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 1-8 of 8 results.

A071247 Numbers n such that A078142(n) = A078142(n+1), where A078142(n) is the sum of the differences of the distinct prime factors p of n and the next square larger than p.

Original entry on oeis.org

7, 12, 19, 29, 86, 96, 99, 121, 132, 138, 153, 154, 164, 183, 192, 220, 230, 234, 251, 274, 280, 286, 353, 390, 444, 455, 476, 484, 539, 589, 651, 675, 704, 730, 774, 785, 813, 850, 867, 944, 965, 1069, 1152, 1216, 1238, 1250, 1266, 1280, 1299, 1308, 1333
Offset: 1

Views

Author

Jason Earls, Nov 20 2002

Keywords

Links

Crossrefs

Cf. A078142.

Programs

  • Mathematica
    s[n_] := Total[Ceiling[Sqrt[(p = FactorInteger[n][[;;,1]])]]^2 - p]; s0 = 0; seq = {}; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n - 1]]; s1 = s2, {n, 2, 1334}]; seq (* Amiram Eldar, Dec 05 2019 *)

A078327 Numbers k such that A078142(k) = A006530(k).

Original entry on oeis.org

2, 4, 6, 8, 12, 15, 16, 18, 24, 32, 36, 45, 48, 54, 64, 72, 75, 96, 105, 108, 110, 128, 135, 144, 162, 192, 216, 220, 225, 256, 288, 315, 324, 375, 384, 385, 405, 432, 440, 486, 512, 525, 550, 576, 648, 675, 735, 768, 864, 880, 935, 945, 972, 1024, 1100, 1125
Offset: 1

Views

Author

Jason Earls, Nov 24 2002

Keywords

Comments

Numbers k such that the sum of the differences of the distinct prime factors p of k and the next square larger than p is equal to the largest prime factor of k.
Are there any other consecutive terms in this sequence other than 15,16 and 384,385?

Links

Crossrefs

Cf. A006530, A078142, A078328, A145445.

Programs

  • PARI
    is(k) = {if(k<2, return(0)); my(f=factor(k)[, 1]); sum(i=1, #f, (sqrtint(f[i])+1)^2-f[i]) == vecmax(f); } \\ Jinyuan Wang, Apr 17 2020

Extensions

Offset changed to 1 by Jinyuan Wang, Apr 17 2020

A073938 Numbers n such that A078142(n) = A078142(n+1) = A078142(n+2), where A078142(n) is the sum of the differences of the distinct prime factors p of n and the next square larger than p.

Original entry on oeis.org

153, 3009, 3288, 5170, 5364, 11186, 11295, 11395, 12874, 13545, 16288, 17892, 27760, 28118, 34187, 38907, 47650, 55282, 63455, 64972, 65290, 95886, 104718, 106793, 110944, 155573, 163964, 169644, 172081, 187164, 202607, 203255, 204609
Offset: 1

Views

Author

Jason Earls, Nov 20 2002

Keywords

Comments

Are there infinitely many k-tuples in A078142?

Links

Crossrefs

Cf. A071247, A078142.

Programs

  • Mathematica
    s[n_] := Total[Ceiling[Sqrt[(p = FactorInteger[n][[;; , 1]])]]^2 - p]; s1 = s2 = 0; seq = {}; Do[s3 = s[n]; If[s1 == s2 == s3, AppendTo[seq, n - 2]]; s1 = s2; s2 = s3, {n, 3, 2*10^5}]; seq (* Amiram Eldar, Dec 08 2019 *)

A073939 Least m such that A078142(m) gives the n-th prime, where A078142(n) is the sum of the differences of the distinct prime factors p of n and the next square larger than p.

Original entry on oeis.org

2, 6, 11, 22, 53, 106, 83, 166, 173, 227, 293, 863, 443, 857, 853, 971, 1097, 2194, 1229, 1373, 2746, 2837, 2221, 2027, 2819, 3499, 4253, 3257, 3491, 3251, 4229, 4493, 5639, 6917, 6907, 7949, 6899, 7937, 7229, 7927, 11057, 10223, 9413, 10211, 9803
Offset: 1

Views

Author

Jason Earls, Nov 20 2002

Keywords

Examples

			a(4)=22 because A078142(22)=7, the fourth prime and this is the first time 7 occurs.

Links

Crossrefs

Cf. A078142.

Programs

  • Mathematica
    s[n_] := Total[Ceiling[Sqrt[(p = FactorInteger[n][[;; , 1]])]]^2 - p]; max=45; seq=Table[0,{max}]; c = 0; n=1; While[cAmiram Eldar, Dec 08 2019 *)

A076828 Record high values in A078142.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 17, 20, 23, 28, 29, 32, 33, 40, 41, 42, 48, 52, 54, 57, 58, 65, 67, 72, 74, 77, 80, 89, 91, 92, 98, 102, 108, 112, 113, 114, 117, 122, 126, 127, 132, 138, 140, 143, 148, 150, 152, 153, 161, 168, 171, 173, 180, 182, 188, 191, 197, 203, 209
Offset: 1

Views

Author

Jason Earls, Nov 21 2002

Keywords

Links

Crossrefs

Cf. A078142.

Programs

  • Mathematica
    s[n_] := Total[Ceiling[Sqrt[(p = FactorInteger[n][[;; , 1]])]]^2 - p]; seq={}; sm = -1; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, s1]], {n, 1, 10^4}]; seq (* Amiram Eldar, Dec 08 2019 *)
f[n_]:=Module[{difs=Transpose[FactorInteger[n]][[1]]},Total[Ceiling[Sqrt[difs]]^2-difs]];DeleteDuplicates[Array[f,12000],GreaterEqual] (* Harvey P. Dale, Sep 07 2022 *)

A076830 Least square s such that A078142(s) is equal to the n-th prime.

Original entry on oeis.org

4, 36, 121, 484, 2809, 11236, 6889, 27556, 29929, 51529, 85849, 744769, 196249, 734449, 727609, 942841, 1203409, 4813636, 1510441, 1885129, 7540516, 8048569, 4932841, 4108729, 7946761, 12243001, 18088009, 10608049, 12187081, 10569001
Offset: 1

Views

Author

Jason Earls, Nov 21 2002

Keywords

Examples

			a(1) = 4 since 4 = 2^2 is the least square s such that A078142(s) = 2, the first prime.
a(2) = 36 since 36 = 6^2 is the least square s such that A078142(s) = 3, the second prime.

Links

Crossrefs

Cf. A078142.

Programs

  • Mathematica
    s[n_] := Total[Ceiling[Sqrt[(p = FactorInteger[n][[;; , 1]])]]^2 - p]; m = 30; seq = Table[0, {m}]; c = 0; k = 1; While[c < m, n = s[k^2]; If[PrimeQ[n] && (i = PrimePi[n]) <= m && seq[[i]] == 0, c++; seq[[i]] = k^2]; k++]; seq (* Amiram Eldar, Dec 11 2019 *)

Extensions

Name edited by Amiram Eldar, Dec 11 2019

A078328 Least m such that A078142(m) = A006530(m) = n-th prime.

Original entry on oeis.org

2, 6, 15, 105, 110, 2145, 935, 3553, 163438, 618222, 1681130, 1314610, 7478810, 56528230, 2533253470, 872029246, 84271102135, 459400861870, 58425959482, 247074041822, 949284476474, 327711206734538, 1447872915170, 11714608131830, 12499486876662610, 63324881704174
Offset: 1

Views

Author

Jason Earls, Nov 24 2002

Keywords

Comments

Is this sequence infinite?

Examples

			a(3)=15 because A078142(15) = A006530(15) = 5 and this is the first time 5 occurs.

Crossrefs

Cf. A078142, A006530, A078327.

Extensions

More terms from Jinyuan Wang, Apr 17 2020

A078340 Least k such that A078142(n) = A078142(n+k).

Original entry on oeis.org

2, 6, 3, 9, 6, 1, 8, 18, 9, 4, 1, 5, 11, 11, 7, 21, 3, 1, 13, 3, 7, 9, 12, 3, 5, 54, 11, 1, 14, 11, 15, 2, 34, 5, 12, 48, 3, 7, 3, 25, 3, 7, 13, 7, 10, 2, 6, 15, 28, 4, 7, 36, 7, 3, 36, 3, 56, 16, 2, 2, 3, 6, 15, 12, 4, 7, 3, 3, 3, 11, 24, 3, 33, 9, 10, 11, 2, 49, 13, 162, 13, 44, 7, 24, 1
Offset: 2

Views

Author

Jason Earls, Nov 22 2002

Keywords

Crossrefs

Cf. A078142.
Showing 1-8 of 8 results.

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