A085371 -id:A085371 - cp's OEIS frontend

A085370 Niven (or Harshad) numbers that are not divisible by 3.

1, 2, 4, 5, 7, 8, 10, 20, 40, 50, 70, 80, 100, 110, 112, 133, 140, 152, 190, 200, 209, 220, 224, 230, 247, 266, 280, 308, 320, 322, 364, 370, 392, 400, 407, 410, 440, 448, 460, 476, 481, 500, 506, 511, 512, 518, 550, 592, 605, 629, 640, 644, 700, 704, 715

Offset: 1

Reinhard Zumkeller, Jun 26 2003

a(n) mod A007953(a(n)) = 0 and A007953(a(n)) mod 3 > 0;

Intersection of A005349 and A001651.

Eric Weisstein's World of Mathematics, Harshad Number

Cf. A085371.

A379240 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j, where f(n) = [A003415(n), A085731(n)] if A359550(n) = 1, otherwise f(n) = n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 28, 40, 41, 21, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 46, 55, 2, 56, 57, 58, 2, 59, 41, 60, 61, 62, 2, 63, 37, 64

Offset: 1

Antti Karttunen, Dec 19 2024

nonn

It is conjectured that this is also the lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A085731(i) = A085731(j) and A376418(i) = A376418(j), for all i, j >= 1, i.e., the restricted growth sequence transform of the triple [A003415(n), A085731(n), A376418(n)]. This is true if for every pair of i and j for which i <> j, and A376418(i) = A376418(j) > 0, the ordered pairs [A003415(i), A085731(i)] and [A003415(j), A085731(j)] differ from each other.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A003415, A048103, A083345, A085371, A100716, A359550, A369051, A376418.
Differs from A344025 first at n=140, where a(140) = 97, while A344025(140) = 92.
Differs from A369046 first at n=171, where a(171) = 63, while A369046(171) = 121.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); };
Aux379240(n) = if(!A359550(n), n, my(d=A003415(n)); [d, gcd(d,n)]);
v379240 = rgs_transform(vector(up_to, n, Aux379240(n)));
A379240(n) = v379240[n];

For all i, j >= 1:
  a(i) = a(j) => A369051(i) = A369051(j) => A083345(i) = A083345(j),
  a(i) = a(j) => A376418(i) = A376418(j).

cp's OEIS Frontend

A085370 Niven (or Harshad) numbers that are not divisible by 3.

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