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.

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).

A354096 a(n) = A354092(sigma(A354091(n))).

Original entry on oeis.org

1, 3, 1, 31, 3, 3, 1, 39, 13, 9, 9, 31, 7, 3, 3, 295, 3, 39, 2, 93, 1, 27, 6, 39, 133, 21, 2, 31, 21, 9, 1, 1953, 9, 9, 3, 403, 19, 6, 7, 117, 3, 3, 5, 279, 39, 18, 27, 295, 57, 399, 3, 217, 6, 6, 27, 39, 2, 63, 9, 93, 31, 3, 13, 19531, 21, 27, 11, 93, 6, 9, 21, 507, 37, 57, 133, 62, 9, 21, 2, 885, 25, 9, 18, 31, 9
Offset: 1

Views

Author

Antti Karttunen, May 17 2022

Keywords

Links

Crossrefs

Cf. A000203, A003627, A007949, A010872, A329963, A354091, A354092, A354093, A354095.
Cf. also A326042, A354088 for variants.

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); };
A354092(n) = { my(f=factor(n)); for(k=1,#f~, if(2==(f[k,1]%3), if(2==f[k,1], f[k,1]--, forstep(i=primepi(f[k,1])-1,0,-1,if(2==(prime(i)%3), f[k,1]=prime(i); break))))); factorback(f); };
A354096(n) = A354092(sigma(A354091(n)));

Formula

Multiplicative with a(p^e) = A354092((q^(e+1)-1)/(q-1)), where q = A003627(1+n) if p = A003627(n), otherwise q = p.
a(n) = A354092(A354093(n)) = A354095(A354091(n)) = A354092(A000203(A354091(n))).
For all n >= 1, A010872(a(n)) = A010872(A354095(n)).
For all k in A329963, A007949(a(k)) = A007949(sigma(k)) = A354100(k) = 0.

A354205 a(n) = sigma(A354202(n)), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)).

Original entry on oeis.org

1, 6, 8, 31, 14, 48, 12, 156, 57, 84, 20, 248, 18, 72, 112, 781, 30, 342, 24, 434, 96, 120, 32, 1248, 183, 108, 400, 372, 38, 672, 44, 3906, 160, 180, 168, 1767, 42, 144, 144, 2184, 54, 576, 48, 620, 798, 192, 60, 6248, 133, 1098, 240, 558, 62, 2400, 280, 1872, 192, 228, 68, 3472, 74, 264, 684, 19531, 252, 960, 72
Offset: 1

Views

Author

Antti Karttunen, May 23 2022

Keywords

Links

Crossrefs

Cf. A000203, A000290 (positions of odd terms), A000720, A354200, A354202, A354204, A354206.
Cf. A003973, A354089, A354093 for variants.

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))));
A354205(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); sigma(factorback(f)); };
\\ Alternatively:
A354205(n) = sumdiv(n,d,A354202(d));

Formula

Multiplicative with a(p^e) = (q^(e+1)-1)/(q-1) where q = A354200(A000720(p)).
a(n) = A000203(A354202(n)).
a(n) = Sum_{d|n} A354202(d).

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).

A354095 a(n) = A354092(sigma(n)).

Original entry on oeis.org

1, 3, 1, 7, 3, 3, 1, 6, 13, 9, 3, 7, 7, 3, 3, 31, 9, 39, 2, 21, 1, 9, 3, 6, 31, 21, 2, 7, 6, 9, 1, 63, 3, 27, 3, 91, 19, 6, 7, 18, 21, 3, 5, 21, 39, 9, 3, 31, 57, 93, 9, 49, 27, 6, 9, 6, 2, 18, 6, 21, 31, 3, 13, 127, 21, 9, 11, 63, 3, 9, 9, 78, 37, 57, 31, 14, 3, 21, 2, 93, 25, 63, 21, 7, 27, 15, 6, 18, 18, 117, 7
Offset: 1

Views

Author

Antti Karttunen, May 17 2022

Keywords

Links

Crossrefs

Cf. A000203, A007949, A010872, A354092, A354093, A354096.
Cf. also A350073.

Programs

  • PARI
    A354095(n) = { my(f=factor(sigma(n))); for(k=1,#f~, if(2==(f[k,1]%3), if(2==f[k,1], f[k,1]--, forstep(i=primepi(f[k,1])-1,0,-1,if(2==(prime(i)%3), f[k,1]=prime(i); break))))); factorback(f); };

Formula

Multiplicative with a(p^e) = A354092(1 + p + p^2 + ... + p^e).
a(n) = A354092(A000203(n)).
For all n >= 1, A010872(a(n)) = A010872(A354096(n)), A007949(a(n)) = A007949(A000203(n)).
Showing 1-5 of 5 results.

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