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.

User: Donovan Johnson

Donovan Johnson's wiki page.

Donovan Johnson has authored 61 sequences. Here are the ten most recent ones:

A230636 Numbers n such that m + (sum of digits in base-4 representation of m) = n has exactly four solutions.

Original entry on oeis.org

16777234, 33554451, 50331668, 83886099, 100663316, 117440533, 150994964, 167772181, 184549398, 218103829, 234881046, 251658263, 268435476, 268435478, 285212691, 301989908, 318767125, 352321556, 369098773, 385875990, 419430421, 436207638, 452984855, 486539286
Offset: 1

Views

Author

Donovan Johnson and N. J. A. Sloane, Oct 31 2013

Keywords

Links

Crossrefs

Related base-4 sequences: A053737, A230631, A230632, A010064, A230633, A230634, A230635, A230636, A230637, A230638, A010065 (trajectory of 1)

A228466 Smallest odd number expressible in exactly n ways as p + 2*m^2 where p is 1 or a prime and m >= 0.

Original entry on oeis.org

5777, 1, 3, 13, 19, 55, 61, 169, 139, 271, 181, 391, 439, 559, 661, 619, 829, 859, 1069, 1081, 1459, 1489, 1609, 1741, 1951, 2029, 2509, 2341, 3631, 3769, 3331, 3961, 4525, 4189, 4261, 5281, 4801, 6229, 6361, 5911, 6439, 7111, 6319, 13081, 9931, 8869, 10321
Offset: 0

Views

Author

Donovan Johnson, Aug 22 2013

Keywords

Examples

			a(3) = 13 = 5+2*2^2 = 11+2*1^2 = 13+2*0^2. 13 is the smallest odd number expressible in exactly 3 ways.
a(4) = 19 = 1+2*3^2 = 11+2*2^2 = 17+2*1^2 = 19+2*0^2. 19 is the smallest odd number expressible in exactly 4 ways.
a(5) = 55 = 5+2*5^2 = 23+2*4^2 = 37+2*3^2 = 47+2*2^2 = 53+2*1^2. 55 is the smallest odd number expressible in exactly 5 ways.

Links

Crossrefs

Cf. A007697, A016067, A046921.

Programs

  • Mathematica
    (* finds terms < mx *) upto[mx_] := Block[{r = Floor[1+mx/2], k, t, p, s = {}}, t = 0*Range@r; p = Prime@ Range@ PrimePi@ mx; p[[1]] = 1; t[[# + Range[0, Sqrt[r - #]]^2]]++ & /@ ((1 + p)/2); k = 0; While[(r = Position[t, k, 1, 1]) != {}, k++; AppendTo[s, 2 r[[1,1]] - 1]]; s]; upto[10^5] (* Giovanni Resta, Aug 23 2013 *)
  • PARI
    /* finds terms up to a(1000) */ mx=10602619; v=vector(mx); nn=vector(1000); p=vector(701940); p[1]=1; pr=2; for(j=2, 701940, pr=nextprime(pr+1); p[j]=pr); for(m=0, 2302, m2=2*m^2; for(j=1, 701940, s=m2+p[j]; if(s<=mx, v[s]++, next(2)))); forstep(j=1, mx, 2, if(v[j]==0, write("b228466.txt", 0 " " j); j=mx)); forstep(j=1, mx, 2, if(v[j]>0, if(v[j]<=1000, if(nn[v[j]]==0, nn[v[j]]=j)))); for(n=1, 1000, write("b228466.txt", n " " nn[n]))

A226480 Squarefree numbers n such that (sopf(n)*d(n))^2 = sigma(n) where sopf(n) = sum of prime factors of n and d(n) = number of divisors of n.

Original entry on oeis.org

22446139, 26116291, 28097023, 30236557, 31090489, 31124341, 49941589, 61137673, 62224039, 66960589, 71334867, 71585139, 82266591, 83045869, 92346023, 92837591, 105183961, 114762567, 123563821, 130399138, 131494219, 134156197, 134867722, 135095767, 136026037
Offset: 1

Views

Author

Donovan Johnson, Jun 09 2013

Keywords

Comments

Suggested by N. J. A. Sloane.

Examples

			n = 22446139 = 31*67*101*107. sopf(n) = 31+67+101+107 = 306. d(n) = 16. (sopf(n)*d(n))^2 = (306*16)^2 = 23970816 = sigma(n).

Links

Crossrefs

Cf. A000005, A000203, A006532, A008472, A126028, A226479.

A226479 Numbers n such that (sopf(n)*d(n))^2 = sigma(n) where sopf(n) = sum of distinct prime factors of n (A008472) and d(n) = number of divisors of n.

Original entry on oeis.org

22446139, 26116291, 28097023, 30236557, 31090489, 31124341, 39618558, 41628195, 49941589, 51777957, 61137673, 62224039, 66960589, 71096795, 71334867, 71585139, 72304400, 82266591, 83045869, 92346023, 92837591, 105183961, 114762567, 117908994, 123563821
Offset: 1

Views

Author

Donovan Johnson, Jun 09 2013

Keywords

Comments

Suggested by N. J. A. Sloane.

Examples

			n = 22446139 = 31*67*101*107. sopf(n) = 31+67+101+107 = 306. d(n) = 16. (sopf(n)*d(n))^2 = (306*16)^2 = 23970816 = sigma(n).

Links

Crossrefs

Cf. A000005, A000203, A006532, A008472, A126028, A226480.

A224988 Even RMS numbers: even numbers n such that root mean square of divisors of n is an integer.

Original entry on oeis.org

2217231104, 6221622528, 9644780288, 12127073024, 15377570560, 15520617728, 22426778880, 25138541824, 34766068480, 43551357696, 49424655104, 56022543104, 67513462016, 84107119360, 84889511168, 90906475264, 107642993920, 156987452160, 174347951360, 175969792768
Offset: 1

Views

Author

Donovan Johnson, Apr 25 2013

Keywords

Comments

Even numbers from A140480.
The first 20 terms are all divisible by 30976. 30976 = 2^8*11^2.
a(21) > 2*10^11.
All the 83 terms up to 10^13 are divisible by 30976. - Giovanni Resta, Oct 29 2019

Examples

			n = 2217231104 (even). sigma_2(n) = 6616291782395055852. n has 108 divisors. 6616291782395055852/108 = 247511537^2.

Links

Crossrefs

Cf. A000005, A001157, A140480.

Programs

  • PARI
    forstep(n=2, 10^10, 2, s=sigma(n, 2); nd=numdiv(n); if(s%nd==0, if(issquare(s\nd), print(n))))
  • PARI
    isok(n) = my(s=sigma(n, 2), nd=numdiv(n)); if(s%nd==0, issquare(s\nd), 0); \\ program adapted by Michel Marcus, Oct 29 2019

Formula

Even numbers n such that A001157(n)/A000005(n) is a square.

A224987 Numbers such that Liouville's function (A002819) and the little omega analog to Liouville's function (A174863) are equal.

Original entry on oeis.org

1, 2, 3, 20, 21, 22, 23, 24, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 13092, 13093, 13094, 13095, 13096, 13097, 13098, 13099, 13100, 13101, 13102, 13103, 13104, 13105, 13106, 13107, 13232, 13233, 13234, 13235, 13239, 13240, 13241, 13242
Offset: 1

Views

Author

Donovan Johnson, Apr 22 2013

Keywords

Comments

Numbers n such that A002819(n) = A174863(n). There are 9056 terms <= 10^12 (the largest is 16959554). For n from 16959555 to 10^12, A002819(n) < A174863(n).

Examples

			n = 43:
A002819(n) = sum_{k = 1..n} (-1)^bigomega(k) = -3.
A174863(n) = sum_{k = 1..n} (-1)^omega(k) = -3.
A002819(43) = A174863(43) = -3.

Links

Crossrefs

Cf. A001221, A001222, A008836, A002819, A174863.

Programs

  • Mathematica
    PrimeOmega[n_] := Plus @@ FactorInteger[n][[All, 2]]; PrimeNu[n_] := Length[FactorInteger[n]]; Reap[For[s1 = 0; s2 = 0; n = 1, n < 15000, n++, s1 = s1 + (-1)^PrimeOmega[n]; s2 = s2 + (-1)^PrimeNu[n]; If[s1 == s2, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, May 03 2013, after Pari *)
  • PARI
    s1=0; s2=0; c=0; for(n=1, 16959554, s1=s1+(-1)^bigomega(n); s2=s2+(-1)^omega(n); if(s1==s2, c++; write("b224987.txt", c " " n)))

A223613 Numbers k whose abundance is 24: sigma(k) - 2*k = 24.

Original entry on oeis.org

112, 6592, 124672, 33452032, 82364907508, 36893488040044920832, 9444732964021303508992
Offset: 1

Views

Author

Donovan Johnson, Mar 23 2013

Keywords

Comments

Suggested by N. J. A. Sloane and Robert G. Wilson v.
Every number of the form 2^(j-1)*(2^j - 25), where 2^j - 25 is prime, is a term (cf. A379020). - Jon E. Schoenfield, Jun 02 2019
Also contains 269113900901410149234393088. - Max Alekseyev, May 28 2025

Examples

			For k = 82364907508, sigma(k) - 2*k = 24.

Crossrefs

Cf. A000203, A033880, A379020.

Programs

  • PARI
    for(n=1, 10^8, if(sigma(n)-2*n==24, print1(n ", ")))

Extensions

a(6) from Jon E. Schoenfield confirmed and a(7) added by Max Alekseyev, Jun 06 2025

A223612 Numbers k whose abundance is 22: sigma(k) - 2*k = 22.

Original entry on oeis.org

1312, 29824, 8341504, 134029312, 34356723712
Offset: 1

Views

Author

Donovan Johnson, Mar 23 2013

Keywords

Comments

Suggested by N. J. A. Sloane and Robert G. Wilson v.
a(6) > 10^12.
a(6) > 10^13. - Giovanni Resta, Mar 29 2013
a(6) > 10^18. - Hiroaki Yamanouchi, Aug 23 2018
Any term x of this sequence can be combined with any term y of A223606 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. [Proof: If x = a(n) and y = A223606(m), then sigma(x) = 2x+22 and sigma(y) = 2y-22. Thus, sigma(x)+sigma(y) = (2x+22)+(2y-22) = 2x+2y = 2(x+y), which implies that (sigma(x)+sigma(y))/(x+y) = 2(x+y)/(x+y) = 2.] - Timothy L. Tiffin, Sep 13 2016
a(6) <= 2361183240644548624384. Every number of the form 2^(j-1)*(2^j - 23), where 2^j - 23 is prime, is a term. - Jon E. Schoenfield, Jun 02 2019

Examples

			For k = 34356723712, sigma(k) - 2*k = 22.

Crossrefs

Cf. A000203, A033880, A223606 (deficiency 22).

Programs

  • Magma
    [n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq 22]; // Vincenzo Librandi, Sep 14 2016
  • Mathematica
    Select[Range[1, 10^6], DivisorSigma[1, #] - 2 # == 22 &] (* Vincenzo Librandi, Sep 14 2016 *)
  • PARI
    for(n=1, 10^8, if(sigma(n)-2*n==22, print1(n ", ")))

Extensions

Name edited by Timothy L. Tiffin, Sep 10 2023

A223611 Numbers k whose abundance is 20: sigma(k) - 2*k = 20.

Original entry on oeis.org

176, 1376, 3230, 3770, 6848, 114256, 125696, 544310, 561824, 740870, 2075648, 4199030, 4607296, 8436950, 33468416, 134045696, 199272950, 624032630, 1113445430, 1550860550, 85905593344, 2199001235456, 35184284008448, 10805836895078390, 103285638050111990
Offset: 1

Views

Author

Donovan Johnson, Mar 23 2013

Keywords

Comments

Suggested by N. J. A. Sloane and Robert G. Wilson v.
a(22) > 10^12.
a(23) > 10^13. - Giovanni Resta, Mar 29 2013
a(29) > 10^18. - Hiroaki Yamanouchi, Aug 23 2018
Any term x of this sequence can be combined with any term y of A223607 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. - Timothy L. Tiffin, Sep 13 2016
Every number of the form 2^(j-1)*(2^j - 21), where 2^j - 21 is prime, is a term. - Jon E. Schoenfield, Jun 02 2019

Examples

			For k = 544310, sigma(k) - 2*k = 20.

Links

Crossrefs

Cf. A000203, A033880, A223607 (deficiency 20).

Programs

  • Magma
    [n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq 20]; // Vincenzo Librandi, Sep 14 2016
  • Mathematica
    Select[Range[1, 10^8], DivisorSigma[1, #] - 2 # == 20 &] (* Vincenzo Librandi, Sep 14 2016 *)
  • PARI
    for(n=1, 10^8, if(sigma(n)-2*n==20, print1(n ", ")))

Extensions

a(22) from Giovanni Resta, Mar 29 2013
a(23)-a(25) from Hiroaki Yamanouchi, Aug 23 2018

A223610 Numbers k whose abundance is 18: sigma(k) - 2*k = 18.

Original entry on oeis.org

208, 6976, 8415, 31815, 351351, 2077696, 20487159, 159030135, 536559616, 2586415095, 137433972736, 2199003332608, 2305842988812599296
Offset: 1

Views

Author

Donovan Johnson, Mar 23 2013, at suggestion of N. J. A. Sloane and Robert G. Wilson v

Keywords

Comments

Any term x of this sequence can be combined with any term y of A223608 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. - Timothy L. Tiffin, Sep 13 2016
Every number of the form 2^(j-1)*(2^j - 19), where 2^j - 19 is prime, is a term (cf. A096819). - Jon E. Schoenfield, Jun 02 2019

Examples

			For k = 159030135, sigma(k) - 2*k = 18.

Crossrefs

Cf. A000203, A033880, A096819, A223608 (deficiency 18).

Programs

  • Magma
    [n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq 18]; // Vincenzo Librandi, Sep 14 2016
  • Mathematica
    Select[Range[1, 10^8], DivisorSigma[1, #] - 2 # == 18 &] (* Vincenzo Librandi, Sep 14 2016 *)
  • PARI
    for(n=1, 10^8, if(sigma(n)-2*n==18, print1(n ", ")))

Extensions

a(12) from Giovanni Resta, Mar 29 2013
a(13) from Jon E. Schoenfield confirmed and added by Max Alekseyev, Jun 03 2025

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

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