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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A063961 Numbers k such that z(k) = j(k), where z(k) = sopf(k - d(k)), j(k) = d(sopf(k) + k), sopf(k) = A008472(k) and d(k) = A000005(k).

Original entry on oeis.org

6, 24, 40, 516, 532, 679, 1219, 1581, 1790, 2196, 2212, 3060, 3182, 4120, 4266, 5816, 9084, 9812, 11648, 11911, 13532, 16488, 16904, 17016, 17436, 20448, 20460, 21129, 23962, 25356, 26016, 34307, 34856, 41624, 42348, 44392, 48420, 50696
Offset: 1

Views

Author

Jason Earls, Sep 04 2001

Keywords

Links

Programs

  • PARI
    sumprime(n,s,fac,i) = fac=factor(n); for(i=1,matsize(fac)[1],s=s+fac[i,1]); return(s); z(n)=sumprime(n-numdiv(n)); d(n)=numdiv(sumprime(n)+n); for(n=1,10^6, if(d(n)==z(n),print(n)))
  • PARI
    sumprime(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) } z(n)= { sumprime(n - numdiv(n)) } d(n)= { numdiv(sumprime(n) + n) } { n=0; for (m=1, 10^9, if(d(m)==z(m), write("b063961.txt", n++, " ", m); if (n==375, break)) ) } \\ Harry J. Smith, Sep 04 2009

A063969 Numbers k such that sopf(k) = sopf(k+3), where sopf(k) = A008472(k).

Original entry on oeis.org

7, 60, 147, 407, 470, 1053, 1175, 3431, 3822, 5126, 5960, 6280, 6321, 6897, 7200, 8687, 9243, 10760, 12614, 15093, 16153, 18080, 18818, 19668, 20433, 20976, 24648, 26826, 30804, 44016, 45878, 46221, 47423, 55965, 58506, 58682, 59645, 63897
Offset: 1

Views

Author

Jason Earls, Sep 05 2001

Keywords

Links

Crossrefs

Cf. A008472, A006145.

Programs

  • Mathematica
    Select[Range[65000],Total[Transpose[FactorInteger[#]][[1]]] == Total[ Transpose[FactorInteger[#+3]][[1]]]&] (* Harvey P. Dale, Jan 19 2013 *)
  • PARI
    sopf(n,s,fac,i)=fac=factor(n); for(i=1,matsize(fac)[1],s=s+fac[i,1]); return(s);
j=[]; for(n=1,100000, if(sopf(n)==sopf(n+3),j=concat(j,n))); j
  • PARI
    sopf(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) }
{ n=0; for (m=1, 10^9, if(sopf(m)==sopf(m + 3), write("b063969.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 04 2009

A064010 Numbers k such that sopf(k) = d(k) where d(k) = A001223(k) and sopf(k) = A008472(k).

Original entry on oeis.org

2, 64, 135, 154, 168, 256, 350, 512, 539, 1029, 1350, 1875, 2106, 2268, 2646, 2673, 2736, 2976, 4375, 6000, 6076, 6517, 6880, 7680, 9680, 10092, 10584, 14000, 14406, 14580, 14976, 17500, 18522, 20412, 26000, 26068, 26112, 26620, 27216, 28812
Offset: 1

Views

Author

Jason Earls, Sep 07 2001

Keywords

Links

Crossrefs

Cf. A001223, A008472.

Programs

  • PARI
    sopf(n,s,fac,i)=fac=factor(n); for(i=1,matsize(fac)[1], s=s+fac[i,1]); return(s);
d(n) = prime(n+1)-prime(n);
j=[]; for(n=1,50000, if(sopf(n)==d(n),j=concat(j,n))); j
  • PARI
    sopf(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) }
d(n)= { prime(n + 1) - prime(n) }
{ default(primelimit, 13500000); n=0; for (m=1, 10^9, if (sopf(m)==d(m), write("b064010.txt", n++, " ", m); if (n==150, break)) ) } \\ Harry J. Smith, Sep 05 2009

A064112 Numbers k such that sopf(k) = 2*sopf(k+1), where sopf(k) = A008472.

Original entry on oeis.org

99, 155, 689, 1106, 1517, 1524, 3014, 3403, 3479, 5809, 6478, 6723, 8606, 9143, 9454, 9797, 10126, 11771, 12283, 12382, 13112, 13969, 14150, 17422, 19303, 22184, 24856, 27466, 28119, 28529, 35927, 36464, 37512, 41904, 44505, 45016, 45506
Offset: 1

Views

Author

Jason Earls, Sep 08 2001

Keywords

Examples

			sopf(689)=66, 2*sopf(690)=66.

Links

Crossrefs

Cf. A008472 (sopf).

Programs

  • PARI
    sopf(n, s, fac, i) = fac=factor(n); for(i=1,matsize(fac)[1],s=s+fac[i,1]); return(s);
j=[]; for(n=1,50000, if(sopf(n)==2*sopf(n+1),j=concat(j,n))); j
  • PARI
    sopf(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) }
{ n=0; sm=sopf(1); for (m=1, 10^9, sp=sopf(m + 1); if (sm==2*sp, write("b064112.txt", n++, " ", m); if (n==600, break)); sm=sp ) } \\ Harry J. Smith, Sep 07 2009

A064444 Numbers k such that pi(k) = sopf(k) where sopf(k) is sum of distinct prime factors of k (A008472).

Original entry on oeis.org

1, 4, 12, 28, 30, 52, 55, 65, 68, 76, 95, 145, 155, 185, 205, 822, 894, 2779, 2863, 8392, 23481, 24093, 24237, 64270, 174691, 174779, 1301989, 1302457, 3523478, 9554955, 9555045, 9556455, 70111213, 70111247, 189960426, 514269523, 514269599, 10246934786
Offset: 1

Views

Author

Jason Earls, Oct 02 2001

Keywords

Comments

No further terms < 800000. - Klaus Brockhaus, Oct 05 2001

Links

Crossrefs

Cf. A008472, A000720.

Programs

  • Mathematica
    sopf[n_] := If[n==1, 0, Total[First /@ FactorInteger[n]]]; Select[Range[10^4], PrimePi@ # == sopf@ # &] (* Giovanni Resta, Mar 28 2017 *)
  • PARI
    pi(x, c=0) = forprime(p=2,x,c++); c sopf(n, fac) = fac=factor(n); sum(i=1,matsize(fac)[1],fac[i,1]) j=[]; for(n=1,25000, if(pi(n)==sopf(n),j=concat(j,n))); j
  • PARI
    sopf(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) } { n=0; for (m=1, 10^9, if (primepi(m)==sopf(m), write("b064444.txt", n++, " ", m); if (n==100, break)) ) } \\ Harry J. Smith, Sep 14 2009

Extensions

More terms from Klaus Brockhaus, Oct 05 2001
a(27)-a(29) from Harry J. Smith, Sep 14 2009
a(30)-a(38) from Giovanni Resta, Mar 28 2017

A064675 Numbers k such that sopfr(k) = sopf(k+1), where sopf(k) = A008472(k) and sopfr(k) = A001414(k).

Original entry on oeis.org

5, 27, 77, 714, 836, 948, 1449, 4185, 4624, 5405, 5560, 8476, 8855, 10175, 16932, 17080, 18655, 20450, 20600, 21183, 26642, 28809, 31524, 35631, 37828, 37881, 40081, 47544, 48203, 49240, 52155, 52554, 53192, 63344, 63426, 63665, 79118, 80800, 81576, 83780
Offset: 1

Views

Author

Jason Earls, Oct 10 2001

Keywords

Links

Crossrefs

Cf. A001414 (sopfr), A008472 (sopf).

Programs

  • PARI
    sopf(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) }
sopfr(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]*f[i, 2]); return(s) }
{ n=0; for (m=1, 10^9, if (sopfr(m)==sopf(m + 1), write("b064675.txt", n++, " ", m); if (n==500, break)) ) } \\ Harry J. Smith, Sep 21 2009
  • Python
    from sympy import factorint
def aupton(terms):
  alst, k, sopfk, sopfrk, sopfkp1, sopfrkp1 = [], 2, 2, 3, 2, 3
  while len(alst) < terms:
    if sopfrk == sopfkp1: alst.append(k)
    k += 1
    fkp1 = factorint(k+1)
    sopfk, sopfkp1 = sopfkp1, sum(p for p in fkp1)
    sopfrk, sopfrkp1 = sopfrkp1, sum(p*fkp1[p] for p in fkp1)
  return alst
print(aupton(40)) # Michael S. Branicky, May 27 2021

A064678 Numbers k such that sopf(k) = sopfr(k+1), where sopf(k) = A008472(k) and sopfr(k) = A001414(k).

Original entry on oeis.org

5, 15, 77, 99, 104, 153, 369, 492, 539, 714, 1330, 2491, 4191, 5405, 5831, 5959, 6556, 6579, 6723, 8463, 9424, 12221, 12351, 12726, 13419, 14587, 21716, 24432, 24880, 24895, 26642, 30267, 31487, 33019, 35456, 38324, 43215, 43802, 44831, 45524
Offset: 1

Views

Author

Jason Earls, Oct 10 2001

Keywords

Links

Crossrefs

Cf. A001414 (sopfr), A008472 (sopf).

Programs

  • PARI
    sopf(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) } sopfr(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]*f[i, 2]); return(s) } { n=0; for (m=1, 10^9, if (sopf(m)==sopfr(m + 1), write("b064678.txt", n++, " ", m); if (n==500, break)) ) } \\ Harry J. Smith, Sep 22 2009

A067008 Numbers k such that Sum_{j=1..k} A008472(j) divides k!.

Original entry on oeis.org

2, 5, 7, 10, 13, 16, 19, 20, 31, 34, 37, 40, 41, 44, 46, 48, 51, 53, 55, 56, 62, 64, 65, 67, 68, 69, 73, 74, 76, 79, 81, 82, 83, 84, 85, 86, 87, 89, 95, 96, 97, 99, 100, 101, 106, 107, 108, 109, 111, 112, 115, 119, 121, 122, 123, 124, 133, 134, 135, 136, 137, 138, 143
Offset: 1

Views

Author

Benoit Cloitre, Oct 03 2002

Keywords

Links

Crossrefs

Cf. A008472.

Programs

  • Mathematica
    tot[n_] := tot[n] = If[n < 2, 0, tot[n - 1] + Plus @@ FactorInteger[n][[;; , 1]]]; ef[n_, p_] := Block[{c=0, m=n}, While[m > 0, m = Floor[m/p]; c += m]; c]; ok[n_] := n > 1 && AllTrue[ FactorInteger@ tot@n, #[[1]] <= n && ef[n, #[[1]]] >= #[[2]] &]; Select[ Range@ 1000, ok] (* Giovanni Resta, Dec 13 2019 *)
  • PARI
    for (j=2,143,if(!(j!%sum(k=1,j,vecsum(factor(k)[,1]))),print1(j,", "))) \\ Hugo Pfoertner, Dec 13 2019

Formula

a(n) seems to be asymptotic to c*n with c=2.3.....

A075659 Sum of prime divisors (A008472) is a power of an integer with exponent greater than 1.

Original entry on oeis.org

14, 15, 28, 39, 45, 46, 55, 56, 66, 75, 87, 92, 94, 98, 112, 117, 132, 135, 155, 158, 183, 184, 186, 188, 196, 198, 203, 224, 225, 247, 255, 261, 264, 275, 285, 290, 291, 295, 299, 316, 322, 323, 334, 351, 354, 357, 368, 372, 375, 376, 392, 396, 405, 418, 429
Offset: 1

Views

Author

Floor van Lamoen, Sep 23 2002

Keywords

Links

Crossrefs

Cf. A008472, A001597.

Programs

  • PARI
    isok(n) = {my(f=factor(n)); ispower(sum(k=1, #f~, f[k,1]));} \\ Michel Marcus, Sep 09 2017

Extensions

More terms from Matthew Conroy, Oct 15 2002
Offset corrected by Michel Marcus, Sep 09 2017

A076382 Numbers n such that sum of digits in base 4 is a divisor of sum of prime divisors (A008472).

Original entry on oeis.org

2, 3, 4, 8, 9, 16, 26, 32, 42, 64, 65, 78, 81, 84, 86, 92, 94, 95, 104, 114, 115, 119, 128, 130, 143, 146, 156, 161, 168, 170, 178, 186, 209, 212, 215, 228, 234, 244, 256, 258, 259, 260, 287, 294, 308, 312, 319, 322, 326, 332, 335, 336, 338, 340, 342, 343, 344
Offset: 1

Views

Author

Floor van Lamoen, Oct 08 2002

Keywords

Crossrefs

Cf. A075657, A076380 - A076387.

Programs

  • Maple
    A076382 := proc(n) local i,j,t,t1, sod, sopd; t := NULL; for i from 2 to n do t1 := i; sod := 0; while t1 <> 0 do sod := sod + (t1 mod 4); t1 := floor(t1/4); od; sopd := 0; j := 1; while ithprime(j) <= i do if i mod ithprime(j) = 0 then sopd := sopd+ithprime(j); fi; j := j+1; od; if sopd mod sod = 0 then t := t,i; fi; od; t; end;
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