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 1-6 of 6 results.

A348773 First differences of A307632.

Original entry on oeis.org

2, 42, 6, 1321, 12, 2352, 18, 2924, 24, 77922, 32, 4822, 42, 2310, 48, 81212, 60, 19730, 68, 331637, 74, 340640, 84, 11158, 98, 13838, 104, 13690, 110, 14476, 128, 709992, 138, 17990, 150, 19518, 158, 20830, 168, 2277394, 180, 62350, 192, 82484, 198, 76962, 212, 84852, 228, 15407670, 234, 87388, 242, 90636
Offset: 1

Views

Author

N. J. A. Sloane, Nov 06 2021

Keywords

Links

Crossrefs

Cf. A307720, A307730, A307632.
Bisections: A348774, A348775.

A307720 Lexicographically earliest sequence of positive integers in which, for all positive k, there are exactly k pairs of consecutive terms whose product is k.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 2, 4, 2, 4, 2, 4, 2, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 5, 1, 5, 1, 5, 1, 7, 1, 7, 1, 7, 1, 7, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 7, 3, 7, 3, 7, 3, 7, 3, 7, 3, 7, 3, 7, 3, 7, 3, 7, 3, 7, 3, 8, 2, 8
Offset: 1

Views

Author

Eric Angelini and Jean-Marc Falcoz, Apr 24 2019

Keywords

Comments

All natural integers will appear sooner or later in the sequence (from the definition) - but mostly "later"! Indeed, the sequence increases very slowly: after 100000 terms the smallest term not yet present is 32.
Here is, in the same range, a sample of the count {term, occurrences} so far:
{1,192},{2,396},{3,618},{4,796},{5,1160},{6,1296},{7,2294},{8,2080},{9,2489},{10,2826},{11,3487},{12,1596},{13,2295},{14,1960},{15,2370},{16,2640},{17,4097},{18,2214},{19,4598},{20,2770},{21,3759},{22,4477},{23,5612},{24,4884},{25,5825},{26,6006},{27,6359},{28,4676},{29,5481},{30,3060},{31,1411},{32,0},{33,182},{34,0},{35,315},{36,0},{37,1221},{38,0},{39,214},{40,0},{41,1353},{42,0},{43,1183},{44,0},{45,0},{46,0},{47,1058},{48,0},{49,172},{50,0},{51,0},{52,0},{53,580},...
After 100000 terms, the first products that are not yet present are (the primes): 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, ... and (the composites) 118, 122, 134, ...
Here is again a sample so far (100000 terms computed) of {product, number of occurrences of the product}:
{1,1},{2,2},{3,3},{4,4},{5,5},{6,6},{7,7},{8,8},{9,9},{10,10},{11,11},{12,12},{13,13},{14,14},{15,15},{16,16},{17,17},{18,18},{19,19},{20,20},{21,21},{22,22},{23,23},{24,24},{25,25},{26,26},{27,27},{28,28},{29,29},{30,30},{31,31},{32,32},{33,33},{34,34},{35,35},{36,36},{37,37},{38,38},{39,39},{40,40},{41,41},{42,42},{43,43},{44,44},{45,45},{46,46},{47,47},{48,48},{49,49},{50,50},{51,51},{52,52},{53,53},{54,54},{55,55},{56,56},{57,57},{58,58},{59,0},{60,60},{61,0},{62,62},{63,63},{64,64},{65,65},{66,66},{67,0},{68,68},{69,69},{70,70},{71,0},{72,72},{73,0},{74,74},{75,75},{76,76},{77,77},{78,78},{79,0},{80,80},{81,81},{82,82},{83,0},{84,84},{85,85},{86,86},{87,87},{88,88},{89,0},{90,90},{91,91},{92,92},{93,93},{94,94},{95,95},{96,96},{97,0},{98,98},{99,99},{100,100},{101,0},{102,102},{103,0},{104,104},{105,105},{106,106},{107,0},{108,108},{109,0},{110,110},{111,111},{112,112},{113,0},{114,114},{115,115},{116,116},{117,117},{118,0},{119,119},{120,120},{121,121},{122,0},{123,123},{124,124},{125,125},{126,126},{127,0},{128,128},{129,129},{130,130},{131,0},{132,132},{133,133},{134,0},{135,135},{136,136},{137,0},{138,138},{139,0},{140,140},{141,141},{142,0},...
Comment from N. J. A. Sloane, Oct 19 2021: (Start)
Theorem. This sequence can also be defined by a greedy algorithm. That is, let b(1)=1, and for n >= 1, let b(n+1) be the smallest positive integer k such that m = k*b(n) has appeared at most n-1 times in the list [b(i)*b(i+1): i=1..n-1]. Then b(n) = a(n) for all n >= 1.
(Note that for n=1 the list is empty, and so we take k = b(1) = 1.)
Remark: The theorem is not obvious and requires a proof, given in a link below. "Lexicographically earliest" sequences often require some backtracking, but the point of the theorem is that no backtracking is needed here.
The proof also shows that there are infinitely many 1's in the sequence, and that each k appears k times in the sequence of products a(i)*a(i+1). (End)

Examples

			The sequence starts with 1,1,2,1,3,1,3,2,2,2,2,2,3,...
The product a(n)*a(n+1) = 1 is true exactly once [this is the product a(1) * a(2) = 1 * 1 = 1];
The product a(n)*a(n+1) = 2 is true exactly twice [these are the products a(2) * a(3) = 1 * 2 = 2 and a(3) * a(4) = 2 * 1 = 2];
The product a(n)*a(n+1) = 3 is true exactly three times [these are the products a(4) * a(5) = 1 * 3 = 3 ; a(5) * a(6) = 3 * 1 = 3, and a(6) * a(7) = 1 * 3 = 3];
...
The product a(n)*a(n+1) = 4 is true exactly four times [these are the products a(8) * a(9) = 2 * 2 = 4 ; a(9) * a(10) = 2 * 2 = 4 ; a(10) * a(11) = 2 * 2 = 4 ; a(11) * a(12) = 2 * 2 = 4] ; and so on.

Links

Crossrefs

Cf. A307707 (same idea, but with the sum of contiguous terms instead of the product), A307730 (the products), A307630 (when n appears), A307631 (indices of records), A307632 (indices of primes), A348241 and A348242 (bisections), A307633 and A307634 (RUNS transforms of bisections), A348446 (bisection differences), A348458 (partial sums).
See also A307747.

Programs

  • Mathematica
    nmax = 1000; time = {0}; v = 1;
A307720 = Reap[For[n = 1, n <= nmax, n++, Sow[v]; For[o = 1, True, o++, While[Length[time] < o*v, time = Join[time, Table[0, {Length[time]}]]]; If[time[[o*v]]+1 <= o*v, time[[o*v]]++; v = o; Break[]]]]][[2, 1]] (* Jean-François Alcover, Oct 23 2021, after Rémy Sigrist's PARI program *)
  • PARI
    \\ See Links section.
  • Python
    from itertools import islice
from collections import Counter
def A307720(): # generator of terms. Greedy algorithm
    yield 1
    c, b = Counter(), 1
    while True:
        k, kb = 1, b
        while c[kb] >= kb:
            k += 1
            kb += b
        c[kb] += 1
        b = k
        yield k
A307720_list = list(islice(A307720(),100)) # Chai Wah Wu, Oct 21 2021

Extensions

Definition revised slightly by Allan C. Wechsler, Apr 24 2019
Example clarified by Rémy Sigrist, Oct 24 2021

A307630 Index at which n first appears in A307720.

Original entry on oeis.org

1, 3, 5, 27, 47, 99, 53, 137, 177, 1024, 1374, 2474, 1386, 3326, 3662, 5274, 3738, 6290, 3756, 8954, 9374, 12878, 6680, 9682, 9850, 10324, 11010, 14578, 6704, 78506, 84626, 106968, 88474, 127682, 86802, 143544, 84658, 160664, 97850, 274079, 89480, 326195
Offset: 1

Views

Author

N. J. A. Sloane, Apr 25 2019

Keywords

Links

Crossrefs

Cf. A307720, A307631, A307632, A307633, A307634, A307730, A307747.

Extensions

More terms from Hans Havermann, Apr 25 2019

A348774 A348773(2*n+1).

Original entry on oeis.org

2, 6, 12, 18, 24, 32, 42, 48, 60, 68, 74, 84, 98, 104, 110, 128, 138, 150, 158, 168, 180, 192, 198, 212, 228, 234, 242, 258, 270, 278, 284, 308, 314, 332, 348, 354, 368, 380, 390, 402, 420, 432, 440, 450, 462, 468, 488, 500, 510, 524, 548, 564, 572, 588, 600, 608, 618, 632, 644, 654
Offset: 0

Views

Author

N. J. A. Sloane, Nov 07 2021

Keywords

Comments

The first differences are 4, 6, 6, 6, 8, ... and apart from the initial term4, appears to coincide with A155067, the differences between successive odd-indexed primes. If confirmed, this will be one of the few formulas known for A307720.
The other bisection of A348773, A348775, seems much more mysterious.

Links

Crossrefs

Cf. A155067, A307720, A307730, A307632, A348773, A348775.

A348775 A348773(2*n).

Original entry on oeis.org

42, 1321, 2352, 2924, 77922, 4822, 2310, 81212, 19730, 331637, 340640, 11158, 13838, 13690, 14476, 709992, 17990, 19518, 20830, 2277394, 62350, 82484, 76962, 84852, 15407670, 87388, 90636, 408240, 14526794, 7023466, 272792, 117864, 293946, 40034157, 386674, 168172, 136472, 40847194, 729008, 768646
Offset: 1

Views

Author

N. J. A. Sloane, Nov 07 2021

Keywords

Links

Crossrefs

Cf. A307720, A307730, A307632, A348773, A348774.

A348247 Where prime(n) first appears in A307730.

Original entry on oeis.org

2, 4, 47, 52, 1374, 1385, 3738, 3755, 6680, 6703, 84626, 84657, 89480, 89521, 91832, 91879, 173092, 173151, 192882, 192949, 524587, 524660, 865301, 865384, 876543, 876640, 890479, 890582, 904273, 904382, 918859, 918986, 1628979, 1629116, 1647107, 1647256, 1666775
Offset: 1

Views

Author

N. J. A. Sloane, Oct 17 2021

Keywords

Links

Crossrefs

Cf. A307720, A307730, A307632, A348246.

Programs

  • Python
    from collections import Counter
from sympy import prime
def A348247(n):
    c, b, p, i = Counter(), 1, prime(n), 1
    while True:
        k, kb = 1, b
        while c[kb] >= kb:
            k += 1
            kb += b
        if kb == p:
            return i
        c[kb] += 1
        b = k
        i += 1 # Chai Wah Wu, Oct 21 2021

Extensions

More terms from Chai Wah Wu, Oct 21 2021
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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