A116360 Smallest number having exactly n partitions into products of two successive primes (A006094), or -1 if no such number exists.
1, 6, 30, 60, 90, 105, 120, 135, 143, 158, 155, 167, 173, 182, 185, 207, 197, 203, 212, 215, 221, 231, 227, 233, 239, 242, 256, 245, 251, 261, 257, 260, 263, 266, 282, 272, 275, 278, 281, 291, -1, 287, 290, 293, 296, 309, 312, 302, 305, 319, 308, 314, -1, 317, 322, 320
Offset: 0
Keywords
A337853 a(n) is the number of partitions of n as the sum of two Niven numbers.
0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 4, 4, 4, 3, 2, 4, 3, 3, 4, 3, 3, 5, 3, 4, 5, 4, 4, 7, 4, 5, 6, 5, 3, 7, 4, 4, 6, 4, 2, 7, 3, 4, 5, 4, 3, 7, 3, 4, 5, 4, 3, 8, 3, 4, 6, 3, 3, 6, 2, 5, 6, 5, 3, 8, 4, 4, 6
Offset: 0
Comments
a(n) >= 1 for n >= 2 ?.
For n <= 200000, a(n) = 1 only for n = 2, 3, 299, (2 = 1 + 1, 3 = 1 + 2, 299 = 1 + 288) and a(n) = 2 only for n in {4, 5, 35, 59, 79, 95, 97, 149, 169, 179, 389}.
Examples
0 and 1 cannot be decomposed as the sum of two Niven numbers, so a(0) = a(1) = 0. 4 = 1 + 3 = 2 + 2 and 1, 2, 3 are in A005349, so a(4) = 2. 15 = 3 + 12 = 5 + 10 = 6 + 9 = 7 + 8 and 3, 5, 6, 7, 8, 9, 10, 12 are in A005349, so a(15) = 4.
Links
- Rémy Sigrist, Table of n, a(n) for n = 0..10000
Crossrefs
Programs
-
Magma
niven:=func
; [#RestrictedPartitions(n,2,{k: k in [1..n-1] | niven(k)}): n in [0..100]]; -
Mathematica
m = 100; nivens = Select[Range[m], Divisible[#, Plus @@ IntegerDigits[#]] &]; a[n_] := Length[IntegerPartitions[n, {2}, nivens]]; Array[a, m, 0] (* Amiram Eldar, Sep 27 2020 *)
A116358 Numbers that cannot be written as sum of products of two successive primes.
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 37, 38, 40, 43, 44, 46, 49, 52, 55, 58, 61, 64, 67, 73, 79
Offset: 1
A116359 Numbers that can be written as sum of products of two successive primes.
6, 12, 15, 18, 21, 24, 27, 30, 33, 35, 36, 39, 41, 42, 45, 47, 48, 50, 51, 53, 54, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107
Offset: 1
Keywords
Examples
45 = 2*3 + 2*3 + 2*3 + 2*3 + 2*3 + 3*5 = = 3*5 + 3*5 + 3*5, therefore 45 is a term: A116357(45)=2.
Comments
Examples
Links
Crossrefs
Extensions