A246702 -id:A246702 - cp's OEIS frontend

A246717 Numbers of the form 2n - 1 such that A246702(n) = 2.

7, 17, 23, 35, 41, 47, 49, 71, 77, 79, 95, 97, 103, 115, 137, 143, 167, 175, 191, 193, 199, 209, 235, 239, 245, 263, 271, 289, 295, 299, 311, 313, 319, 335, 343, 359, 367, 371, 383, 395, 401, 407, 409, 413, 415, 437, 449, 463, 475, 479, 487, 503, 515, 517, 521, 529, 535, 539, 551, 569, 575, 581, 583, 599, 607, 611, 647, 649, 667, 695, 707

Offset: 1

Juri-Stepan Gerasimov, Nov 15 2014

nonn

From Antti Karttunen, Nov 15 2014: (Start)
Equally: Odd numbers n for which A246702((n+1)/2) = 2.
Primes in this sequence: 7, 17, 23, 41, 47, 71, 79, ... seem to be A115591.
A249819 gives the composite terms.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..105

Cf. A002326, A115591, A249819, A167791, A246702.

isA246717(n) = { if(!(n%2), return(0), my(u, s=0); u = n^2; for(k=1, u-1, if(!(((2^k)-1)%u), s++;if(s > 2, return(0)))); return(2==s)); }
n = 0; i = 0; while(i < 105, n++; if(isA246717(n), i++; write("b246717.txt", i, " ", n))); \\ From Antti Karttunen, Nov 15 2014
(Scheme, with Antti Karttunen's IntSeq-library)
(define A246717 (MATCHING-POS 1 1 (lambda (n) (and (odd? n) (= 2 (A246702 (/ (+ 1 n) 2)))))))

Terms corrected by Antti Karttunen, Nov 15 2014

A246703 Where records occur in A246702.

Original entry on oeis.org

1, 2, 4, 8, 11, 28, 53, 78, 83, 98, 103, 137, 233, 305, 308, 445, 452, 547, 1640, 2733, 3826, 7105, 11477, 21314, 49732, 149195, 745973, 1026968, 2536307, 3765548, 4326641, 5756285, 6415364, 7188773, 7907309, 8205698, 9593808, 13431331, 17268854, 24943900, 28781423, 40293992, 67156653, 74831699, 86344268, 120881975

Offset: 1

Juri-Stepan Gerasimov, Nov 15 2014

RECORDS transform of A246702.

Cf, A246702.

Corrected by R. J. Mathar, Nov 15 2014

Terms a(13) onward from Max Alekseyev, Oct 11 2023

A246755 Numbers of the form 2k - 1 such that A246702(k) = 3.

Original entry on oeis.org

15, 33, 43, 45, 69, 75, 87, 99, 109, 135, 141, 157, 159, 177, 207, 213, 225, 229, 249, 261, 277, 283, 297, 303, 307, 321, 363, 375, 393, 405, 423, 447, 477, 499, 501, 519, 531, 537, 573, 591, 621, 639, 643, 675, 681, 691, 717, 733, 739, 747, 783, 789, 807, 811

Offset: 1

Juri-Stepan Gerasimov, Nov 15 2014

nonn

Composites in this sequence: 15, 33, 45, 69, 75, 87, 99, 135, 141, 159, 177, 207, 213, 225, 249, 261, 297, 303, 321, 363, 375, 393, 405, 423, 447, 477, ...

			A246702(8) = 3 for the first time, hence a(1) = 2*8 - 1 = 15.

Cf. Numbers of the form 2k - 1 such that A246702(k) = m: number 1 (m = 0), A167791 (m = 1), A246717 (m = 2), this sequence (m = 3), A001133 (primes in this sequence).

is(k) = (m=Mod(k%2, k*k)) && sum(i=1, k*k-1, m*=2; m==1) == 3; \\ Jinyuan Wang, May 15 2020

More terms from and terms corrected by Jinyuan Wang, May 15 2020

A249819 Composite natural numbers n for which there are exactly two distinct 0 < k < n^2 such that 2^k - 1 is divisible by n^2.

Original entry on oeis.org

35, 49, 77, 95, 115, 143, 175, 209, 235, 245, 289, 295, 299, 319, 335, 343, 371, 395, 407, 413, 415, 437, 475, 515, 517, 529, 535, 539, 551, 575, 581, 583, 611, 649, 667, 695, 707, 749, 767, 815, 835, 847, 851, 869, 875, 893, 895, 913, 917, 923, 995, 1007

Offset: 1

Antti Karttunen, Nov 15 2014

nonn

Equally: odd composite numbers n for which A246702((n+1)/2) = 2.

			35 = 5*7 is an odd composite. Only cases where 2^k - 1 (with k in range 1 .. 35^2 - 1 = 1 .. 1224) is a multiple of 35 are k = 420 and k = 840, thus 35 is included in this sequence.

Composite terms in A246717.
Seems also to be a subsequence of A038509.
Cf. A246702.

isA249819 := proc(n)
    if isprime(n) or n=1 then
        false;
    else
        ct := 0 ;
        for k from 1 to n^2-1 do
            if modp(2 &^ k-1,n^2) = 0 then
                ct := ct+1 ;
            end if;
            if ct > 2 then
                return false;
            end if;
        end do:
        return is(ct=2) ;
    end if;
end proc:
for n from 1 to 1100 do
    if isA249819(n) then
        printf("%d,\n",n) ;
    end if;
end do: # R. J. Mathar, Nov 16 2014

A246719 Smallest natural number m for which there are exactly n distinct values k such that 0 < k < m^2 and 2^k - 1 is divisible by m^2.

Original entry on oeis.org

1, 3, 7, 15, 113, 65, 31, 91, 73, 39, 21, 331, 267, 55, 217, 435, 203, 697, 127, 703, 565, 429, 451, 231, 595, 253, 105, 327, 171, 1045, 1335, 255, 385, 497, 341, 1295, 219, 455, 155, 1417, 969, 165, 2143, 861, 357, 453, 555, 2821, 195, 1477, 301, 205, 2091

Offset: 0

Juri-Stepan Gerasimov, Nov 15 2014

nonn

Smallest odd number of the form 2q - 1 such that A246702(q) = n.

Additional terms include: a(426) = 1705, a(451) = 903, a(516) = 2067, a(536) = 2145, a(563) = 2255, a(566) = 2265, a(593) = 2373, a(761) = 3045, a(770) = 3081, a(786) = 2359, a(1333) = 2667, and a(3282) = 1093. - Kevin P. Thompson, Nov 26 2021

			The first occurrence of 3 in the sequence A246702 occurs at n = 8. Therefore, a(3) = 2n - 1 = 2*8 - 1 = 15.

Kevin P. Thompson, Table of n, a(n) for n = 0..350, with missing terms.

Cf. Numbers of the form 2n - 1 such that A246702(n) = i: number 1 (i = 0), A167791 (i = 1), A246717 (i = 2), A246755 (i = 3).

NumK[m_]:=NumK[m]=(m2=m^2;nk=0;Do[If[Mod[2^i,m2]==1,nk++],{i,m2-1}];nk)
nterms=10;Table[m=0;While[NumK[++m]!=n];m,{n,0,nterms-1}] (* Paolo Xausa, Nov 30 2021 *)

isok(m, n) = {my(v = vector(m^2-1, k, Mod(2, m^2)^k == 1)); vecsum(v) == n;}
a(n) = {my(m=1); while (!isok(m, n), m++); m;} \\ Michel Marcus, Nov 27 2021

Name corrected by Antti Karttunen, Nov 18 2014

Multiple corrections and new terms a(17)-a(52) from Kevin P. Thompson, Nov 26 2021

cp's OEIS Frontend

A246717 Numbers of the form 2n - 1 such that A246702(n) = 2.

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A246703 Where records occur in A246702.

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A246755 Numbers of the form 2k - 1 such that A246702(k) = 3.

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A249819 Composite natural numbers n for which there are exactly two distinct 0 < k < n^2 such that 2^k - 1 is divisible by n^2.

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A246719 Smallest natural number m for which there are exactly n distinct values k such that 0 < k < m^2 and 2^k - 1 is divisible by m^2.

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