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.

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A360358 Numbers k such that A360327(k) = A360327(k+1) > 1.

Original entry on oeis.org

714, 6603, 16115, 18920, 23154, 24530, 39984, 41360, 42789, 51204, 56814, 58190, 59619, 60995, 65229, 66605, 68034, 69410, 73644, 79304, 82059, 84249, 84864, 86240, 94655, 101375, 101694, 103070, 107304, 108680, 121374, 125510, 126125, 126939, 135128, 135354, 137329
Offset: 1

Views

Author

Amiram Eldar, Feb 04 2023

Keywords

Comments

Numbers k such that A360327(k) = A360327(k+1) = 1 are terms of A360357.

Examples

			714 is a term since A360327(714) = A360327(715) = 72 > 1.

Links

Crossrefs

Cf. A360327, A360357.
Similar sequences: A002961, A064115, A064125, A293183, A306985, A360359.

Programs

  • Mathematica
    f[p_, e_] := If[PrimeQ[PrimePi[p]], (p^(e+1)-1)/(p-1), 1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; s1 = s[1]; n = 2; c = 0; While[c < 40, s2 = s[n]; If[s1 == s2 > 1, c++; AppendTo[seq, n - 1]]; s1 = s2; n++]; seq
  • PARI
    s(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), (p[i]^(e[i]+1)-1)/(p[i]-1), 1));}
lista(nmax) = {my(s1 = s(1), s2); for(n=2, nmax, s2=s(n); if(s2 > 1 && s1 == s2, print1(n-1, ", ")); s1 = s2); }

A360359 Numbers k such that A360331(k) = A360331(k+1).

Original entry on oeis.org

69, 574, 713, 781, 2394, 2506, 5699, 5750, 6499, 6509, 8441, 19250, 26529, 32130, 36549, 38065, 41749, 41929, 43239, 48025, 50301, 53037, 53382, 59178, 59822, 61754, 66906, 67689, 70277, 71198, 81620, 94000, 100775, 119214, 124640, 127442, 134665, 153202, 154908
Offset: 1

Views

Author

Amiram Eldar, Feb 04 2023

Keywords

Examples

			69 is a term since A360331(69) = A360331(70) = 24.

Links

Crossrefs

Cf. A360331.
Similar sequences: A002961, A064115, A064125, A293183, A306985, A360358.

Programs

  • Mathematica
    f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, (p^(e+1)-1)/(p-1)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; s1 = s[1]; n = 2; c = 0; While[c < 40, s2 = s[n]; If[s1 == s2, c++; AppendTo[seq, n - 1]]; s1 = s2; n++]; seq
  • PARI
    s(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), 1, (p[i]^(e[i]+1)-1)/(p[i]-1)));}
lista(nmax) = {my(s1 = s(1), s2); for(n=2, nmax, s2=s(n); if(s1 == s2, print1(n-1, ", ")); s1 = s2); }

A372693 Numbers k such that A372692(k) = A372692(k+1) > 1.

Original entry on oeis.org

7380, 18755, 24804, 25631, 26299, 27467, 32799, 44891, 49196, 49725, 50940, 53603, 59652, 64386, 71027, 79739, 85788, 89300, 94275, 103212, 105056, 105875, 124992, 129348, 132011, 138060, 141899, 147100, 149435, 155484, 158147, 164196, 170324, 175571, 181620, 184283
Offset: 1

Views

Author

Amiram Eldar, May 10 2024

Keywords

Comments

The numbers k such that A372692(k) = A372692(k+1) = 1 are in A372690.

Links

Crossrefs

Cf. A036537, A372690, A372692.
Subsequence of A068781.
A372694 is a subsequence.
Similar sequences: Cf. A002961, A064125, A293183, A306985, A343819, A348346.

Programs

  • Mathematica
    f[p_, e_] := p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], ?(# == 0 &)])); s[1] = 1; s[n] := s[n] = Times @@ (Flatten@ (f @@@ FactorInteger[n]) + 1);
Select[Range[10^5], (s1 = s[#]) > 1 && s1 == s[# + 1] &]
  • PARI
    s(n) = {my(f = factor(n), k); prod(i = 1, #f~, k = apply(x -> 1 - x, binary(f[i, 2])); prod(j = 1, #k, if(k[j], f[i, 1]^(2^(#k-j)) + 1, 1)));}
lista(kmax) = {my(s1 = s(1), s2); for(k = 2, kmax, s2 = s(k); if(s1 > 1 && s1 == s2, print1(k - 1, ", ")); s1 = s2);}

A064348 Numbers k such that k and k+1 have the same sum of unitary divisors and the same number of divisors.

Original entry on oeis.org

14, 44, 1334, 1634, 2295, 2685, 3195, 17255, 33998, 42818, 45716, 79316, 84134, 122073, 124676, 125811, 166934, 239499, 289454, 294151, 383594, 440013, 544334, 605985, 649154, 655005, 736515, 1163624, 1325511, 1364104
Offset: 1

Views

Author

Jason Earls, Oct 15 2001

Keywords

Links

Crossrefs

Cf. A000005, A034448, A064125.

Programs

  • Mathematica
    g[1]={1,1}; g[n_] := {Times @@ ((f = FactorInteger[n])[[;;,2]] + 1), Times @@ (1 + Power @@@ f)}; s={}; g1={0,0}; Do[g2=g[n]; If[g1==g2, AppendTo[s,n-1]]; g1=g2, {n, 1, 50000}]; s (* Amiram Eldar, Jun 19 2019 *)
  • PARI
    {usigma(n, s=1, fac, i) = fac=factor(n); for(i=1,matsize(fac)[1], s=s*(1+fac[i,1]^fac[i,2]) ); return(s); } for(n=1,10^6, if(usigma(n)==usigma(n+1) && numdiv(n)==numdiv(n+1),print1(n,",")))
  • PARI
    usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) } { n=0; s=0; d=0; for (m=1, 10^9, us=usigma(m + 1); ud=numdiv(m + 1); if(s==us && d==ud, write("b064348.txt", n++, " ", m); if (n==100, break)); s=us; d=ud ) } \\ Harry J. Smith, Sep 12 2009

A064729 Numbers k such that k and k+1 have the same sum of unitary and nonunitary divisors.

Original entry on oeis.org

14, 957, 1334, 1634, 2685, 20145, 33998, 42818, 74918, 79826, 79833, 84134, 111506, 122073, 138237, 147454, 166934, 201597, 274533, 289454, 347738, 383594, 416577, 440013, 544334, 605985, 649154, 655005, 1642154, 1857513, 2168906, 2284814
Offset: 1

Views

Author

Jason Earls, Oct 17 2001

Keywords

Links

Crossrefs

Cf. A034448, A048146, A064125.

Programs

  • Mathematica
    g[1]={1, 1}; g[n_] := { Times @@ (1 + Power @@@ (f = FactorInteger[n])), Times @@ ((f[[;; , 1]]^(f[[;;,2]]+1)- 1)/(f[[;;,1]]-1))}; s={}; g1={0, 0}; Do[g2=g[n]; If[g1==g2, AppendTo[s, n-1]]; g1=g2, {n, 1, 50000}]; s (* Amiram Eldar, Jun 19 2019 *)
  • PARI
    {usigma(n, s=1, fac, i) = fac=factor(n); for(i=1,matsize(fac)[1],s=s*(1+fac[i,1]^fac[i,2])); return(s); } nu(n) = sigma(n)-usigma(n); for(n=1,10^7, if(usigma(n)==usigma(n+1) && nu(n)==nu(n+1), print1(n,",")))
  • PARI
    usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) } nu(n)= { sigma(n) - usigma(n) } { n=0; for (m = 1, 10^10, if(usigma(m)==usigma(m + 1) && nu(m)==nu(m + 1), write("b064729.txt", n++, " ", m); if (n==190, break)) ) } \\ Harry J. Smith, Sep 24 2009

Extensions

a(27)-a(32) from Harry J. Smith, Sep 24 2009

A286837 Numbers n such that usigma(n) = usigma(2*n+1) where usigma(n) = A034448(n).

Original entry on oeis.org

1386, 6790, 8130, 18618, 21378, 27654, 38874, 60030, 64020, 71058, 89178, 92130, 97014, 117114, 118902, 180438, 182226, 224058, 247044, 396078, 495114, 510906, 528510, 723486, 855966, 979098, 1007562, 1012380, 1032360, 1141194, 1302906, 1410294
Offset: 1

Views

Author

Altug Alkan, Aug 01 2017

Keywords

Comments

46495995 = 3*5*7*13*23*1481 is the smallest odd term of this sequence.

Links

Crossrefs

Cf. A034448, A064125, A272553.

Programs

  • Mathematica
    usigma[1] = 1;  usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[10^5], usigma[#] == usigma[2#+1] &] (* Amiram Eldar, Aug 04 2019 *)
  • PARI
    a034448(n) = sumdivmult(n, d, if(gcd(d, n/d)==1, d));
isok(n) = a034448(n)==a034448(2*n+1); \\ after Charles R Greathouse IV at A034448

A324367 Numbers k such that s(k) = s(k+1) where s(k) is the sum of divisors of k that are larger than sqrt(k) (A238535).

Original entry on oeis.org

45, 62, 15795, 355022, 14257705, 28856174, 2324581982, 103321586193
Offset: 1

Views

Author

Amiram Eldar, Sep 03 2019

Keywords

Comments

a(9) > 2*10^11. - Giovanni Resta, Sep 06 2019

Examples

			45 is in the sequence since A238535(45) = A238535(46) = 69.

Crossrefs

Cf. A002961, A064115, A064125, A164522, A164557, A238535, A293183, A306985, A324295.

Programs

  • Mathematica
    s[n_] := DivisorSum[n, # &, # > Sqrt[n] &]; seq={}; s1 = 0; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n - 1]]; s1 = s2, {n, 2, 10000}]; seq

Extensions

a(8) from Giovanni Resta, Sep 06 2019

A327875 Numbers k such that s(k) = s(k+1) where s(k) is the sum of unitary, squarefree divisors of k, including 1 (A092261).

Original entry on oeis.org

8, 14, 288, 675, 735, 957, 1334, 1634, 2685, 2871, 5750, 8055, 9800, 12104, 12167, 20145, 33998, 42818, 71994, 74918, 79826, 79833, 84134, 111506, 122073, 138237, 144990, 147454, 166934, 201597, 235224, 274533, 289454, 324423, 332928, 347738, 383594, 400315
Offset: 1

Views

Author

Amiram Eldar, Sep 28 2019

Keywords

Links

Crossrefs

Cf. A002961, A064115, A064125, A092261, A164522, A164557, A293183, A306985, A324295, A325175.

Programs

  • Mathematica
    f[p_, e_] := If[e==1, p+1, 1]; s[n_] := Times @@ f @@@ FactorInteger[n]; s1=0; seq = {}; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n-1]]; s1 = s2, {n,1,10000}]; seq
  • PARI
    s(n)={sumdiv(n, d, d*issquarefree(d)*(gcd(d, n/d) == 1))}
{ for(k=1, 10^6, if(s(k)==s(k+1), print1(k, ", "))) } \\ Andrew Howroyd, Sep 28 2019

Formula

8 is in the sequence since A092261(8) = A092261(9) = 1.

A332315 Numbers k such that k and k + 1 have the same norm of the sum of divisors in Gaussian integers.

Original entry on oeis.org

30514, 36777, 43978, 3474262, 5745125, 10628554, 16567494, 40831527, 58008301, 111798477, 142981839, 288834504, 392413941, 580867202, 650141557, 944224497, 967593411, 1874210882, 6306287377, 6442064745, 7377567197, 8121464245
Offset: 1

Views

Author

Amiram Eldar, Feb 09 2020

Keywords

Comments

The first term, 30514, is also a number k such that k and k + 1 have the sum divisors in Gaussian integers: -54720 + 48960*i (where i is the imaginary unit). What is the next term with this property?
No more terms below 1.5*10^10.

Examples

			30514 is a term since A103230(30514) = A103230(30515) = 5391360000.

Crossrefs

Cf. A002961, A064125, A103228, A103229, A103230, A293183, A306985.

Programs

  • Mathematica
    csigma[n_] :=(Abs @ DivisorSigma[1, n, GaussianIntegers -> True])^2; seq = {}; n1 = csigma[1]; Do[n2 = csigma[n]; If[n1 == n2, AppendTo[seq, n - 1]]; n1 = n2, {n, 2, 5*10^5}]; seq

A332475 Numbers k such that k and k + 1 have the same norm of the sum of unitary divisors in Gaussian integers (A332474).

Original entry on oeis.org

5, 11, 37, 1738, 2772, 6600, 42251, 49913, 57816, 104754, 220324, 288350, 364452, 792156, 1711932, 1971475, 2607049, 2793473, 3211933, 3521148, 3526312, 4012736, 5805149, 5918276, 6522320, 6542147, 6635436, 7612267, 12604600, 14844791, 17078848, 19024332, 21177516
Offset: 1

Views

Author

Amiram Eldar, Feb 13 2020

Keywords

Examples

			5 is a term since A332474(5) = A332474(6) = 80.

Links

Crossrefs

Cf. A002961, A064125, A293183, A306985, A332315, A332472, A332473, A332474.

Programs

  • Mathematica
    f[p_, e_] := If[Abs[p] == 1, 1, (p^e + 1)]; normUsigma[n_] := Abs[Times @@ f @@@ FactorInteger[n, GaussianIntegers -> True]]^2; seq = {}; u1 = normUsigma[1]; Do[u2 = normUsigma[n]; If[u1 == u2, AppendTo[seq, n - 1]]; u1 = u2, {n, 2, 10^6}]; seq
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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