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.

A351446 Numbers k for which A003958(sigma(k)) = A003958(k), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

Original entry on oeis.org

1, 6, 10, 26, 28, 49, 54, 74, 122, 126, 146, 294, 314, 386, 408, 490, 496, 554, 626, 680, 794, 842, 914, 1082, 1226, 1232, 1274, 1322, 1346, 1466, 1514, 1560, 1754, 1768, 1994, 2186, 2306, 2402, 2426, 2474, 2642, 2646, 2762, 2906, 3242, 3314, 3360, 3506, 3626, 3672, 3746, 3808, 3866, 3986, 4034
Offset: 1

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Author

Antti Karttunen, Feb 12 2022

Keywords

Comments

Numbers k for which A351442(k) = A003958(k), or equally, for which k = A351444(k) = A322582(k) + A351442(k).

Links

Crossrefs

Cf. A000203, A003958, A322582, A351442, A351447.
Fixed points of A351444, positions of zeros in A351445.
Subsequences: A000396, A351443 (odd terms), A351440, A336702 (numbers k for which A064989(sigma(k)) = A064989(k)).

Programs

A353634 Nondeficient numbers k such that phi(k) = phi(sigma(k)) and A003958(k) = A003958(sigma(k)).

Original entry on oeis.org

234728, 280904, 461168, 463112, 604136, 742664, 909872, 996008, 1065896, 1191944, 1204424, 1224392, 1465256, 1527656, 1620008, 1757288, 1758536, 1956848, 1985672, 2081768, 2102984, 2358824, 2376296, 2405552, 2518568, 2543528, 2589704, 2670824, 2820584, 2899208, 2912936, 3014024, 3151304, 3196232, 3374696, 3432104
Offset: 1

Views

Author

Antti Karttunen, May 04 2022

Keywords

Links

Crossrefs

Intersection of A023196 and A353635.
Cf. A000010, A000203, A003958, A006872, A062401, A351440, A351442, A351446.

Programs

  • PARI
    A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
isA353634(n) = { my(s=sigma(n)); if(s<(2*n),return(0)); ((eulerphi(s)==eulerphi(n)) && (A003958(s)==A003958(n))); };

A353635 Numbers k such that phi(k) = phi(sigma(k)) and A003958(k) = A003958(sigma(k)).

Original entry on oeis.org

1, 26, 74, 122, 146, 314, 386, 554, 626, 794, 842, 914, 1082, 1226, 1322, 1346, 1466, 1514, 1754, 1994, 2186, 2306, 2402, 2426, 2474, 2642, 2762, 2906, 3242, 3314, 3506, 3746, 3866, 3986, 4034, 4274, 4682, 4946, 5114, 5186, 5594, 5714, 5834, 6122, 6434, 6506, 6626, 7034, 7466, 8042, 8114, 8354, 8522, 8546, 8714, 8882
Offset: 1

Views

Author

Antti Karttunen, May 04 2022

Keywords

Comments

Question 1: Are there any odd terms after the initial 1?
Interestingly, most of the terms seem to belong to a set where the abundancy index (ratio sigma(n)/n) converges towards 3/2. But there are exceptions, see A353634 for example.

Links

Crossrefs

Intersection of A006872 and A351446. A353634 lists the nondeficient terms.
Cf. A000010, A000203, A003958, A062401, A351440, A351442.

Programs

  • PARI
    A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
isA353635(n) = { my(s=sigma(n)); ((eulerphi(s)==eulerphi(n)) && (A003958(s)==A003958(n))); };

A353746 Numbers k for which phi(sigma(k)) + A064989(sigma(k)) is equal to phi(k) + A064989(k).

Original entry on oeis.org

1, 114, 618, 2428, 3868, 11706, 16012, 23946, 2031108, 2938902, 3531102, 10475862, 98250486, 205600756
Offset: 1

Views

Author

Antti Karttunen, May 06 2022

Keywords

Comments

Numbers k such that A353747(k) = A353747(sigma(k)), or equally that A062401(k) + A350073(k) = A000010(k) + A064989(k).
When A003961(x) is substituted for k, the condition becomes: numbers x such that phi(A003973(x))+A326042(x) = A003972(x)+x, i.e. A353747(A003973(x)) - x = A003972(x).
a(15) > 1442840576, if it exists.

Links

Crossrefs

Cf. A000010, A000203, A062401, A064989, A350073, A353747.
Cf. A003961, A003972, A003973, A326042.
Cf. also A348748, A348749, A351440.

Programs

  • PARI
    A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
isA353746(n) = { my(s=sigma(n)); ((eulerphi(s)+A064989(s))==(eulerphi(n)+A064989(n))); };
Showing 1-4 of 4 results.

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