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.

A354088 Sum of divisors function conjugated by Pythagorean prime shift: a(n) = A348747(sigma(A348746(n))).

Original entry on oeis.org

1, 1, 2, 5, 7, 2, 1, 3, 31, 7, 2, 10, 4, 1, 14, 121, 6, 31, 3, 35, 2, 2, 2, 6, 106, 4, 10, 5, 19, 14, 1, 35, 4, 6, 7, 155, 14, 3, 8, 21, 8, 2, 11, 10, 217, 2, 2, 242, 38, 106, 12, 20, 31, 10, 14, 3, 6, 19, 6, 70, 29, 1, 31, 1069, 28, 4, 13, 30, 4, 7, 4, 93, 12, 14, 212, 15, 2, 8, 3, 847, 781, 8, 14, 10, 42, 11, 38
Offset: 1

Views

Author

Antti Karttunen, May 17 2022

Keywords

Comments

This is variant of A326042, and like that sequence, also this one is multiplicative.

Links

Crossrefs

Cf. A000203, A326042, A348744, A348746, A348747, A354089.
Cf. also A326042, A354096 for variants.

Programs

  • PARI
    A348746(n) = { my(f=factor(n)); for(k=1,#f~, if(2==f[k,1], f[k,1]=3, if(3==f[k,1], f[k,1]=5, if(1==(f[k,1]%4), for(i=1+primepi(f[k,1]),oo,if(1==(prime(i)%4), f[k,1]=prime(i); break)))))); factorback(f); };
A348747(n) = { my(f=factor(n)); for(k=1,#f~, if(f[k,1]<=3, f[k,1]--, if(5==f[k,1], f[k,1]=3, if(1==(f[k,1]%4), forstep(i=primepi(f[k,1])-1,0,-1,if(1==(prime(i)%4), f[k,1]=prime(i); break)))))); factorback(f); };
A354088(n) = A348747(sigma(A348746(n)));

Formula

Multiplicative with a(p^e) = A348747((q^(e+1)-1)/(q-1)), where q = A348744(A000720(p)).
a(n) = A348747(A354089(n)) = A348747(A000203(A348746(n))).

A354089 Sum of divisors function applied to Pythagorean prime shift: a(n) = sigma(A348746(n)).

Original entry on oeis.org

1, 4, 6, 13, 14, 24, 8, 40, 31, 56, 12, 78, 18, 32, 84, 121, 30, 124, 20, 182, 48, 48, 24, 240, 183, 72, 156, 104, 38, 336, 32, 364, 72, 120, 112, 403, 42, 80, 108, 560, 54, 192, 44, 156, 434, 96, 48, 726, 57, 732, 180, 234, 62, 624, 168, 320, 120, 152, 60, 1092, 74, 128, 248, 1093, 252, 288, 68, 390, 144, 448, 72
Offset: 1

Views

Author

Antti Karttunen, May 17 2022

Keywords

Links

Crossrefs

Inverse Möbius transform of A348746.
Cf. A000203, A000720, A348744, A354088.
Cf. A003973, A354093 for variants.

Programs

  • PARI
    A348746(n) = { my(f=factor(n)); for(k=1,#f~, if(2==f[k,1], f[k,1]=3, if(3==f[k,1], f[k,1]=5, if(1==(f[k,1]%4), for(i=1+primepi(f[k,1]),oo,if(1==(prime(i)%4), f[k,1]=prime(i); break)))))); factorback(f); };
A354089(n) = sigma(A348746(n));

Formula

Multiplicative with a(p^e) = (q^(e+1)-1)/(q-1) where q = A348744(A000720(p)).
a(n) = A000203(A348746(n)).
a(n) = Sum_{d|n} A348746(d).

A108548 Fully multiplicative with a(prime(j)) = A108546(j), where A108546 is the lexicographically earliest permutation of primes such that after 2 the forms 4*k+1 and 4*k+3 alternate, and prime(j) is the j-th prime in A000040.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11, 14, 15, 16, 17, 18, 19, 20, 21, 26, 29, 24, 25, 22, 27, 28, 23, 30, 37, 32, 39, 34, 35, 36, 31, 38, 33, 40, 41, 42, 43, 52, 45, 58, 53, 48, 49, 50, 51, 44, 47, 54, 65, 56, 57, 46, 61, 60, 59, 74, 63, 64, 55, 78, 73, 68, 87, 70, 67, 72
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 10 2005

Keywords

Comments

Multiplicative with a(2^e) = 2^e, else if p is the m-th prime then a(p^e) = q^e where q is the m/2-th prime of the form 4*k + 3 (A002145) for even m and a(p^e) = r^e where r is the (m-1)/2-th prime of the form 4*k + 1 (A002144) for odd m. - David A. Corneth, Apr 25 2022
Permutation of the natural numbers with fixed points A108549: a(A108549(n)) = A108549(n).

Links

Crossrefs

Cf. A002144, A002145, A049084, A108546, A108549 (fixed points), A332808 (inverse permutation).
Cf. also A332815, A332817 (this permutation applied to Doudna tree and its mirror image), also A332818, A332819.
Cf. also A267099, A332212 and A348746 for other similar mappings.

Programs

  • Mathematica
    terms = 72;
A111745 = Module[{prs = Prime[Range[2 terms]], m3, m1, min},
     m3 = Select[prs, Mod[#, 4] == 3&];
     m1 = Select[prs, Mod[#, 4] == 1&];
     min = Min[Length[m1], Length[m3]];
     Riffle[Take[m3, min], Take[m1, min]]];
A108546[n_] := If[n == 1, 2, A111745[[n - 1]]];
A049084[n_] := PrimePi[n]*Boole[PrimeQ[n]];
a[n_] := If[n == 1, 1, Module[{p, e}, Product[{p, e} = pe; A108546[A049084[p]]^e, {pe, FactorInteger[n]}]]];
Array[a, terms] (* Jean-François Alcover, Nov 19 2021, using Harvey P. Dale's code for A111745 *)
  • PARI
    up_to = 26927; \\ One of the prime fixed points.
A108546list(up_to) = { my(v=vector(up_to), p,q); v[1] = 2; v[2] = 3; v[3] = 5; for(n=4,up_to, p = v[n-2]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[n] = q); (v); };
v108546 = A108546list(up_to);
A108546(n) = v108546[n];
A108548(n) = { my(f=factor(n)); f[,1] = apply(A108546,apply(primepi,f[,1])); factorback(f); }; \\ Antti Karttunen, Apr 25 2022

Extensions

Name edited by Antti Karttunen, Apr 25 2022

A348736 a(n) = n - A326042(n), where A326042(n) = A064989(sigma(A003961(n))).

Original entry on oeis.org

0, 1, 1, -7, 4, 4, 5, 5, -20, 9, 6, -10, 9, 12, 13, -33, 14, -11, 17, 9, 17, 17, 17, 18, -9, 22, 5, 6, 28, 28, 14, -23, 23, 31, 33, -283, 27, 36, 31, 37, 34, 38, 41, -11, 16, 40, 39, -50, -36, 16, 45, 8, 47, 32, 50, 50, 53, 57, 30, 38, 48, 45, 5, -1027, 61, 56, 63, 35, 57, 68, 40, -15, 70, 64, 7, 54, 67, 70, 69, 31
Offset: 1

Views

Author

Antti Karttunen, Nov 02 2021

Keywords

Links

Crossrefs

Cf. A000203, A003961, A064989, A326042, A348746 [= n - A161942(n)], A348940.
Cf. A348737, A348738 (positions of positive terms), A348739 (of negative terms).

Programs

  • PARI
    A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));
A348736(n) = (n-A326042(n));

Formula

a(n) = n - A064989(A161942(A003961(n))).

A354202 Fully multiplicative with a(p^e) = A354200(A000720(p))^e.

Original entry on oeis.org

1, 5, 7, 25, 13, 35, 11, 125, 49, 65, 19, 175, 17, 55, 91, 625, 29, 245, 23, 325, 77, 95, 31, 875, 169, 85, 343, 275, 37, 455, 43, 3125, 133, 145, 143, 1225, 41, 115, 119, 1625, 53, 385, 47, 475, 637, 155, 59, 4375, 121, 845, 203, 425, 61, 1715, 247, 1375, 161, 185, 67, 2275, 73, 215, 539, 15625, 221, 665, 71, 725
Offset: 1

Views

Author

Antti Karttunen, May 23 2022

Keywords

Comments

Permutation of A007310. Preserves the prime signature.

Links

Crossrefs

Cf. A007310 (terms sorted into ascending order), A354200, A354203 (left inverse), A354204 (Möbius transform), A354205 (inverse Möbius transform).
Cf. also A003961, A108548, A267099, A332818, A348746, A354091 for similar constructions.

Programs

  • PARI
    A354200(n) = if(1==n,5,my(p=prime(n), m=p%4); forprime(q=1+p,,if(m==(q%4),return(q))));
A354202(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); factorback(f); };

A354091 Fully multiplicative prime shift where the primes of the form 3k+2 are replaced by the next larger such prime, and primes of the form 3k and 3k+1 stay as they are.

Original entry on oeis.org

1, 5, 3, 25, 11, 15, 7, 125, 9, 55, 17, 75, 13, 35, 33, 625, 23, 45, 19, 275, 21, 85, 29, 375, 121, 65, 27, 175, 41, 165, 31, 3125, 51, 115, 77, 225, 37, 95, 39, 1375, 47, 105, 43, 425, 99, 145, 53, 1875, 49, 605, 69, 325, 59, 135, 187, 875, 57, 205, 71, 825, 61, 155, 63, 15625, 143, 255, 67, 575, 87, 385, 83, 1125
Offset: 1

Views

Author

Antti Karttunen, May 17 2022

Keywords

Comments

Permutation of odd numbers. Preserves the prime signature.

Examples

			The primes in A003627 are replaced by the next prime in that sequence, as: 2 -> 5 -> 11 -> 17 -> 23 -> 29 -> 41 -> ..., while other kinds of primes (A002476) stay intact, thus for 60 = 2^2 * 3^1 * 5^1, we have a(60) = 5^2 * 3^1 * 11^1 = 825.

Links

Crossrefs

Cf. A003627, A007645, A074941, A353815.
Cf. A354092 (left inverse), A354093 (inverse Möbius transform), A354094 (Möbius transform), A354095, A354096.
Cf. also A003961, A332818, A348746 for similar constructions.

Programs

  • PARI
    A354091(n) = { my(f=factor(n)); for(k=1,#f~, if(2==(f[k,1]%3), for(i=1+primepi(f[k,1]),oo,if(2==(prime(i)%3), f[k,1]=prime(i); break)))); factorback(f); };

Formula

Fully multiplicative with a(A003627(n)) = A003627(1+n), a(A007645(n)) = A007645(n).
For all n >= 1, A354092(a(n)) = n.
For all n >= 1, A046523(a(n)) = A046523(n) and A074941(a(n)) = A074941(n).

A348744 Lexicographically earliest bijection from primes to odd primes where each prime of the form 4k+1 is mapped to the next larger prime that is of the same form.

Original entry on oeis.org

3, 5, 13, 7, 11, 17, 29, 19, 23, 37, 31, 41, 53, 43, 47, 61, 59, 73, 67, 71, 89, 79, 83, 97, 101, 109, 103, 107, 113, 137, 127, 131, 149, 139, 157, 151, 173, 163, 167, 181, 179, 193, 191, 197, 229, 199, 211, 223, 227, 233, 241, 239, 257, 251, 269, 263, 277, 271, 281, 293, 283, 313, 307, 311, 317, 337, 331, 349, 347
Offset: 1

Views

Author

Antti Karttunen, Nov 02 2021

Keywords

Comments

Terms of A002144 map each to the next term there, as: 5 -> 13 -> 17 -> 29 -> 37 -> 41, etc., and the remaining positions are filled with the terms of A002145: 3, 7, 11, 19, 23, 31, 43, etc., which gives the result that 2 is mapped to 3, 3 is mapped 5, and the rest of 4k+3 primes are fixed.

Crossrefs

Cf. A000040, A002144, A002145, A348745, A348746.
Cf. also A108546, A332211.

Programs

  • PARI
    up_to = 10000;
A348744list(up_to) = { my(v=vector(up_to), xs=Map(), i=2, p, q); mapput(xs,v[1]=3,1); for(n=2,up_to, p = prime(n); if(1==(p%4), for(k=1+n,oo,q=prime(k);if((1==(q%4))&&!mapisdefined(xs,q),v[n]=q;break)), while(mapisdefined(xs,prime(i)), i++); v[n] = prime(i)); mapput(xs,v[n],n)); (v); };
v348744 = A348744list(up_to);
A348744(n) = v348744[n];

Formula

a(n) = A348746(A000040(n)).

A348747 Fully multiplicative with a(2) = 1, a(3) = 2, a(5) = 3, a(A002144(1+n)) = A002144(n) and a(A002145(1+n)) = a(A002145(1+n)) for all n >= 1, where A002144 and A002145 give the primes of the form 4k+1 and 4k+3 respectively.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 7, 1, 4, 3, 11, 2, 5, 7, 6, 1, 13, 4, 19, 3, 14, 11, 23, 2, 9, 5, 8, 7, 17, 6, 31, 1, 22, 13, 21, 4, 29, 19, 10, 3, 37, 14, 43, 11, 12, 23, 47, 2, 49, 9, 26, 5, 41, 8, 33, 7, 38, 17, 59, 6, 53, 31, 28, 1, 15, 22, 67, 13, 46, 21, 71, 4, 61, 29, 18, 19, 77, 10, 79, 3, 16, 37, 83, 14, 39, 43, 34, 11, 73
Offset: 1

Views

Author

Antti Karttunen, Nov 02 2021

Keywords

Crossrefs

Left inverse of A348746.
Cf. A000265, A000720, A002144, A002145, A348745.
Cf. also A064989, A332819 for similar maps.

Programs

  • PARI
    A348747(n) = { my(f=factor(n)); for(k=1,#f~, if(f[k,1]<=3, f[k,1]--, if(5==f[k,1], f[k,1]=3, if(1==(f[k,1]%4), forstep(i=primepi(f[k,1])-1,0,-1,if(1==(prime(i)%4), f[k,1]=prime(i); break)))))); factorback(f); };

Formula

Fully multiplicative with a(p) = A348745(A000720(p)).
a(A348746(n)) = n.
a(2n) = a(A000265(n)) = a(n).
Showing 1-8 of 8 results.

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