A177689 -id:A177689 - cp's OEIS frontend

A370121 Triangle read by rows: T(n,k) = A002110(n) + A002110(k), 0 <= k <= n; sums of two primorials, not necessarily distinct.

2, 3, 4, 7, 8, 12, 31, 32, 36, 60, 211, 212, 216, 240, 420, 2311, 2312, 2316, 2340, 2520, 4620, 30031, 30032, 30036, 30060, 30240, 32340, 60060, 510511, 510512, 510516, 510540, 510720, 512820, 540540, 1021020, 9699691, 9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200, 19399380, 223092871, 223092872

Offset: 0

Antti Karttunen, Feb 29 2024

After the initial 2, numbers with either one 2 or two 1's in their primorial base representation (A049345), with all the other digits zeros.

			Triangle begins as:
        2;
        3,       4;
        7,       8,      12;
       31,      32,      36,      60;
      211,     212,     216,     240,     420;
     2311,    2312,    2316,    2340,    2520,    4620;
    30031,   30032,   30036,   30060,   30240,   32340,   60060;
   510511,  510512,  510516,  510540,  510720,  512820,  540540,  1021020;
  9699691, 9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200, 19399380;

Cf. A002110, A049345, A087112, A276086, A276150, A370129 (arithmetic derivative applied to this triangle).
Cf. A006862 (left edge), A088860 (right edge).
Cf. A177689 (same triangle without the right edge), A370134 (without the leftmost column).
Subsequence of A370132.
Cf. also A173786.

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))); };

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

For n >= 1, A276086(a(n)) = A087112(1+n).

A177711 Natural numbers which are not sums of one or more distinct primorials.

Original entry on oeis.org

4, 5, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 34, 35, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Jonathan Vos Post, May 11 2010

Numbers with a digit larger than one in primorial base representation, A049345. Numbers k for which A276086(k) is not squarefree. - Antti Karttunen, Feb 17 2025

			1 and 2 are not in the sequence, as they are the first and second primorials, 0# and 1#. 3 is not in the sequence, as 3 = 1+2. Neither 4 nor 5 can be the sum of distinct primorials (i.e. 4=2+2 or 5 = 2+2+1 repeat a primorial). 6 is not in the sequence, as it is 3#. 7 and 8 are not in the sequence as 7 = 6+1 and 8 = 6+2. 9 is not in the sequence, as 9 = 6+2+1.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to primorial base.

Cf. A002110, A049345, A177689, A177697, A177708, A177709, A276086.
Complement of A276156.
Positions of terms > 1 in A328114.
Subsequences: A380535, A381034.

is_A177711(n) = { my(p=2); while(n, if(n%p > 1, return(1)); n = n\p; p = nextprime(1+p)); (0); }; \\ Antti Karttunen, Feb 17 2025

COMPLEMENT of {Primorial numbers A002110 UNION A177689 Sums of 2 distinct primorials UNION Sums of 3 distinct primorials A177697 UNION Sums of 4 distinct primorials A177709 UNION ...}.

{k such that A328114(k) > 1}. - Antti Karttunen, Feb 17 2025

A177697 Sums of 3 distinct primorials.

Original entry on oeis.org

9, 33, 37, 38, 213, 217, 218, 241, 242, 246, 2313, 2317, 2318, 2341, 2342, 2346, 2521, 2522, 2526, 2550, 30033, 30037, 30038, 30061, 30062, 30066, 30241, 30242, 30246, 30270, 32341, 32342, 32346, 32370, 32550, 510513, 510517, 510518, 510541, 510542

Offset: 1

Jonathan Vos Post, May 11 2010

This is to numbers that are the sum of 3 different primes (A124867) as primorials (A002110) are to primes (A000040). The subsequence of primes among these sums of 3 distinct primorials begins: 37, 241, 2341, 2521, 30241, 32341, 512821, 540541.

			9 = 6+2+1
33 = 30+2+1
37 = 30+6+1
38 = 30+6+2
213 = 210+2+1

Cf. A000040, A002110, A006862, A038609, A124867, A177689.

Take[Total/@Subsets[Join[{1},FoldList[Times,Prime[Range[10]]]],{3}]// Union,40] (* Harvey P. Dale, Nov 07 2017 *)

{a(n)} = {A002110(i) + A002110(j) + A002110(k) for i =/= j, i =/= k, j =/= k}.

A177709 Sums of 4 distinct primorials.

Original entry on oeis.org

39, 219, 243, 247, 248, 2319, 2343, 2347, 2348, 2523, 2527, 2528, 2551, 2552, 2556, 30039, 30063, 30067, 30068, 30243, 30247, 30248, 30271, 30272, 30276, 32343, 32347, 32348, 32371, 32372, 32376, 32551, 32552, 32556, 32580, 510519, 510543

Offset: 1

Jonathan Vos Post, May 11 2010

This is to numbers that are the sum of 4 different primes (A177708) as primorials (A002110) are to primes (A000040). The subsequence of primes among these sums of 4 distinct primorials begins: 2347, 2551, 30271, 32371, 510751. The subsequence of nontrivial powers a^b with b>1 begin: a(3) = 243, a(24) = 30276 = 30030+210+30+6 = 2^2 x 3^2 x 29^2.

			a(1) = 39 = 30+6+2+1
a(2) = 219 = 210+6+2+1
a(3) = 243 = 210+30+2+1 = 3^5
a(4) = 247 = 210+30+6+1
a(5) = 248 = 210+30+6+2.

Cf. A000040, A002110, A006862, A038609, A124867, A177689, A177697, A177708.

{a(n)} = {A002110(i) + A002110(j) + A002110(k)+ A002110(L) for distinct i, j, k, L}.

Corrected (2348 inserted) by R. J. Mathar, May 15 2010

cp's OEIS Frontend

A370121 Triangle read by rows: T(n,k) = A002110(n) + A002110(k), 0 <= k <= n; sums of two primorials, not necessarily distinct.

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A177711 Natural numbers which are not sums of one or more distinct primorials.

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A177697 Sums of 3 distinct primorials.

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A177709 Sums of 4 distinct primorials.

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