cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A330813 Numbers k that are Niven numbers in a record number of bases 1 <= b <= k.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 18, 24, 36, 48, 60, 72, 96, 120, 144, 168, 180, 240, 336, 360, 480, 600, 630, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 4320, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 30240, 40320, 45360, 50400, 55440, 60480, 75600, 83160
Offset: 1

Views

Author

Amiram Eldar, Jan 01 2020

Keywords

Comments

Indices of records of A080221.

Examples

			4 is a term since it is a Niven number in 4 bases: 1, 2, 3, 4, while the numbers below 4 are Niven numbers in less than 4 bases.

Links

Crossrefs

Cf. A005349, A049445, A080221.

Programs

  • Mathematica
    nivenQ[n_, b_] := Divisible[n, Total @ IntegerDigits[n, b]]; basesCount[n_] := 1 + Sum[Boole @ nivenQ[n, b], {b, 2, n}]; bmax = 0; seq = {}; Do[b = basesCount[n]; If[b > bmax ,bmax = b; AppendTo[seq,n]],{n,1,1000}];seq

A334344 Binary Moran numbers: numbers k such that k divided by its binary weight (A000120) is a prime number.

Original entry on oeis.org

2, 6, 10, 21, 34, 55, 69, 92, 115, 116, 155, 172, 185, 205, 212, 222, 246, 284, 295, 318, 321, 332, 355, 356, 366, 395, 404, 438, 452, 474, 498, 514, 535, 556, 565, 596, 606, 623, 652, 749, 788, 822, 835, 865, 889, 905, 973, 978, 1041, 1052, 1076, 1086, 1108, 1124
Offset: 1

Views

Author

Amiram Eldar, Apr 23 2020

Keywords

Examples

			2 is a term since its binary weight is 1 and 2/1 = 2, which is a prime number.
6 (110 in binary) has a binary weight of 2 and 6/2 = 3, which is prime, so 6 is also in the sequence. Likewise 10 (1010 in binary) also has a binary weight of 2, and 10/2 = 5, which is prime, so 10 is also in the sequence.
14 (1110 in binary) has binary weight of 3. But 14/3 is not prime, so 14 is not in the sequence.

Links

Crossrefs

Subsequence of A049445.
Cf. A000120, A001101.

Programs

  • Maple
    q:= n-> (p-> is(p, integer) and isprime(p))(n/add(i, i=Bits[Split](n))):
select(q, [$1..1200])[];  # Alois P. Heinz, Apr 23 2020
  • Mathematica
    Select[Range[1000], PrimeQ[# / DigitCount[#, 2, 1]] &]
  • PARI
    isok(m) = iferr(isprime(m/hammingweight(m)), E, 0); \\ Michel Marcus, Apr 24 2020
  • Scala
    def isPrime(num: Int): Boolean = Math.abs(num) match {
  case 0 => false; case 1 => false; case n => (2 to Math.floor(Math.sqrt(n)).toInt) forall (p => n % p != 0)
}
(1 to 1000).filter{ n => val binWt = Integer.bitCount(n); (n % binWt) == 0 && isPrime(n / binWt) } // Alonso del Arte, Apr 23 2020

A356640 a(n) is the least number k such that the least base in which k is a Niven number is n, i.e., A356552(k) = n, or -1 if no such k exists.

Original entry on oeis.org

1, 3, 50, 5, 44, 7, 161, 119, 201, 11, 253, 13, 494, 226, 1444, 17, 799, 19, 437, 1189, 957, 23, 1081, 2263, 755, 767, 927, 29, 932, 31, 1147, 5141, 1191, 1226, 2009, 37, 1517, 1522, 1641, 41, 1927, 43, 2021, 2026, 2164, 47, 2491, 4559, 5001, 2602, 2757, 53, 2972
Offset: 2

Views

Author

Amiram Eldar, Aug 19 2022

Keywords

Examples

			a(3) = 3 since 3 is a Niven number in base 3 and in no other base smaller than 3. 1 and 2 are also Niven numbers in base 3, but they are also Niven numbers in base 2.

Links

Crossrefs

Cf. A005349, A049445, A064150, A064438, A064481, A356552.
Similar sequence: A249634.

Programs

  • Mathematica
    f[n_] := Module[{b = 2}, While[! Divisible[n, Plus @@ IntegerDigits[n, b]], b++]; b]; A356640[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] - 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; A356640[50, 10^4]

Formula

a(p) = p for an odd prime p.

A363788 Even primitive binary Niven numbers: even terms of A363787.

Original entry on oeis.org

6, 10, 18, 34, 60, 66, 92, 108, 116, 126, 130, 156, 172, 180, 204, 212, 222, 228, 246, 258, 284, 300, 308, 318, 332, 340, 356, 366, 378, 396, 404, 414, 420, 438, 452, 462, 474, 486, 498, 514, 540, 556, 564, 588, 596, 606, 612, 630, 652, 660, 676, 708, 726, 780
Offset: 1

Views

Author

Amiram Eldar, Jun 22 2023

Keywords

Comments

The odd terms of A363787 are all the odd binary Niven numbers (A144302).
This sequence is infinite. E.g., A052548(k) = 2^k + 2 is a term for all k >= 2.

Links

Crossrefs

Subsequence of A049445 and A363787.
Equals A363787 \ A144302.
Cf. A052548, A358255 (decimal analog).

Programs

  • Mathematica
    binNivQ[n_] := Divisible[n, DigitCount[n, 2, 1]]; q[n_] := binNivQ[n] && ! (EvenQ[n] && binNivQ[n/2]); Select[Range[2, 1000, 2], q]
  • PARI
    isbinniv(n) = !(n % hammingweight(n));
is(n) = !(n%2) && isbinniv(n) && !isbinniv(n/2);

A363790 Numbers k such that k and k+1 are both primitive binary Niven numbers (A363787).

Original entry on oeis.org

115, 155, 204, 284, 355, 395, 404, 555, 564, 595, 675, 804, 835, 846, 1075, 1124, 1164, 1182, 1266, 1315, 1434, 1555, 1604, 1686, 1795, 1938, 2075, 2124, 2195, 2244, 2315, 2324, 2358, 2435, 2595, 3084, 3204, 3282, 3366, 4124, 4195, 4206, 4235, 4244, 4364, 4458
Offset: 1

Views

Author

Amiram Eldar, Jun 22 2023

Keywords

Examples

			115 is a term since 115 and 116 are both primitive binary Niven numbers.

Links

Crossrefs

Subsequence of A049445, A330931 and A363787.
Subsequences: A363791, A363792.

Programs

  • Mathematica
    binNivQ[n_] := Divisible[n, DigitCount[n, 2, 1]]; q[n_] := binNivQ[n] && ! (EvenQ[n] && binNivQ[n/2]); Select[Range[5000], q[#] && q[# + 1] &]
  • PARI
    isbinniv(n) = !(n % hammingweight(n));
isprim(n) = isbinniv(n) && !(!(n%2) && isbinniv(n/2));
is(n) = isprim(n) && isprim(n+1);

A363791 Starts of runs of 3 consecutive integers that are primitive binary Niven numbers (A363787).

Original entry on oeis.org

4184046, 5234670, 6285294, 7861230, 8123886, 8255214, 8255215, 8320878, 8353710, 8370126, 8379247, 12238830, 12451631, 12572622, 13623246, 13629935, 14515182, 14646510, 14673870, 14673871, 14679342, 15040494, 15335375, 15449071, 15531759, 15708078, 15986543, 16178670
Offset: 1

Views

Author

Amiram Eldar, Jun 22 2023

Keywords

Examples

			4184046 is a term since 4184046, 4184047 and 4184048 are all primitive binary Niven numbers.

Links

Crossrefs

Subsequence of A049445, A330931, A330932, A363787 and A363790.
A363792 is a subsequence.

Programs

  • Mathematica
    binNivQ[n_] := Divisible[n, DigitCount[n, 2, 1]]; primBinNivQ[n_] := binNivQ[n] && ! (EvenQ[n] && binNivQ[n/2]);
seq[kmax_] := Module[{tri = primBinNivQ /@ Range[3], s = {}, k = 4}, While[k < kmax, If[And @@ tri, AppendTo[s, k - 3]]; tri = Join[Rest[tri], {primBinNivQ[k]}]; k++]; s]; seq[10^7]
  • PARI
    isbinniv(n) = !(n % hammingweight(n));
isprim(n) = isbinniv(n) && !(!(n%2) && isbinniv(n/2));
lista(kmax) = {my(tri = vector(3, i, isprim(i)), k = 4); while(k < kmax, if(vecsum(tri) == 3, print1(k-3, ", ")); tri = concat(vecextract(tri, "^1"), isprim(k)); k++); }

A363792 Starts of runs of 4 consecutive integers that are primitive binary Niven numbers (A363787).

Original entry on oeis.org

8255214, 14673870, 29092590, 33185646, 41743854, 47697390, 48069486, 56348622, 56999790, 58116078, 59604462, 60534702, 60813774, 61837038, 62581230, 64069614, 64999854, 65371950, 66581262, 66674286, 75232494, 83418606, 86767470, 88069806, 92255886, 95418702, 96441966, 99511758, 99604782
Offset: 1

Views

Author

Amiram Eldar, Jun 22 2023

Keywords

Comments

There are no runs of 5 or more consecutive integers that are primitive binary Niven numbers (see the second comment in A330933).

Examples

			8255214 is a term since 8255214, 8255215, 8255216 and 8255217 are all primitive binary Niven numbers.

Links

Crossrefs

Subsequence of A049445, A330931, A330932, A330933, A363787, A363790 and A363791.

Programs

  • Mathematica
    binNivQ[n_] := Divisible[n, DigitCount[n, 2, 1]]; primBinNivQ[n_] := binNivQ[n] && ! (EvenQ[n] && binNivQ[n/2]);
seq[kmax_] := Module[{quad = primBinNivQ /@ Range[4], s = {}, k = 5}, While[k < kmax, If[And @@ quad, AppendTo[s, k - 4]]; quad = Join[Rest[quad], {primBinNivQ[k]}]; k++]; s]; seq[3*10^7]
  • PARI
    isbinniv(n) = !(n % hammingweight(n));
isprim(n) = isbinniv(n) && !(!(n%2) && isbinniv(n/2));
lista(kmax) = {my(quad = vector(4, i, isprim(i)), k = 5); while(k < kmax, if(vecsum(quad) == 4, print1(k-4, ", ")); quad = concat(vecextract(quad, "^1"), isprim(k)); k++); }

A376619 a(n) is the least odd number k such that A376615(k) = n, or -1 if no such number exists.

Original entry on oeis.org

3, 21, 345, 10625, 74375, 860625, 84189105, 1599592995, 23993894925
Offset: 1

Views

Author

Amiram Eldar, Sep 30 2024

Keywords

Comments

Without the restriction to odd numbers the corresponding sequence is 3*2^(n-1) = A007283(n-1).
All the terms above 3 are odd binary Niven numbers (A144302).
a(10) > 10^13, if it exists.

Examples

			  n | The n iterations
  --+------------------------------------------------------
  1 | 3 -> 3/2
  2 | 21 -> 7 -> 7/3
  3 | 345 -> 69 -> 23 -> 23/4
  4 | 10625 -> 2125 -> 425 -> 85 -> 85/4
  5 | 74375 -> 10625 -> 2125 -> 425 -> 85 -> 85/4
  6 | 860625 -> 95625 -> 10625 -> 2125 -> 425 -> 85 -> 85/4

Crossrefs

Cf. A007283, A049445, A144302, A376615, A376616, A376617.

Programs

  • Mathematica
    s[n_] := s[n] = Module[{bw = DigitCount[n, 2, 1]}, If[bw == 1, 0, If[!Divisible[n, bw], 1, 1 + s[n/bw]]]]; seq[len_] := Module[{v = Table[0, {len}], c = 0, k = 3, i}, While[c < len, i = s[k]; If[v[[i]] == 0, c++; v[[i]] = k]; k += 2]; v]; seq[5]
  • PARI
    s(n) = {my(w = hammingweight(n)); if(w == 1, 0, if(n % w, 1, 1 + s(n/w)));}
lista(len) = {my(v = vector(len), c = 0, k = 3, i); while(c < len, i = s(k); if(v[i] == 0, c++; v[i] = k); k += 2); v;}

A385484 Records in A385482.

Original entry on oeis.org

1, 2, 3, 5, 10, 12, 42, 84, 88, 90, 99, 130, 165, 184, 187, 209, 221, 252, 299, 434, 450, 459, 486, 525, 555, 611, 675, 702, 726, 858, 899, 975, 984, 1034, 1036, 1104, 1107, 1197, 1275, 1357
Offset: 1

Views

Author

Amiram Eldar, Jun 30 2025

Keywords

Crossrefs

Cf. A049445, A144363 (decimal analog), A385482, A385483 (indices of records), A385487.

Programs

  • Mathematica
    f[n_] := Module[{m = n, k = 1}, While[!Divisible[m, DigitSum[m, 2]], m += n; k++]; k];
seq[lim_] := Module[{s = {}, fm = -1, fi}, Do[fi = f[i]; If[fi > fm, fm = fi; AppendTo[s, fi]], {i, 1, lim}]; s]; seq[10^4]
  • PARI
    f(n) = {my(m = n, k = 1); while(m % hammingweight(m), m += n; k++); k;}
list(lim) = my(fm = -1, fi); for(i = 1, lim, fi = f(i); if(fi > fm, fm = fi; print1(fi, ", ")));

Formula

a(n) = A385482(A385483(n)).

A117890 Numbers k such that number of non-leading 0's in binary representation of k divides k.

Original entry on oeis.org

2, 4, 5, 6, 10, 11, 12, 13, 14, 16, 18, 22, 23, 24, 26, 27, 28, 29, 30, 36, 40, 42, 46, 47, 48, 54, 55, 58, 59, 60, 61, 62, 65, 75, 76, 78, 80, 84, 88, 90, 94, 95, 99, 100, 102, 104, 105, 108, 110, 111, 112, 114, 118, 119, 120, 122, 123, 124, 125, 126, 132, 140, 144, 145
Offset: 1

Views

Author

Leroy Quet, Mar 30 2006

Keywords

Comments

Contains primes of A095078(n) as a subset. Intersection of a(n) with A049445(n) is A117891(n). - R. J. Mathar, Apr 03 2006

Examples

			24 is 11000 in binary. This binary representation has three 0's and 3 divides 24. So 24 is in the sequence.

Links

Crossrefs

Cf. A049445, A117891.
Cf. A023416.

Programs

  • C
    #include 
int main(int argc, char *argv[]) { for(int n=1; n< 500; n++) { int digs=0; int nshifted=n; while(nshifted) { digs += 1- nshifted & 1; nshifted >>= 1; } if ( digs) if( n % digs == 0 ) printf("%d, ",n); } } // R. J. Mathar, Apr 03 2006
  • Haskell
    a117890 n = a117890_list !! (n-1)
a117890_list = [x | x <- [1..], let z = a023416 x, z > 0, mod x z == 0]
-- Reinhard Zumkeller, Mar 31 2015

Formula

a(n) <= A117891(n). - R. J. Mathar, Apr 03 2006
a(n) mod A023416(a(n)) = 0. - Reinhard Zumkeller, Nov 22 2007

Extensions

More terms from R. J. Mathar, Apr 03 2006
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