 2316, 2340, 2520, 4620, 30032, 30036, 30060, 30240, 32340, 60060, 510512, 510516, 510540, 510720, 512820, 540540, 1021020, 9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200, 19399380, 223092872, 223092876, 223092900, 223093080, 223095180, 223122900, 223603380

Offset: 1

Antti Karttunen, Mar 07 2024

			Triangle begins as:
        4;
        8,      12;
       32,      36,      60;
      212,     216,     240,     420;
     2312,    2316,    2340,    2520,    4620;
    30032,   30036,   30060,   30240,   32340,   60060;
   510512,  510516,  510540,  510720,  512820,  540540, 1021020;
  9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200, 19399380;

A370121 without its leftmost column. Subsequence of  A370132.
Cf. A002110, A049345, A087112, A276150, A370135.
Cf. A088860 (right edge).

nn = 20; MapIndexed[Set[P[First[#2] - 1], #1] &, FoldList[Times, 1, Prime@ Range[nn + 1]]]; Table[(P[n] + P[k]), {n, nn}, {k, n}] (* Michael De Vlieger, Mar 08 2024 *)

A002110(n) = prod(i=1,n,prime(i));
A370134(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2); (A002110(1+c) + A002110(1+n - binomial(c + 1, 2))); };

For n >= 1, A276150(a(n)) = 2.

A373844 Triangle read by rows: T(n,k) = A276086(1 + A002110(n) + A002110(k)), 1 <= k <= n, where A276086 is the primorial base exp-function.

Original entry on oeis.org

18, 30, 50, 42, 70, 98, 66, 110, 154, 242, 78, 130, 182, 286, 338, 102, 170, 238, 374, 442, 578, 114, 190, 266, 418, 494, 646, 722, 138, 230, 322, 506, 598, 782, 874, 1058, 174, 290, 406, 638, 754, 986, 1102, 1334, 1682, 186, 310, 434, 682, 806, 1054, 1178, 1426, 1798, 1922, 222, 370, 518, 814, 962, 1258, 1406, 1702, 2146, 2294, 2738

Offset: 1

Antti Karttunen, Jun 21 2024

Triangle giving all products of three primes, of which one is even (2) and two are odd (not necessarily distinct), so that the product is of the form 4m+2.

The only terms such that T(n, k) > A373845(n, k) > 1 are 30, 42, 110 at positions T(2,1), T(3,1), T(4,2), and the corresponding terms in A373845 are 6, 14, 38.

			Triangle begins as:
   18,
   30,  50,
   42,  70,  98,
   66, 110, 154, 242,
   78, 130, 182, 286, 338,
  102, 170, 238, 374, 442,  578,
  114, 190, 266, 418, 494,  646,  722,
  138, 230, 322, 506, 598,  782,  874, 1058,
  174, 290, 406, 638, 754,  986, 1102, 1334, 1682,
  186, 310, 434, 682, 806, 1054, 1178, 1426, 1798, 1922,
  222, 370, 518, 814, 962, 1258, 1406, 1702, 2146, 2294, 2738,
etc.

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 rows of the triangle

Cf. A002110, A276086, A370121, A373845.

Cf. also A087112, A369979, A373848.

A002110(n) = prod(i=1,n,prime(i));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A373844(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2, x=A002110(1+n - binomial(c + 1, 2))); A276086(1+(A002110(1+c)+x)); };

For n, k >= 1, T(n, k) = A276086(1+A370121(n, k)).

For n, k >= 1, T(n, k) = 2*A087112(n+1, k+1).

A347047 Smallest squarefree semiprime whose prime indices sum to n.

Original entry on oeis.org

6, 10, 14, 21, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526

Offset: 3

Gus Wiseman, Aug 22 2021

nonn

Compared to A001747, we have 21 instead of 22 and lack 2 and 4.
Compared to A100484 (shifted) we have 21 instead of 22 and lack 4.
Compared to A161344, we have 21 instead of 22 and lack 4 and 8.
Compared to A339114, we have 11 instead of 9 and lack 4.
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.
A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

			The initial terms and their prime indices:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   21: {2,4}
   26: {1,6}
   34: {1,7}
   38: {1,8}
   46: {1,9}

The opposite version (greatest instead of smallest) is A332765.
These are the minima of rows of A338905.
The nonsquarefree version is A339114 (opposite: A339115).
A001358 lists semiprimes (squarefree: A006881).
A024697 adds up semiprimes by weight (squarefree: A025129).
A056239 adds up prime indices, row sums of A112798.
A246868 gives the greatest squarefree number whose prime indices sum to n.
A320655 counts factorizations into semiprimes (squarefree: A320656).
A338898, A338912, A338913 give the prime indices of semiprimes.
A338899, A270650, A270652 give the prime indices of squarefree semiprimes.
A339116 groups squarefree semiprimes by greater factor, sums A339194.
A339362 adds up prime indices of squarefree semiprimes.
Cf. A001221, A087112, A089994, A098350, A176504, A338900, A338901, A338904, A338907/A338908, A339005, A339191.

Table[Min@@Select[Table[Times@@Prime/@y,{y,IntegerPartitions[n,{2}]}],SquareFreeQ],{n,3,50}]

from sympy import prime, sieve
def a(n):
    p = [0] + list(sieve.primerange(1, prime(n)+1))
    return min(p[i]*p[n-i] for i in range(1, (n+1)//2))
print([a(n) for n in range(3, 58)]) # Michael S. Branicky, Sep 05 2021

cp's OEIS Frontend

A369979 Three-dimensional array giving all products of three (not necessarily distinct) odd primes.

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A370129 Triangle read by rows: T(n,k) = A003415(A002110(n)+A002110(k)), 0 <= k <= n; arithmetic derivatives of the sums of two primorial numbers.

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A077553 Triangle in which the n-th row contains n distinct composite numbers with the least product and has least number of prime divisors. No member of a row is a multiple of another member of the row.

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A077554 Final terms of rows of A077553.

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A143215 a(n) = prime(n) * Sum_{i=1..n} prime(i).

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A077555 Product of terms of n-th row of A077553.

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A219603 a(n) = prime(n) * prime(2*n-1).

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A370134 Triangle read by rows: T(n,k) = A002110(n) + A002110(k), 1 <= k <= n; sums of two primorials > 1, not necessarily distinct.

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