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 1-5 of 5 results.

A005069 Sum of odd primes dividing n.

Original entry on oeis.org

0, 0, 3, 0, 5, 3, 7, 0, 3, 5, 11, 3, 13, 7, 8, 0, 17, 3, 19, 5, 10, 11, 23, 3, 5, 13, 3, 7, 29, 8, 31, 0, 14, 17, 12, 3, 37, 19, 16, 5, 41, 10, 43, 11, 8, 23, 47, 3, 7, 5, 20, 13, 53, 3, 16, 7, 22, 29, 59, 8, 61, 31, 10, 0, 18, 14, 67, 17, 26, 12, 71, 3, 73, 37, 8, 19, 18, 16, 79
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Primes counted without multiplicity. - Harvey P. Dale, Aug 28 2019

Links

Crossrefs

Cf. A000265, A005066, A005067, A005068, A005078, A005082, A008472, A020639, A028234, A059841.

Programs

  • Mathematica
    a = {0, 0}; For[n = 3, n < 80, n++, su = 0; b = FactorInteger[n]; For[i = 1, i < Length[b] + 1, i++, If[OddQ[b[[i, 1]]], su = su + b[[i, 1]]]]; AppendTo[a, su]]; a (* Stefan Steinerberger, Jun 02 2007 *)
Array[DivisorSum[#, # &, And[PrimeQ@ #, OddQ@ #] &] &, 79] (* Michael De Vlieger, Jul 11 2017 *)
Join[{0},Table[Total[FactorInteger[n][[All,1]]/.(2->0)],{n,2,100}]] (* Harvey P. Dale, Aug 28 2019 *)
  • PARI
    a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%2) == 1, p)); \\ Michel Marcus, Jul 11 2017
  • Scheme
    (define (A005069 n) (cond ((= 1 n) 0) ((even? n) (A005069 (/ n 2))) (else (+ (A020639 n) (A005069 (A028234 n)))))) ;; Antti Karttunen, Jul 10 2017

Formula

Additive with a(p^e) = 0 if p = 2, p otherwise.
G.f.: Sum_{k>=2} prime(k)*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Dec 24 2016
From Antti Karttunen, Jul 10 & 11 2017: (Start)
a(1) = 0; after which, for even n: a(n) = a(n/2), for odd n: a(n) = A020639(n) + a(A028234(n)).
a(n) = A008472(A000265(n)) = A008472(n) - 2*A059841(n).
a(n) = A005078(n) + A005082(n).
(End)

Extensions

More terms from Stefan Steinerberger, Jun 02 2007

A005083 Sum of squares of primes = 3 mod 4 dividing n.

Original entry on oeis.org

0, 0, 9, 0, 0, 9, 49, 0, 9, 0, 121, 9, 0, 49, 9, 0, 0, 9, 361, 0, 58, 121, 529, 9, 0, 0, 9, 49, 0, 9, 961, 0, 130, 0, 49, 9, 0, 361, 9, 0, 0, 58, 1849, 121, 9, 529, 2209, 9, 49, 0, 9, 0, 0, 9, 121, 49, 370, 0, 3481, 9, 0, 961, 58, 0, 0, 130, 4489, 0, 538, 49, 5041, 9, 0, 0, 9, 361, 170, 9, 6241, 0, 9, 0, 6889, 58
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000290, A005063, A005079, A005082, A005084, A005085, A059841.

Programs

  • Mathematica
    Array[DivisorSum[#, #^2 &, And[PrimeQ@ #, Mod[#, 4] == 3] &] &, 84] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 4] == 3, p^2, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)
  • PARI
    a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%4) == 3, p^2)); \\ Michel Marcus, Jul 11 2017
  • Scheme
    (define (A005083 n) (if (= 1 n) 0 (+ (if (= 3 (modulo (A020639 n) 4)) (A000290 (A020639 n)) 0) (A005083 (A028234 n))))) ;; Antti Karttunen, Jul 11 2017

Formula

Additive with a(p^e) = p^2 if p = 3 (mod 4), 0 otherwise.
a(n) = A005063(n) - A005079(n) - 4*A059841(n). - Antti Karttunen, Jul 11 2017

Extensions

More terms from Antti Karttunen, Jul 11 2017

A005085 Sum of 4th powers of primes = 3 mod 4 dividing n.

Original entry on oeis.org

0, 0, 81, 0, 0, 81, 2401, 0, 81, 0, 14641, 81, 0, 2401, 81, 0, 0, 81, 130321, 0, 2482, 14641, 279841, 81, 0, 0, 81, 2401, 0, 81, 923521, 0, 14722, 0, 2401, 81, 0, 130321, 81, 0, 0, 2482, 3418801, 14641, 81, 279841, 4879681, 81, 2401, 0, 81, 0, 0, 81, 14641, 2401, 130402, 0, 12117361, 81, 0, 923521, 2482, 0, 0, 14722, 20151121, 0, 279922
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000583, A005065, A005081, A005082, A005083, A005084, A059841.

Programs

  • Mathematica
    Array[DivisorSum[#, #^4 &, And[PrimeQ@ #, Mod[#, 4] == 3] &] &, 69] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 4] == 3, p^4, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)
  • PARI
    a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%4) == 3, p^4)); \\ Michel Marcus, Jul 11 2017
  • Scheme
    (define (A005085 n) (if (= 1 n) 0 (+ (if (= 3 (modulo (A020639 n) 4)) (A000583 (A020639 n)) 0) (A005085 (A028234 n))))) ;; Antti Karttunen, Jul 11 2017

Formula

Additive with a(p^e) = p^4 if p = 3 (mod 4), 0 otherwise.
a(n) = A005065(n) - A005081(n) - 16*A059841(n). - Antti Karttunen, Jul 11 2017

Extensions

More terms from Antti Karttunen, Jul 11 2017

A005078 Sum of primes = 1 mod 4 dividing n.

Original entry on oeis.org

0, 0, 0, 0, 5, 0, 0, 0, 0, 5, 0, 0, 13, 0, 5, 0, 17, 0, 0, 5, 0, 0, 0, 0, 5, 13, 0, 0, 29, 5, 0, 0, 0, 17, 5, 0, 37, 0, 13, 5, 41, 0, 0, 0, 5, 0, 0, 0, 0, 5, 17, 13, 53, 0, 5, 0, 0, 29, 0, 5, 61, 0, 0, 0, 18, 0, 0, 17, 0, 5, 0, 0, 73, 37, 5, 0, 0, 13, 0, 5, 0, 41, 0, 0, 22, 0, 29, 0, 89, 5, 13, 0, 0, 0, 5, 0, 97, 0, 0, 5, 101
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A005069, A005079, A005080, A005081, A005082, A008472, A059841.

Programs

Formula

Additive with a(p^e) = p if p = 1 (mod 4), 0 otherwise.
a(n) = A008472(n) - A005082(n) - 2*A059841(n). - Antti Karttunen, Jul 11 2017

Extensions

More terms from Antti Karttunen, Jul 11 2017

A005084 Sum of cubes of primes = 3 mod 4 dividing n.

Original entry on oeis.org

0, 0, 27, 0, 0, 27, 343, 0, 27, 0, 1331, 27, 0, 343, 27, 0, 0, 27, 6859, 0, 370, 1331, 12167, 27, 0, 0, 27, 343, 0, 27, 29791, 0, 1358, 0, 343, 27, 0, 6859, 27, 0, 0, 370, 79507, 1331, 27, 12167, 103823, 27, 343, 0, 27, 0, 0, 27, 1331, 343, 6886, 0, 205379, 27, 0, 29791, 370, 0, 0, 1358, 300763, 0, 12194, 343, 357911
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000578, A005064, A005080, A005082, A005083, A005085, A059841.

Programs

  • Mathematica
    Array[DivisorSum[#, #^3 &, And[PrimeQ@ #, Mod[#, 4] == 3] &] &, 71] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 4] == 3, p^3, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)
  • PARI
    a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%4) == 3, p^3)); \\ Michel Marcus, Jul 11 2017
  • Scheme
    (define (A005084 n) (if (= 1 n) 0 (+ (if (= 3 (modulo (A020639 n) 4)) (A000578 (A020639 n)) 0) (A005084 (A028234 n))))) ;;  Antti Karttunen, Jul 11 2017

Formula

Additive with a(p^e) = p^3 if p = 3 (mod 4), 0 otherwise.
a(n) = A005064(n) - A005080(n) - 8*A059841(n). - Antti Karttunen, Jul 11 2017

Extensions

More terms from Antti Karttunen, Jul 11 2017
Showing 1-5 of 5 results.

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

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