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.

A326130 a(n) = gcd(A000120(n), A294898(n)) = gcd(A000120(n), sigma(n)-A005187(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 1, 2, 1, 3, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 4, 5, 1, 2, 2, 1, 1, 3, 1, 2, 2, 1, 3, 2, 1, 4, 4, 1, 2, 1, 1, 2, 3, 4, 4, 1, 1, 2, 2, 1, 4, 5, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 1, 3, 1, 2, 3, 2, 4, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 4, 2, 5, 4, 1, 1, 2, 2, 3, 1, 2, 1, 4, 4, 1, 1, 2
Offset: 1

Views

Author

Antti Karttunen, Jun 09 2019

Keywords

Links

Crossrefs

Cf. A000120, A000203, A005187, A033879, A326131.
Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326140, A326144.

Programs

Formula

a(n) = gcd(A000120(n), A294898(n)) = gcd(A000120(n), A000203(n)-A005187(n)).

A326064 Odd composite numbers n, not squares of primes, such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors.

Original entry on oeis.org

117, 775, 10309, 56347, 88723, 2896363, 9597529, 12326221, 12654079, 25774633, 29817121, 63455131, 105100903, 203822581, 261019543, 296765173, 422857021, 573332713, 782481673, 900952687, 1129152721, 3350861677, 3703086229, 7395290407, 9347001661, 9350506057
Offset: 1

Views

Author

Antti Karttunen, Jun 06 2019

Keywords

Comments

Nineteen initial terms factored:
n a(n) factorization A060681(a(n))/A318505(a(n))
1: 117 = 3^2 * 13, (3)
2: 775 = 5^2 * 31, (10)
3: 10309 = 13^2 * 61, (39)
4: 56347 = 29^2 * 67, (58)
5: 88723 = 17^2 * 307, (136)
6: 2896363 = 41^2 * 1723, (820)
7: 9597529 = 73^2 * 1801, (1314)
8: 12326221 = 59^2 * 3541, (1711)
9: 12654079 = 113^2 * 991, (904)
10: 25774633 = 71^2 * 5113, (2485)
11: 29817121 = 97^2 * 3169, (2328)
12: 63455131 = 89^2 * 8011, (3916)
13: 105100903 = 101^2 * 10303, (5050)
14: 203822581 = 157^2 * 8269, (6123)
15: 261019543 = 349^2 * 2143, (2094)
16: 296765173 = 131^2 * 17293, (8515)
17: 422857021 = 233^2 * 7789, (6757)
18: 573332713 = 331^2 * 5233, (4965)
19: 782481673 = 167^2 * 28057, (13861).
Note how the quotient (in the rightmost column) seems always to be a multiple of non-unitary prime factor and less than the unitary prime factor.
For p, q prime, if p^2+p+1 = kq and k+1|p-1, then p^2*q is in this sequence. - Charlie Neder, Jun 09 2019

Links

Crossrefs

Subsequence of A326063.
Cf. A032742, A060681, A246282, A318505.
Cf. also A228058, A325981, A326131, A326141.

Programs

  • Mathematica
    Select[Range[15, 10^6 + 1, 2], And[! PrimePowerQ@ #1, Mod[#1 - #2, #2 - #3] == 0] & @@ {#, DivisorSigma[1, #] - #, Divisors[#][[-2]]} &] (* Michael De Vlieger, Jun 22 2019 *)
  • PARI
    A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A060681(n) = (n-A032742(n));
A318505(n) = if(1==n,0,(sigma(n)-A032742(n))-n);
isA326064(n) = if((n%2)&&(2!=isprimepower(n)), my(s=A032742(n), t=sigma(n)-s); (gcd(t-n, n-A032742(n)) == t-n), 0);

Extensions

More terms from Amiram Eldar, Dec 24 2020

A326133 Numbers n for which sigma(n) > A005187(n).

Original entry on oeis.org

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 110, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264, 270, 272, 276, 280, 282, 288, 294, 300
Offset: 1

Views

Author

Antti Karttunen, Jun 11 2019

Keywords

Comments

Differs from A023196 for the first time at the 28th term, which here is 110, which is not included in A023196.
Note that as there is at least one odd number (815634435) in A326138, it means that A005231 does not contain all odd terms of this sequence.

Links

Crossrefs

Positions of negative terms in A294898.
Cf. A000203, A005187, A023196.
Cf. A000396, A005231, A083207, A111592, A326131, A326138 (subsequences).

Programs

  • Mathematica
    Select[Range[300], DivisorSigma[1, #] > 2*# - DigitCount[2*#, 2, 1] &] (* Amiram Eldar, Aug 06 2023 *)
  • PARI
    A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
isA326133(n) = (sigma(n)>A005187(n));

A326141 Odd numbers n for which A318879(n) is not zero and A318879(n) divides A318878(n); odd numbers such that A326140(n) = A318879(n).

Original entry on oeis.org

105, 195, 4785, 22515, 56865, 228285, 237315, 484245, 671853, 1838145, 1946955, 3446895, 4522695, 12955245, 37730865, 52475055, 53568885, 87612975
Offset: 1

Views

Author

Antti Karttunen, Jun 09 2019

Keywords

Comments

Not a subsequence of A036798, even though many terms are members.
Questions: Are all terms multiples of three? Multiples of 3^(2k+1) but not of 3^(2k)? Are any of the terms included in A228058, A326137?

Links

Crossrefs

Cf. A033879, A083254, A036798, A228058, A318878, A318879, A326137, A326140.
Cf. also A325981, A326131.

Programs

  • PARI
    isA326141(n) = if(!(n%2),0, my(t=0, u=0); fordiv(n,d, d -= 2*eulerphi(d); if(d<0, t -= d, u += d)); (gcd(t,u)==u));

A326132 Numbers n for which A294898(n) is not zero and A294898(n) divides A000120(n); numbers for which A326130(n) = abs(A294898(n)).

Original entry on oeis.org

5, 6, 7, 14, 15, 28, 44, 52, 110, 152, 184, 496, 592, 884, 1012, 1155, 2144, 2272, 8128, 8384, 12008, 18632, 18904, 33664, 63248, 70564, 85936, 100804, 116624, 318250, 527872, 1090912, 1360810, 1503370, 1788490, 2085710, 2102272, 3477608, 4495808, 8394752, 15370304, 16102808, 26347688, 29322008, 33550336, 73995392
Offset: 1

Views

Author

Antti Karttunen, Jun 11 2019

Keywords

Links

Crossrefs

Cf. A000360, A326131 (subsequences), A326133.
Cf. A000120, A000203, A005187, A294898, A295296.

Programs

  • PARI
    A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
isA326132(n) = { my(t=sigma(n)-A005187(n)); (gcd(hammingweight(n), t) == abs(t)); };

A326138 Numbers k such that A005187(k) < sigma(k) <= 2k, where A005187(k) = 2k - {binary weight of k}.

Original entry on oeis.org

6, 28, 110, 496, 884, 8128, 18632, 85936, 116624, 391612, 15370304, 17619844, 33550336, 73995392, 815634435, 3915380170, 5556840416, 6800695312, 8589869056, 42783299288, 80999455688, 137438691328, 217898810368, 546409576448, 1081071376208, 1661355408388
Offset: 1

Views

Author

Antti Karttunen, Jun 13 2019

Keywords

Comments

Non-abundant numbers whose deficiency (A033879) is less than their binary weight (A000120).
No other terms below < 2^31.

Examples

			815634435 = 3*5*7*11*547*1291 is included as in base-2 (A007088) it is written as 110000100111011001100000000011_2, thus A000120(815634435) = 12, while its nonnegative deficiency (A033879) is 2*815634435 - sigma(815634435) = 6 < 12.

Links

Crossrefs

Cf. A000120, A000203, A000396 (subsequence), A005187, A033879, A294898, A295296 (deficiency equals binary weight), A326131, A326132.
Intersection of A263837 and A326133.
Cf. also A087485, A141548, A188597.

Programs

Extensions

a(16)-a(26) from Giovanni Resta, Jun 16 2019
Showing 1-6 of 6 results.

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