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 1-10 of 67 results. Next

A346242 Dirichlet inverse of A324198, where A324198(n) = gcd(n, A276086(n)).

Original entry on oeis.org

1, -1, -3, 0, -1, 5, -1, 0, 6, -3, -1, -2, -1, 1, -9, 0, -1, -16, -1, 4, 3, 1, -1, 0, -24, 1, -12, 0, -1, 43, -1, 0, 3, 1, -5, 14, -1, 1, 3, 0, -1, -11, -1, 0, 54, 1, -1, 0, -6, 32, 3, 0, -1, 44, -3, -6, 3, 1, -1, -50, -1, 1, -24, 0, 1, -5, -1, 0, 3, -15, -1, -4, -1, 1, 96, 0, -5, -5, -1, 0, 24, 1, -1, 8, -3, 1, 3, 0, -1
Offset: 1

Views

Author

Antti Karttunen, Jul 13 2021

Keywords

Links

Crossrefs

Cf. A276086, A324198, A346243.
Cf. A008966 (parity of terms), A005117 (positions of odd terms), A013929 (of even terms), A045344 (of -1's, at least a subset of them), A354810 (of 0's), A354811 (of 1's), A354812 (of 2's), A354813 (of 3's), A354814 (of 4's), A354822 (of -2's).
Cf. also A354347, A354348, A354823, A354824.

Programs

  • PARI
    up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
v346242 = DirInverseCorrect(vector(up_to,n,A324198(n)));
A346242(n) = v346242[n];

Formula

a(n) = A346243(n) - A324198(n).
From Antti Karttunen, Jun 09 2022: (Start)
a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA324198(n/d) * a(d).
For all n >= 1, A000035(a(n)) = A008966(n).
For all n >= 1, a(A045344(n)) = -1.
(End)

A364286 Composite numbers k for which A324644(k)/A324198(k) = 2.

Original entry on oeis.org

33, 51, 69, 91, 99, 135, 141, 145, 153, 159, 187, 207, 213, 217, 285, 295, 303, 321, 339, 391, 411, 423, 427, 435, 445, 477, 507, 519, 573, 637, 639, 679, 681, 699, 771, 783, 799, 843, 855, 861, 885, 895, 901, 909, 933, 951, 963, 1017, 1041, 1057, 1059, 1071, 1081, 1083, 1147, 1149, 1173, 1185, 1195, 1203, 1207
Offset: 1

Views

Author

Antti Karttunen, Jul 17 2023

Keywords

Comments

See comments in A351458.
All terms are odd. Of the 63 initial terms of A349169, only term 13923 occurs also in this sequence. The first common term with A332458 is 161257. - Antti Karttunen, Mar 10 2024

Links

Crossrefs

Cf. A000203, A276086, A324198, A324644, A332458, A349169, A351458, A371082 (subsequence).
Subsequence of A082686.

Programs

  • Mathematica
    f[x_] := Block[{m, i, n = x, p}, m = i = 1; While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m]; Select[Select[Range[1350], CompositeQ], GCD[#2, #3]/GCD[#1, #3] == 2 & @@ {#, DivisorSigma[1, #], f[#]} &] (* Michael De Vlieger, Mar 10 2024 *)
  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA364286(n) = if(isprime(n), 0, my(u=A276086(n)); (gcd(sigma(n),u)==2*gcd(n,u))); \\ Antti Karttunen, Mar 10 2024

A351080 Lexicographically earliest infinite sequence such that a(i) = a(j) => A324198(i) = A324198(j) and A351083(i) = A351083(j) for all i, j >= 0.

Original entry on oeis.org

1, 2, 2, 3, 4, 2, 2, 5, 6, 3, 7, 2, 4, 2, 2, 8, 9, 2, 10, 2, 11, 3, 2, 2, 12, 13, 2, 3, 4, 2, 2, 2, 4, 3, 14, 15, 16, 2, 2, 17, 18, 2, 19, 2, 12, 8, 2, 2, 4, 19, 13, 3, 12, 2, 10, 20, 21, 3, 2, 2, 4, 2, 2, 22, 12, 2, 2, 2, 4, 3, 23, 2, 24, 2, 2, 25, 4, 26, 2, 2, 27, 3, 2, 2, 28, 20, 2, 3, 6, 2, 10, 29, 12, 3, 2, 2, 4
Offset: 0

Views

Author

Antti Karttunen, Feb 03 2022

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A324198(n), A351083(n)].
For all i, j: a(i) = a(j) => A351084(i) = A351084(j).

Links

Crossrefs

Cf. A324198, A351083, A351084.
Cf. also A351085.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
A351083(n) = gcd(n, A327860(n));
Aux351080(n) = [A324198(n), A351083(n)];
v351080 = rgs_transform(vector(1+up_to,n,Aux351080(n-1)));
A351080(n) = v351080[1+n];

A371082 Composite numbers for which A324644(n)/A324198(n) = 2 and sigma(n) == 2 (mod 4).

Original entry on oeis.org

153, 477, 637, 909, 1017, 1233, 1557, 2097, 3577, 4753, 9457, 10693, 10933, 12393, 13357, 14013, 15337, 17629, 20817, 21097, 21217, 22021, 26353, 29449, 30037, 30717, 31117, 31149, 31797, 32013, 32229, 32337, 32481, 32977, 35557, 35917, 38637, 38725, 41797, 42237, 50029, 53557, 56497, 56677, 56953, 58621, 59437, 60309
Offset: 1

Views

Author

Antti Karttunen, Mar 10 2024

Keywords

Links

Crossrefs

Intersection of A191218 and A364286.
Apparently also the intersection of A228058 and A364286.
Cf. A000203, A276086, A351458.

Programs

  • Mathematica
    f[x_] := Block[{m, i, n = x, p}, m = i = 1; While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m]; Select[Select[Range[2^16], CompositeQ], GCD[#2, #3]/GCD[#1, #3] == Mod[#2, 4] == 2 & @@ {#, DivisorSigma[1, #], f[#]} &] (* Michael De Vlieger, Mar 10 2024 *)
  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA371082(n) = if(isprime(n) || (2!=(sigma(n)%4)), 0, my(u=A276086(n)); (gcd(sigma(n),u)==2*gcd(n,u)));

A386422 Odd numbers k that are closer to being perfect than previous terms and also satisfy the condition that A324644(k)/A324198(k) = 2.

Original entry on oeis.org

3, 33, 99, 135, 855, 2295, 19575, 38745, 63855, 121485, 371925, 3870195, 8109585, 28306005, 36340395, 113215095, 463084245, 672363615, 675916395, 686574735, 1208140395
Offset: 1

Views

Author

Antti Karttunen, Jul 21 2025

Keywords

Comments

Questions: Are there only multiples of 5 after the three initial terms? Are there any common terms with A228058?

Links

Crossrefs

Apart from initial 3, a subsequence of A364286.
Cf. A000203, A000396, A228058, A324198, A324644.
Cf. also A171929, A228059, A386419, A386420, A386421 for similar sequences.

Programs

  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
is_A364286(n) = if(isprime(n), 0, my(u=A276086(n)); (gcd(sigma(n),u)==2*gcd(n,u))); \\ Antti Karttunen, Jul 21 2025
m=-1; n=-1; k=0; while(m!=0, n+=2; if(!((n-1)%(2^25)),print1("("n")")); if(isprime(n) || is_A364286(n), if((m<0) || abs((sigma(n)/n)-2)

A387163 Numbers k such that sigma(k) >= 3*k and A324644(k)/A324198(k) = 3.

Original entry on oeis.org

10065440, 12794600, 22862840, 24806600, 27399680, 30692480, 33904640, 41629280, 41851040, 46803680, 54625760, 54842480, 70384160, 81915680, 83545280, 87311840, 91571480, 93964640, 95221280, 98030240, 101978240, 103527200, 106719200, 110116160, 121983680, 122904320, 137106200, 137359040, 143195360, 143638880, 144491200
Offset: 1

Views

Author

Antti Karttunen, Aug 28 2025

Keywords

Comments

This sequence contains all 3-perfect numbers (A005820) that are not multiples of three: 459818240 (= a(99)), 51001180160, and also any such hypothetical triperfects of the form 4u+2, when 2u+1 is not multiple of 3. See comments in A351458.

Links

Crossrefs

Intersection of A023197 and A387161.
Cf. A000203, A005820, A276086, A324198, A324644, A351458.
Cf. also A387165.

Programs

  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
is_A387163(n) = if(sigma(n)<3*n, 0, my(u=A276086(n)); (gcd(sigma(n), u)==3*gcd(n, u)));

Formula

{k | sigma(k) >= 3*k, A324644(k) = 3*A324198(k)}.

A346243 Sum of A324198 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 6, 0, 1, 9, 2, 0, -1, 0, 2, 6, 1, 0, -15, 0, 9, 6, 2, 0, 1, 1, 2, -9, 1, 0, 44, 0, 1, 6, 2, 2, 15, 0, 2, 6, 5, 0, -4, 0, 1, 69, 2, 0, 1, 1, 57, 6, 1, 0, 45, 2, 1, 6, 2, 0, -49, 0, 2, -3, 1, 2, -4, 0, 1, 6, 20, 0, -3, 0, 2, 171, 1, 2, -4, 0, 5, 27, 2, 0, 15, 2, 2, 6, 1, 0, -235, 2, 1, 6, 2, 2, 1, 0, 13
Offset: 1

Views

Author

Antti Karttunen, Jul 13 2021

Keywords

Links

Crossrefs

Cf. A276086, A324198, A346242.

Programs

  • PARI
    up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
v346242 = DirInverseCorrect(vector(up_to,n,A324198(n)));
A346242(n) = v346242[n];
A346243(n) = (A324198(n)+A346242(n));

Formula

a(n) = A324198(n) + A346242(n).

A358747 Lexicographically earliest infinite sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = [A007814(n), A007949(n), A324198(n)] when n > 1, with f(1) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 5, 7, 8, 9, 5, 10, 5, 2, 11, 12, 5, 13, 5, 14, 3, 2, 5, 15, 16, 2, 17, 4, 5, 6, 5, 18, 3, 2, 19, 20, 5, 2, 3, 21, 5, 22, 5, 4, 23, 2, 5, 24, 19, 25, 3, 4, 5, 26, 27, 28, 3, 2, 5, 10, 5, 2, 29, 30, 5, 6, 5, 4, 3, 31, 5, 32, 5, 2, 33, 4, 19, 6, 5, 34, 35, 2, 5, 36, 27, 2, 3, 7, 5, 13, 19, 4, 3, 2, 5, 37, 5, 38, 8, 14, 5, 6, 5, 7, 39
Offset: 1

Views

Author

Antti Karttunen, Dec 01 2022

Keywords

Comments

For all i, j: A305900(i) = A305900(j) => a(i) = a(j).

Links

Crossrefs

Cf. A007814, A007949, A324198.
Cf. also A305900, A358230.

Programs

  • PARI
    up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007814(n) = valuation(n,2);
A007949(n) = valuation(n,3);
A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
Aux358747(n) = if(1==n,n,[A007814(n), A007949(n), A324198(n)]);
v358747 = rgs_transform(vector(up_to, n,Aux358747(n)));
A358747(n) = v358747[n];

A387161 Numbers k for which A324644(k)/A324198(k) = 3.

Original entry on oeis.org

2, 14, 26, 62, 74, 86, 122, 134, 146, 152, 176, 182, 206, 212, 224, 254, 272, 290, 302, 314, 326, 338, 368, 386, 422, 428, 434, 446, 476, 542, 554, 566, 578, 590, 626, 632, 644, 656, 662, 674, 680, 722, 734, 752, 782, 794, 812, 842, 848, 854, 866, 890, 914, 920, 926, 974, 1046, 1058, 1082, 1088, 1094, 1136, 1154
Offset: 1

Views

Author

Antti Karttunen, Aug 28 2025

Keywords

Links

Crossrefs

Subsequences: A387163 (terms whose abundancy >= 3).
Cf. A000203, A276086, A324198, A324644, A351458.
Cf. also A364286.

Programs

  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
is_A387161(n) = { my(u=A276086(n)); (gcd(sigma(n), u)==3*gcd(n, u)); };

A387165 Nondeficient numbers k for which A324644(k)/A324198(k) = 2.

Original entry on oeis.org

38745, 77805, 78435, 118755, 141075, 157815, 210735, 237195, 241605, 294975, 300105, 323505, 364455, 371925, 390195, 409185, 455715, 475335, 499905, 567945, 607635, 660825, 701415, 733005, 766395, 806085, 809325, 872235, 885465, 891135, 937755, 964845, 978705, 1101555, 1150065, 1201095, 1229445, 1265355, 1293705
Offset: 1

Views

Author

Antti Karttunen, Aug 28 2025

Keywords

Comments

First three nonmultiples of 5 occur at a(138), a(276), a(356) = 4446981, 8909901, 11234223. (Cf. A005231, A064001).

Links

Crossrefs

Intersection of A023196 and A364286.
Cf. A000203, A276086.
Cf. also A371082, A386422, A387163 and A047802, A064001, A115414.

Programs

  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
is_A387165(n) = if(sigma(n)<2*n, 0, my(u=A276086(n)); (gcd(sigma(n),u)==2*gcd(n,u)));

Formula

{k | sigma(k) >= 2*k, A324644(k) = 2*A324198(k)}.
Showing 1-10 of 67 results. Next

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