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 1-6 of 6 results.

A339897 The position of the first n in A339895.

Original entry on oeis.org

1, 8, 27, 54, 162, 343, 961, 1369, 1849, 2809, 3721, 4489, 5329, 6889, 7921, 10201, 11449, 12769, 16129, 18769, 22201, 22801, 26569, 29929, 32761, 36481, 38809, 44521, 49729, 52441, 57121, 58081, 66049, 72361, 76729, 78961, 85849, 96721, 97969, 109561, 120409, 124609, 128881, 139129, 146689, 151321, 160801, 175561
Offset: 0

Views

Author

Antti Karttunen, Dec 21 2020

Keywords

Crossrefs

Cf. A339895.

Formula

a(A339895(n)) = n, for all n >= 0.

A334201 a(n) = A056239(n) - A061395(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 2, 1, 0, 2, 0, 1, 2, 3, 0, 3, 0, 2, 2, 1, 0, 3, 3, 1, 4, 2, 0, 3, 0, 4, 2, 1, 3, 4, 0, 1, 2, 3, 0, 3, 0, 2, 4, 1, 0, 4, 4, 4, 2, 2, 0, 5, 3, 3, 2, 1, 0, 4, 0, 1, 4, 5, 3, 3, 0, 2, 2, 4, 0, 5, 0, 1, 5, 2, 4, 3, 0, 4, 6, 1, 0, 4, 3, 1, 2, 3, 0, 5, 4, 2, 2, 1, 3, 5, 0, 5, 4, 5, 0, 3, 0, 3, 5
Offset: 1

Views

Author

Antti Karttunen, May 11 2020

Keywords

Comments

a(n) is the sum of all other parts of the partition having Heinz number n except one instance of the largest part.

Links

Crossrefs

Cf. A001222, A052126, A056239, A061395, A122111, A318995.
Sum of A339895 and A339896.
Differs from A323077 for the first time at n=169, where a(169) = 6, while A323077(169) = 5.
Cf. also A334107.

Programs

  • Mathematica
    Array[Total[# /. {p_, c_} /; p > 0 :> PrimePi[p] c] - PrimePi@ #[[-1, 1]] &@ FactorInteger[#] &, 105] (* Michael De Vlieger, May 14 2020 *)
  • PARI
    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)};
A334201(n) = if(1==n,0,(bigomega(n)-1)+A334201(A064989(n)));

Formula

a(n) = A056239(n) - A061395(n) = A056239(A052126(n)).
a(n) = A318995(A122111(n)).
a(n) = a(A064989(n)) + A001222(n) - 1.
a(n) = A339895(n) + A339896(n). - Antti Karttunen, Dec 31 2020

A339896 a(n) = A056239(n) - A339894(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 2, 0, 2, 0, 1, 1, 1, 0, 2, 2, 1, 2, 1, 0, 2, 0, 2, 1, 1, 2, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 2, 2, 1, 1, 0, 2, 2, 2, 1, 1, 0, 2, 0, 1, 2, 3, 2, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 2, 2, 0, 2, 3, 1, 0, 2, 2, 1, 1, 2, 0, 2, 2, 1, 1, 1, 2, 3, 0, 2, 2, 3, 0, 2, 0, 2, 2
Offset: 1

Views

Author

Antti Karttunen, Dec 21 2020

Keywords

Links

Crossrefs

Cf. A000523, A056239, A122111, A334201, A339823, A339894, A339895.

Programs

  • PARI
    A000523(n) = if( n<1, 0, #binary(n) - 1); \\ From A000523
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A339896(n) = (A056239(n)-A000523(A122111(n)));

Formula

a(n) = A056239(n) - A339894(n).
a(n) = A334201(n) - A339895(n).
a(n) = A339823(A122111(n)).

A339893 a(n) = A000523(n) - A001222(n); floor(log_2(n)) minus the number of prime divisors of n, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 0, 3, 1, 3, 1, 2, 2, 3, 0, 2, 2, 1, 1, 3, 1, 3, 0, 3, 3, 3, 1, 4, 3, 3, 1, 4, 2, 4, 2, 2, 3, 4, 0, 3, 2, 3, 2, 4, 1, 3, 1, 3, 3, 4, 1, 4, 3, 2, 0, 4, 3, 5, 3, 4, 3, 5, 1, 5, 4, 3, 3, 4, 3, 5, 1, 2, 4, 5, 2, 4, 4, 4, 2, 5, 2, 4, 3, 4, 4, 4, 0, 5, 3, 3, 2, 5, 3, 5, 2, 3
Offset: 1

Views

Author

Antti Karttunen, Dec 21 2020

Keywords

Links

Crossrefs

Cf. A000523, A001222, A029744 (positions of 0's), A339895.
Cf. also A339823, A342657 [= a(A156552(n))].

Programs

  • PARI
    A339893(n) = (#binary(n) - 1 - bigomega(n));

Formula

a(n) = A000523(n) - A001222(n).
a(n) = A339895(A122111(n)).

A339894 a(n) = A000523(A122111(n)).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 2, 3, 3, 5, 3, 6, 4, 4, 2, 7, 3, 8, 4, 5, 5, 9, 3, 4, 6, 4, 5, 10, 4, 11, 3, 6, 7, 5, 4, 12, 8, 7, 4, 13, 5, 14, 6, 5, 9, 15, 4, 6, 5, 8, 7, 16, 5, 6, 5, 9, 10, 17, 5, 18, 11, 6, 3, 7, 6, 19, 8, 10, 6, 20, 5, 21, 12, 6, 9, 7, 7, 22, 5, 5, 13, 23, 6, 8, 14, 11, 6, 24, 6, 8, 10, 12, 15, 9, 4, 25
Offset: 1

Views

Author

Antti Karttunen, Dec 21 2020

Keywords

Links

Crossrefs

Cf. A000523, A122111, A339895, A339896.

Programs

  • PARI
    A000523(n) = if( n<1, 0, #binary(n) - 1);
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A339894(n) = A000523(A122111(n));

Formula

a(n) = A000523(A122111(n)).

A342657 The difference between floor(log_2(.)) of and the number of prime factors in A156552(n) (when counted with multiplicity).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 1, 1, 0, 2, 0, 3, 1, 3, 0, 3, 0, 4, 1, 3, 0, 2, 0, 3, 3, 5, 1, 1, 0, 7, 3, 3, 0, 4, 0, 5, 2, 5, 0, 4, 0, 2, 4, 6, 0, 3, 1, 5, 5, 7, 0, 4, 0, 9, 3, 2, 3, 4, 0, 6, 7, 4, 0, 3, 0, 10, 2, 7, 1, 5, 0, 5, 1, 11, 0, 3, 3, 11, 5, 3, 0, 2, 1, 8, 7, 11, 4, 4, 0, 3, 3, 3, 0, 5, 0, 7, 2
Offset: 2

Views

Author

Antti Karttunen, Mar 18 2021

Keywords

Links

Crossrefs

Cf. A000430 (positions of 0's), A000523, A001222, A003961, A061395, A070939, A156552, A246277, A322993, A325134, A339893, A342655, A342656.
Cf. also A339895.

Programs

  • PARI
    A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res};
A342657(n) = { my(u=A156552(n)); (#binary(u)-bigomega(u))-1; };

Formula

a(n) = A339893(A156552(n)) = A339893(A322993(n)) = A000523(A156552(n)) - A001222(A156552(n)).
a(n) = (A252464(n)-A342655(n))-1 = (A325134(n)-A342655(n)) - 2.
a(p) = a(p^2) = 0 for all primes p. (Second part added Jul 27 2023)
a(A003961(n)) = a(2*A246277(n)) = a(n).
Showing 1-6 of 6 results.

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