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 1-4 of 4 results.

A168128 Numbers Sum_(j=1..k) sigma(j) / sigma(k) when this expression is an integer.

Original entry on oeis.org

1, 2, 23, 247, 699, 55921, 70936, 152730, 478017, 4879815, 12909834588, 223070292558, 583407686653, 601718433498
Offset: 1

Views

Author

Jaroslav Krizek, Nov 18 2009, Dec 04 2009

Keywords

Comments

Numbers A024916(k)/A000203(k) when it is an integer.
Corresponding values of k, sigma(k) and Sum_(j=1..k) sigma(j) are given in A168127, A168129 and A168130. - Jaroslav Krizek, Nov 21 2009

Examples

			a(3)=23 because A024916(29) / A000203(29) = 690 / 30 = 23 is an integer, where 29 is A168127(3).

Crossrefs

Cf. A000203, A024916, A168127, A168129, A168130.

Programs

  • PARI
    m=0; for(k=1, 11413204, s=sigma(k); m=m+s; if(m%s==0, print1(m/s ", "))) \\ Donovan Johnson, Oct 16 2013

Extensions

a(8)-a(10) from Donovan Johnson, Oct 16 2013
a(11)-a(14) from Giovanni Resta, Jun 06 2016
Name edited by Michel Marcus, Feb 22 2020

A168129 Numbers sigma(k) when Sum_(j=1..k) sigma(j) / sigma(k) is an integer.

Original entry on oeis.org

1, 4, 30, 444, 1767, 86250, 97920, 185700, 584640, 21954842, 17654544432, 357550809280, 709346653572, 1061551370880
Offset: 1

Views

Author

Jaroslav Krizek, Nov 18 2009, Dec 04 2009

Keywords

Comments

Numbers A000203(k) when A024916(k) / A000203(k) is an integer.
Corresponding values of k, Sum_(j=1..k) sigma(k) and Sum_(j=1..k) sigma(j)/sigma(k) are given in A168127, A168130 and A168128. - Jaroslav Krizek, Nov 21 2009

Examples

			a(3) = 30 because A024916(29) / A000203(29) = 690 / 30 = 23 is an integer, where 29 is A168127(3).

Crossrefs

Cf. A000203, A024916, A168127, A168128, A168130.

Programs

  • PARI
    v=vector(10); c=0; m=0; for(k=1, 11413204, s=sigma(k); m=m+s; if(m%s==0, c++; v[c]=s)); v=vecsort(v); for(i=1, c, print1(v[i] ", ")) \\ Donovan Johnson, Oct 16 2013

Formula

a(n) = A000203(A168127(n)). - Michel Marcus, Feb 22 2020

Extensions

a(8)-a(10) from Donovan Johnson, Oct 16 2013
a(11)-a(14) from Giovanni Resta, Jun 06 2016
Name edited by Michel Marcus, Feb 22 2020

A168130 Numbers Sum_(j=1..k) sigma(j) when Sum_(j=1..k) sigma(j) / sigma(k) is an integer.

Original entry on oeis.org

1, 8, 690, 109668, 1235133, 5475784320, 6118230000, 28361961000, 279467858880, 107135567314230, 227917248343616414016, 79758963630439261338240, 413838290195487519174516, 638755027963568013738240
Offset: 1

Views

Author

Jaroslav Krizek, Nov 18 2009, Dec 04 2009

Keywords

Comments

Numbers A024916(k) when A024916(k) / A000203(k) is an integer.
Corresponding values of k, sigma(k), and Sum_(j=1..k) sigma(j)/sigma(k) are given in A168127, A168129, and A168128. - Jaroslav Krizek, Nov 21 2009

Examples

			a(3)= 690 = A024916(29) because A024916(29) / A000203(29) = 690 / 30 = 23 is an integer, where 29 is A168127(3).

Crossrefs

Cf. A000203, A024916, A168127, A168128, A168129.

Programs

  • PARI
    m=0; for(k=1, 11413204, s=sigma(k); m=m+s; if(m%s==0, print1(m ", "))) \\ Donovan Johnson, Oct 16 2013

Formula

a(n) = A024916(A168127(n)). - Michel Marcus, Feb 22 2020

Extensions

a(8)-a(10) from Donovan Johnson, Oct 16 2013
a(11)-a(14) from Giovanni Resta, Jun 06 2016
Name edited by Michel Marcus, Feb 22 2020

A227904 Numbers k such that Sum_{j=1..k} antisigma(j) == 0 (mod sigma(k)).

Original entry on oeis.org

1, 2, 39, 78, 100, 126, 434, 501, 1313, 54111, 359466, 523219, 6601441, 8034674, 54092207, 64149290, 158882288, 3016740661, 20951813922, 52815759197, 120508871819
Offset: 1

Views

Author

Paolo P. Lava, Oct 15 2013

Keywords

Comments

Tested up to k = 10^6.
a(22) > 2.1774*10^11. - Kevin P. Thompson, Jan 10 2022

Examples

			Sum_{j=1..39} antisigma(j) = 9408, sigma(39) = 56 and 9408 mod 56 = 0, so 39 is a term.

Crossrefs

Cf. A000203, A024816, A076664, A168127.

Programs

  • Maple
    with(numtheory); P:=proc(q) local a, n; a:=0;
for n from 3 to q do a:=a+n*(n+1)/2-sigma(n);
if (a mod sigma(n))=0 then print(n); fi; od; end: P(10^9);

Extensions

a(13)-a(17) from Donovan Johnson, Oct 15 2013
a(18)-a(21) from Kevin P. Thompson, Jan 10 2022
Showing 1-4 of 4 results.

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