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.

Showing 1-5 of 5 results.

A377455 Numbers k such that k and k+1 are both terms in A377385.

Original entry on oeis.org

1, 1224, 126191, 428519, 649727, 1015416, 1988064, 3425856, 4542740, 4574240, 4743900, 4813668, 5131008, 6899840, 7001315, 7172424, 7356096, 8020583, 10206000, 11146421, 11566800, 11597999, 11693807, 12556700, 13742624, 13745759, 13831487, 14365120, 16939799, 20561400
Offset: 1

Views

Author

Amiram Eldar, Oct 29 2024

Keywords

Examples

			1224 is a term since both 1224 and 1225 are in A377385: 1224/A034968(1224) = 204 and 204/A034968(204) = 34 are integers, and 1225/A034968(1225) = 175 and 175/A034968(175) = 35 are integers.

Links

Crossrefs

Cf. A034968.
Subsequence of A118363, A328205 and A377385.
Subsequences: A377456, A377457.
Analogous sequences: A376793 (binary), A377271 (Zeckendorf).

Programs

  • Mathematica
    fdigsum[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, s += r; m++]; s]; q[k_] := q[k] = Module[{f = fdigsum[k]}, Divisible[k, f] && Divisible[k/f, fdigsum[k/f]]]; Select[Range[2*10^6], q[#] && q[#+1] &]
  • PARI
    fdigsum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); s;}
is1(k) = {my(f = fdigsum(k)); !(k % f) && !((k/f) % fdigsum(k/f));}
lista(kmax) = {my(q1 = is1(1), q2); for(k = 2, kmax, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A377456 Starts of runs of 3 consecutive integers that are all terms of A377385.

Original entry on oeis.org

39998374960, 326660221888, 520935101440, 723006782783, 923072388208, 977932351240, 1134397887874, 1351753892944, 1864828904536, 2171452161023
Offset: 1

Views

Author

Amiram Eldar, Oct 29 2024

Keywords

Examples

			39998374960 is a term since 39998374960, 39998374961 and 39998374962 are all in A377385: 39998374960/A034968(39998374960) = 999959374, and 999959374/A034968(999959374) = 32256754 are integers, 39998374961/A034968(39998374961) = 975570121, and 975570121/A034968(975570121) = 33640349 are integers, and 39998374962/A034968(39998374962) = 1025599358, and 1025599358/A034968(1025599358) = 30164687 are integers.

Crossrefs

Cf. A034968.
Subsequence of A118363, A328205, A377385 and A377455.
Analogous sequences: A376794 (binary), A377273 (Zeckendorf).

Programs

  • PARI
    fdigsum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); s;}
is1(k) = {my(f = fdigsum(k)); !(k % f) && !((k/f) % fdigsum(k/f));}
lista(kmax) = {my(q1 = is1(1), q2 = is1(2), q3); for(k = 3, kmax, q3 = is1(k); if(q1 && q2 && q3, print1(k-2, ", ")); q1 = q2; q2 = q3);}

A377384 a(n) is the number of iterations that n requires to reach a noninteger or a factorial number under the map x -> x / f(x), where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k; a(n) = 0 if n is a factorial number.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 0, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Oct 27 2024

Keywords

Comments

The factorial numbers are fixed points of the map, since f(k!) = 1 for all k >= 0. Therefore they are arbitrarily assigned the value a(k!) = 0.
Each number n starts a chain of a(n) integers: n, n/f(n), (n/f(n))/f(n/f(n)), ..., of them the first a(n)-1 integers are factorial-base Niven numbers (A118363).

Examples

			a(8) = 2 since 8/f(8) = 4 and 4/f(4) = 2 is a factorial number that is reached after 2 iterations.
a(27) = 3 since 27/f(27) = 9, 9/f(9) = 3 and 3/f(3) = 3/2 is a noninteger that is reached after 3 iterations.

Links

Crossrefs

Cf. A000005, A000142, A034968, A118363, A286607, A377385, A377386.
Analogous sequences: A376615 (binary), A377208 (Zeckendorf).

Programs

  • Mathematica
    fdigsum[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, s += r; m++]; s]; a[n_] := a[n] = Module[{s = fdigsum[n]}, If[s == 1, 0, If[!Divisible[n, s], 1, 1 + a[n/s]]]]; Array[a, 100]
  • PARI
    fdigsum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); s;}
a(n) = {my(f = fdigsum(n)); if(f == 1, 0, if(n % f, 1, 1 + a(n/f)));}
  • Python
    def f(n, p=2): return n if n
    Michael S. Branicky, Oct 27 2024

Formula

a(n) = 0 if and only if n is in A000142 (by definition).
a(n) = 1 if and only if n is in A286607.
a(n) >= 2 if and only if n is in A118363 \ A000142 (i.e., n is a factorial-base Niven number that is not a factorial number).
a(n) >= 3 if and only if n is in A377385 \ A000142.
a(n) >= 4 if and only if n is in A377386 \ A000142.
a(n) < A000005(n).

A377386 Factorial-base Niven numbers (A118363) k such that m = k/f(k) and m/f(m) are also factorial-base Niven numbers, where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 36, 40, 48, 54, 72, 80, 96, 108, 120, 135, 144, 180, 192, 240, 280, 288, 360, 384, 432, 480, 576, 594, 600, 720, 840, 864, 1200, 1215, 1225, 1296, 1344, 1440, 1680, 1728, 1800, 2160, 2240, 2352, 2400, 2520, 2592, 2704, 2730, 2880, 3000
Offset: 1

Views

Author

Amiram Eldar, Oct 27 2024

Keywords

Examples

			16 is a term since 16/f(16) = 4 is an integer, 4/f(4) = 2 is an integer, and 2/f(2) = 2 is an integer.

Links

Crossrefs

Subsequence of A118363 and A377385.
A000142 is a subsequence.
Cf. A034968, A377384.
Analogous sequences: A376617 (binary), A377210 (Zeckendorf).

Programs

  • Mathematica
    fdigsum[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, s += r; m++]; s]; q[k_] := Module[{f = fdigsum[k], f2, m, n}, IntegerQ[m = k/f] && Divisible[m, f2 = fdigsum[m]] && Divisible[n = m/f2, fdigsum[n]]]; Select[Range[3000], q]
  • PARI
    fdigsum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); s;}
is(k) = {my(f = fdigsum(k), f2, m); if(k % f, return(0)); m = k/f; f2 = fdigsum(m); !(m % f2) && !((m/f2) % fdigsum(m/f2)); }

A377457 Numbers k such that k and k+1 are both terms in A377386.

Original entry on oeis.org

1, 12563307224, 15897851550, 30412355999, 37706988600, 52576459775, 67673545631, 118533901904, 244316235000, 297265003100, 332110595000, 340800265728, 349358409503, 375624917760, 378624889440, 416375389115, 450026519903, 561162864248, 596004199840, 728643460544
Offset: 1

Views

Author

Amiram Eldar, Oct 29 2024

Keywords

Examples

			12563307224 is a term since both 12563307224 and 12563307225 are in A377386: 12563307224/A034968(12563307224) = 369509036, 369509036/A034968(369509036) = 9723922 and 9723922/A034968(9723922) = 373997 are integers, and 12563307225/A034968(12563307225) = 358951635, 358951635/A034968(358951635) = 7976703 and 7976703/A034968(7976703) = 257313 are integers.

Links

Crossrefs

Cf. A034968.
Subsequence of A118363, A328205, A377385, A377386 and A377455.
Analogous sequences: A376795 (binary), A377272 (Zeckendorf).

Programs

  • PARI
    fdigsum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); s;}
is1(k) = {my(f = fdigsum(k), f2, m); if(k % f, return(0)); m = k/f; f2 = fdigsum(m); !(m % f2) && !((m/f2) % fdigsum(m/f2));}
lista(kmax) = {my(q1 = is1(1), q2); for(k = 2, kmax, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}
Showing 1-5 of 5 results.

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