A347391 -id:A347391 - cp's OEIS frontend

A347394 Positions of 4's in A347381.

21, 26, 30, 32, 33, 36, 37, 40, 42, 44, 48, 49, 50, 54, 60, 65, 75, 80, 84, 90, 91, 112, 120, 125, 126, 153, 162, 175, 176, 198, 208, 220, 231, 252, 272, 275, 304, 325, 343, 368, 400, 425, 475, 546, 575, 725, 765, 775, 11132, 12750, 13167, 31262, 46893, 55660, 63825, 78155, 93500, 171171, 191301, 406406, 483771, 609609

Offset: 1

Antti Karttunen, Aug 31 2021

nonn

Question: Why the sudden rarification of terms after a(48) = 775?

Cf. A347381.

Cf. also A336702, A347391, A347392, A347393.

A347887 Odd numbers k for which A003415(sigma(k^2))-(k^2) is strictly positive and even. Here A003415 is the arithmetic derivative.

Original entry on oeis.org

201, 231, 237, 259, 273, 315, 333, 399, 429, 455, 483, 525, 555, 585, 627, 651, 665, 741, 763, 855, 903, 975, 1057, 1071, 1085, 1113, 1209, 1235, 1351, 1395, 1407, 1505, 1533, 1635, 1659, 1677, 1767, 1785, 1935, 2037, 2079, 2163, 2211, 2265, 2317, 2331, 2345, 2451, 2457, 2479, 2541, 2555, 2583, 2607, 2611, 2613

Offset: 1

Antti Karttunen, Sep 19 2021

nonn

A square root of any hypothetical odd term x in A005820 (triperfect numbers) would be a member of this sequence, because such x should be a term of A342923 [Numbers x such that A342925(x)-x = 3*A003415(x)], and as the right hand side would then certainly be even (A235992 contains all odd squares), the left hand side should also be even. See also comments in A347870 and in A347391.

Cf. A000203, A003415, A005820, A097023, A235991, A235992, A342923, A342925, A342926, A347383, A347391.

Subsequence of A347881 and of A347885. The intersection with A347882 gives A347888.

ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[1, 3000, 2], (d = ad[DivisorSigma[1, #^2]] - #^2) > 0 && EvenQ[d] &] (* Amiram Eldar, Sep 19 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA347887(n) = if(!(n%2),0,my(u=(A003415(sigma(n^2))-(n^2))); ((u>0)&&!(u%2)));

A356321 a(n) = A347381(A005940(1+n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 03 2022

nonn

This sequence tells how near sigma(x) is to each x in Doudna-tree, A005940, with x iterating over the vertices of the tree in the breadth-first fashion. Positions that correspond to perfect numbers or (hypothetical) odd triperfect numbers get values 0 and 1 respectively. 1's occur also elsewhere. (Clarified Jul 03 2023)

See the illustrations in A347391 and in A347392.

Antti Karttunen, Table of n, a(n) for n = 0..65471
Michael De Vlieger, Annotated fan style binary tree diagram labeling A005940(0..511) but using a color function for a(0..16383) where black represents 0, red 1, and magenta the largest value of a(n), n = 0..16383.
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A005940, A070939, A156552, A324054, A347381, A356320.

A000523(n) = logint(n,2);
Abincompreflen(x, y) = if(!x || !y, 0, my(xl=A000523(x), yl=A000523(y), s=min(xl,yl), k=0); x >>= (xl-s); y >>= (yl-s); while(s>=0 && !bitand(1,bitxor(x>>s,y>>s)), s--; k++); (k));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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}; \\ From A156552
A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
A252464(n) = if(1==n,0,(bigomega(n) + A061395(n) - 1));
A347381(n) = (A252464(n)-Abincompreflen(A156552(n), A156552(sigma(n))));
A356321(n) = A347381(A005940(1+n));

a(n) = A070939(n) - A356320(n).

A374464 Numbers k such that the odd parts of k and sigma(A003961(k)) are equal, where A003961 is fully multiplicative function with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 3, 6, 40, 120, 351, 702, 1000, 3000, 14040, 351000

Offset: 1

Antti Karttunen, Jul 11 2024

If x and y are included, and they are coprime (gcd(x,y) = 1), then x*y is also included.
A003961 applied to this sequence gives the odd terms of A374463, which after 1 is a subsequence of A347391.
Conjecture: the sequence is finite.

			351 = 3^3 * 13 is included as sigma(A003961(351)) = sigma(2125) = 2808, with A000265(2808) = A000265(351) = 351.

Cf. A000203, A000265, A003961, A003973, A347391, A374463.

After the initial 1, a subsequence of A348738. Cf. also A326042.

A000265(n) = (n>>valuation(n,2));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA374464(n) = (A000265(sigma(A003961(n)))==A000265(n));

{k | A000265(k) = A000265(sigma(A003961(k)))}.

A347886 Odd numbers k such that sigma(k^2) has an even number of prime factors when counted with multiplicity.

Original entry on oeis.org

1, 7, 9, 11, 13, 15, 19, 23, 25, 29, 31, 43, 47, 51, 53, 63, 67, 73, 77, 79, 81, 83, 85, 91, 97, 99, 103, 105, 111, 113, 117, 121, 123, 127, 133, 135, 143, 147, 157, 161, 163, 165, 171, 175, 177, 179, 183, 185, 191, 195, 197, 199, 203, 205, 207, 209, 211, 213, 217, 223, 225, 227, 233, 239, 241, 243, 245, 247, 251, 253

Offset: 1

Antti Karttunen, Sep 19 2021

nonn

Equally, odd numbers k such that A003415(sigma(k^2)) is even, i.e., k^2 is in A347878. See A235991.

Index entries for sequences related to sigma(n)

Cf. A000203, A001222, A003415, A005820, A235991, A342923, A342925, A347391, A347870, A347878, A347882, A347885 (complement among A005408).

Select[Range[1, 250, 2], EvenQ[PrimeOmega[DivisorSigma[1, #^2]]] &] (* Amiram Eldar, Sep 19 2021 *)

isA347886(n) = ((n%2)&&!(bigomega(sigma(n^2))%2));

A347888 Odd numbers k for which A003415(sigma(k^2))-(k^2) is strictly positive and a multiple of six. Here A003415 is the arithmetic derivative.

Original entry on oeis.org

273, 399, 651, 741, 903, 1209, 1407, 1533, 1659, 1677, 1767, 2037, 2163, 2331, 2451, 2457, 2613, 2667, 2847, 3003, 3081, 3297, 3423, 3591, 3685, 3783, 3819, 3843, 3885, 3999, 4017, 4095, 4161, 4179, 4329, 4345, 4389, 4431, 4503, 4683, 4953, 5061, 5187, 5529, 5691, 5817, 5859, 5871, 5985, 6123, 6231, 6279, 6327, 6357

Offset: 1

Antti Karttunen, Sep 19 2021

nonn

A square root of any hypothetical odd term x (if such numbers exist) in A005820 (triperfect numbers) should be a member of this sequence. See comments in A347882, A347887 and also in A347870 and in A347391.

Of the first 200 terms of A097023, 44 appear also in this sequence, the first ones being 50281, 73535, 379953, etc.

Intersection of A347882 and A347887. Subsequence of A347881 and of A347885.

Cf. A000203, A003415, A005820, A097023, A235991, A235992, A342923, A342925, A342926, A347383, A347391.

ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[1, 6500, 2], (d = ad[DivisorSigma[1, #^2]] - #^2) > 0 && Divisible[d, 6] &] (* Amiram Eldar, Sep 19 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA347888(n) = if(!(n%2),0,my(u=(A003415(sigma(n^2))-(n^2))); ((u>0)&&!(u%6)));

A351448 Odd numbers k for which A003958(sigma(k)) = 2*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

8181, 400869, 1507005, 3918213, 11151837, 22002273, 26669007, 47319957, 58170393, 73843245, 75825981, 83488077, 94338513, 108277641, 119656197, 126889821, 137740257, 163057941, 184758813, 191992437, 199226061, 202842873, 204768225, 220926933, 228160557, 258457473, 264328677, 277602471, 300496797

Offset: 1

Antti Karttunen, Feb 12 2022

nonn

Odd numbers k such that A351442(k) = 2*A003958(k).
Any hypothetical odd term of A005820, if such a term exists, should appear in this sequence, in A347391, and in A016754 (odd squares).
None of the first 33 terms is a square, and all of them except 75825981 and 204768225 are multiples of 81. Note that 81 is one of the terms of A008848 (and of A231484), squares whose sum of divisors is also square (with A000203(81) = 121).

Index entries for sequences related to sigma(n)

Odd terms in A351447.

Cf. A000203, A003958, A005820, A008848, A016754, A231484, A339905, A347391, A351442.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
isA351448(n) = (n%2 && (A003958(sigma(n)) == 2*A003958(n)));

A374199 Divisors of 19679625.

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 17, 21, 25, 27, 35, 45, 49, 51, 63, 75, 85, 105, 119, 125, 135, 147, 153, 175, 189, 225, 245, 255, 315, 343, 357, 375, 425, 441, 459, 525, 595, 675, 735, 765, 833, 875, 945, 1029, 1071, 1125, 1225, 1275, 1323, 1575, 1715, 1785, 2125, 2205, 2295, 2499, 2625, 2975, 3087, 3213, 3375, 3675, 3825, 4165

Offset: 1

Antti Karttunen, Jul 10 2024

Eleven of these divisors are also terms of A347391, and they are: 3, 5, 15, 189, 945, 2125, 6375, 9261, 46305, 401625, 19679625. It is not known whether A347391 contains any other odd terms.

Antti Karttunen, Table of n, a(n) for n = 1..128

Cf. A347391.

Divisors[19679625] (* Paolo Xausa, Jul 10 2024 *)

PARI
```
A374199(n) = divisors(19679625)[n];
```

A374221 Indices of 1's in A374200, where A374200 is the minimum value of A347381 that it obtains among unitary divisors of n larger than 1.

Original entry on oeis.org

3, 4, 5, 12, 15, 20, 21, 24, 33, 35, 36, 39, 40, 44, 45, 48, 51, 52, 55, 57, 60, 65, 68, 69, 75, 76, 80, 85, 87, 92, 93, 95, 96, 100, 105, 108, 111, 115, 116, 120, 123, 124, 129, 132, 135, 141, 145, 147, 148, 155, 156, 159, 160, 164, 165, 168, 172, 177, 180, 183, 185, 188, 189, 192, 195, 196, 201, 204, 205, 212, 213

Offset: 1

Antti Karttunen, Jul 09 2024

nonn

It seems that for all odd terms x in this sequence, A374204(x) = 1, and they form a subsequence of all terms k for which A374204(k) = 1, that might be sequence A086748.

Antti Karttunen, Table of n, a(n) for n = 1..20001

Cf. A000203, A005940, A347381, A374200, A374204, A374220 (characteristic function).

Cf. also A086748 and A347391 (very likely a subsequence).

PARI
```
isA374221 = A374220;
```

A374463 Numbers k such that the odd parts of sigma(k) and A064989(k) are equal, where A064989 shifts the prime factorization one step towards lower primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 5, 15, 189, 945, 2125, 6375, 9261, 42550, 46305, 127650, 401625, 936100, 1191400, 2808300, 3574200, 8041950, 19679625, 22469750, 58378600, 58506250, 67409250, 175135800, 175518750, 176922900, 394055550, 494334500

Offset: 1

Antti Karttunen, Jul 11 2024

Odd terms after 1 form a subsequence of A347391.

If x and y are included, and they are coprime (gcd(x,y) = 1), then x*y is also included.

			945 = 3^3 * 5 * 7 is included as A064989(945) = 2^3 * 3 * 5 = 120, and sigma(945) = 1920 = 2^7 * 3 * 5, with A000265(120) = A000265(1920) = 15.

Cf. A000203, A000265, A064989, A161942, A347391, A374464.

isA374463(n) = (A000265(sigma(n)==A000265(A064989(n))));

{k | A000265(sigma(k)) = A000265(A064989(k))}.

cp's OEIS Frontend

A347394 Positions of 4's in A347381.

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A347887 Odd numbers k for which A003415(sigma(k^2))-(k^2) is strictly positive and even. Here A003415 is the arithmetic derivative.

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A356321 a(n) = A347381(A005940(1+n)).

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A374464 Numbers k such that the odd parts of k and sigma(A003961(k)) are equal, where A003961 is fully multiplicative function with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

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A347886 Odd numbers k such that sigma(k^2) has an even number of prime factors when counted with multiplicity.

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A347888 Odd numbers k for which A003415(sigma(k^2))-(k^2) is strictly positive and a multiple of six. Here A003415 is the arithmetic derivative.

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A351448 Odd numbers k for which A003958(sigma(k)) = 2*A003958(k), where A003958 is multiplicative with a(p^e) = (p-1)^e and sigma is the sum of divisors function.

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A374199 Divisors of 19679625.

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A374221 Indices of 1's in A374200, where A374200 is the minimum value of A347381 that it obtains among unitary divisors of n larger than 1.

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