John L. Drost - cp's OEIS frontend

A329963 Numbers k such that sigma(k) is not divisible by 3.

1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 52, 57, 61, 63, 64, 67, 73, 75, 76, 79, 81, 84, 91, 93, 97, 100, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 133, 139, 144, 148, 151, 156, 157, 163, 171, 172, 175, 181, 183, 189, 192, 193, 199, 201, 208, 211, 217, 219, 223, 225, 228, 229

Offset: 1

John L. Drost, Nov 25 2019

nonn

A number k is in the sequence iff in its prime factorization, all primes p == 1 (mod 3) occur to such a power p^e that e != 2 (mod 3), and all primes == 2 (mod 3) occur to even powers. (3 can occur to any power.) This sequence is similar but not identical to many others; in particular, 343 is in this sequence, but not in A034022. (And here we don't have 196, although it is in A034022). - First sentence corrected and additional notes added by Antti Karttunen, Jul 03 2024, see also Robert Israel's Nov 09 2016 comment in A087943.

The asymptotic density of this sequence is 0 (Dressler, 1975). - Amiram Eldar, Jul 23 2020

Robert Israel, Table of n, a(n) for n = 1..10000
Tewodros Amdeberhan, Victor H. Moll, Vaishavi Sharma, and Diego Villamizar, Arithmetic properties of the sum of divisors, arXiv:2007.03088 [math.NT], 2020. See p. 15 ff. [Note: the "if and only if" condition given in the beginning of Theorem 7.1 is for A003136, not for this sequence. - _Antti Karttunen_, Jul 04 2024]
Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.

Complement of A087943. Positions of zeros in A354100, nonzeros in A074941.
Cf. A000203, A353815 (characteristic function).
Setwise difference A003136 \ A088535.
Subsequences: A002476, A068228, A351537, A374135.
Cf. also A088232.
Not the same as A034022.

[k:k in [1..200]| DivisorSigma(1,k) mod 3 ne 0]; // Marius A. Burtea, Jan 02 2020

select(t -> numtheory:-sigma(t) mod 3 <> 0, [$1..200]); # Robert Israel, Jan 01 2020

Select[Range[200], !Divisible[DivisorSigma[1, #], 3] &] (* Amiram Eldar, Nov 25 2019 *)

isok(k) = (sigma(k) % 3) != 0; \\ Michel Marcus, Nov 26 2019

isA329963 = A353815; \\ Antti Karttunen, Jul 03 2024

More terms from Joshua Oliver, Nov 26 2019

Data section further extended up to a(71), to better differentiate from nearby sequences - Antti Karttunen, Jul 04 2024

A287298 a(n) is the largest square with distinct digits in base n.

Original entry on oeis.org

1, 1, 225, 576, 38025, 751689, 10323369, 355624164, 9814072356, 279740499025, 8706730814089, 23132511879129, 11027486960232964, 435408094460869201, 18362780530794065025, 48470866291337805316, 39207739576969100808801, 1972312183619434816475625, 104566626183621314286288961

Offset: 2

John L. Drost, May 22 2017

a(n) does not always have n digits in base n. If n is 5 mod 8 then a number which contains all the digits in base n is congruent to (n-1)n/2 mod (n-1). It will be then divisible by a single power of 2 and not a square.

a(22) = 340653564758245010607213613056. - Chai Wah Wu, May 24 2017

			a(4)=225 which is 3201 in base 4. Higher squares have at least 5 digits in base 4.

from gmpy2 import isqrt, mpz, digits
def A287298(n): # assumes n <= 62
    m = isqrt(mpz(''.join(digits(i,n) for i in range(n-1,-1,-1)),n))
    m2 = m**2
    d = digits(m2,n)
    while len(set(d)) < len(d):
        m -= 1
        m2 -= 2*m+1
        d = digits(m2,n)
    return int(m2) # Chai Wah Wu, May 24 2017

Added a(16)-a(20) and corrected a(12) by Chai Wah Wu, May 24 2017

A225888 Primes p such that neither 2 nor 3 are primitive roots, but together 2 and 3 generate the nonzero residues mod p.

Original entry on oeis.org

41, 103, 109, 151, 157, 229, 251, 271, 277, 367, 397, 683, 733, 761, 967, 971, 991, 1051, 1069, 1163, 1181, 1289, 1303, 1429, 1471, 1543, 1759, 1783, 1789, 1811, 1879, 2003, 2297, 2411, 2441, 2551, 2749, 2791, 2887, 2917, 3061, 3079, 3109, 3229, 3251, 3301, 3319

Offset: 1

John L. Drost, May 19 2013

nonn

			2 has multiplicative order 20 mod 41, 3 has order 8 mod 41 so neither is a primitive root. The subgroup 2 and 3 generate together will have order lcm(20,8) = 40 so 2 and 3 generate all nonzero residues.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001122, A001123.

is(n)=if(n>40 && isprime(n), my(a=znorder(Mod(2,n)),b); if(a==n-1,return(0)); b=znorder(Mod(3,n)); bCharles R Greathouse IV, May 19 2013

A212605 a(n) is the smallest prime such that it and the previous four primes are all of the form x^2 + n * y^2.

Original entry on oeis.org

2633, 587, 1777, 2633, 239521, 862471, 2017, 208457, 586273, 147451, 4951, 586273, 207073, 612553, 102871, 208457, 301681, 351439, 242447, 2076901, 55948657, 27487, 119503, 9425257, 239521, 5188507, 128467, 75853, 74049413

Offset: 1

John L. Drost, May 22 2012

nonn

			a(7)=2017 since 2017 = 225 + 7*256, 2011 = 1444 + 7*81, 2003 = 1156 + 7*121, 1999 = 1936 + 7*9, and 1997 = 625 + 7*196 are all consecutive primes.

Table[again = True; lim = 10; While[again, lim2 = lim/Sqrt[n]; t = PrimePi[Select[Union[Flatten[Table[x^2 + n y^2, {x, 0, lim}, {y, 0, lim2}]]], # < lim^2 && PrimeQ[#] &]];  pos = Position[Partition[Differences[t], 4, 1], {1, 1, 1, 1}, 1, 1]; If[pos != {}, again = False; ans = Prime[t[[pos[[1, 1]] + 4]]], lim = 10*lim]]; ans, {n, 20}] (* T. D. Noe, May 23 2012 *)

A212604 a(n) is the smallest prime such that it and the previous three primes are all of the form x^2 + n * y^2.

Original entry on oeis.org

409, 577, 1759, 409, 55049, 1783, 127, 20873, 12889, 6529, 4943, 12889, 3461, 138041, 46411, 20873, 115013, 7417, 4919, 158209, 8490721, 7177, 15787, 4967401, 55049, 383393, 76597, 5273, 252541, 10448401, 2543, 577193

Offset: 1

John L. Drost, May 22 2012

nonn

			a(3)=1759 since 1759 = 676 + 3*361, 1753 = 25 + 3*576, 1747 = 1600 + 3*49, 1741 = 289 + 3*484 are all prime.

Table[again = True; lim = 10; While[again, lim2 = lim/Sqrt[n];  t = PrimePi[Select[Union[Flatten[Table[x^2 + n y^2, {x, 0, lim}, {y, 0, lim2}]]], # < lim^2 && PrimeQ[#] &]]; pos = Position[Partition[Differences[t], 3, 1], {1, 1, 1}, 1, 1] ; If[pos != {}, again = False; ans = Prime[t[[pos[[1,1]] + 3]]], lim = 10*lim]]; ans, {n, 20}] (* T. D. Noe, May 23 2012 *)

A212603 a(n) is the smallest prime such that it and the previous two primes are all of the form x^2 + n * y^2.

Original entry on oeis.org

101, 97, 163, 101, 3061, 1777, 113, 2617, 8353, 419, 4937, 8353, 3457, 34729, 8209, 2617, 53201, 2203, 4253, 12301, 54049, 991, 6803, 232801, 3061, 11491, 739, 2237, 32297, 68329, 857, 19801, 12853, 7411, 53299, 28081, 941, 14503, 20107, 88729, 23993, 23251

Offset: 1

John L. Drost, May 22 2012

nonn

			a(2)=97 since 97 = 25 + 2*36, 89 = 81 + 2*4, 83 = 81 + 2*1.

Table[again = True; lim = 10; While[again, lim2 = lim/Sqrt[n];  t = PrimePi[Select[Union[Flatten[Table[x^2 + n y^2, {x, 0, lim}, {y, 0, lim2}]]], # < lim^2 && PrimeQ[#] &]]; i = 1; While[i < Length[t] - 1 && (t[[i]] + 1 < t[[i + 1]] ||  t[[i+1]] + 1 < t[[i+2]]), i++]; If[i < Length[t] - 1, again = False; ans = Prime[t[[i+2]]], lim = 10*lim]]; ans, {n, 42}] (* T. D. Noe, May 23 2012 *)

A212602 a(n) is the smallest prime such that it and the previous prime are both of the form x^2 + n * y^2.

Original entry on oeis.org

17, 3, 37, 17, 409, 79, 11, 97, 673, 251, 53, 673, 17, 239, 211, 97, 353, 337, 23, 521, 1213, 97, 173, 4201, 409, 859, 439, 113, 937, 7369, 293, 2129, 7573, 569, 571, 673, 41, 1567, 997, 409, 1601, 337, 47, 401, 1801, 1783, 1867, 4201, 197, 499, 733, 1301

Offset: 1

John L. Drost, May 22 2012

nonn

			a(1)=17 since 17 = 4^2 + 1^2. 13 = 3^2 + 2^2 and these are the smallest consecutive primes that are the sum of two squares.

Table[again = True; lim = 10; While[again,lim2 = lim/Sqrt[n];  t = PrimePi[Select[Union[Flatten[Table[x^2 + n y^2, {x, 0, lim}, {y, 0, lim2}]]], # < lim^2 && PrimeQ[#] &]]; i = 1; While[i < Length[t] && t[[i]] + 1 < t[[i+1]], i++]; If[i < Length[t], again = False; ans = Prime[t[[i+1]]], lim = 10*lim]]; ans, {n, 60}] (* T. D. Noe, May 23 2012 *)

A178458 Abundant numbers of the form k^2 + 1.

Original entry on oeis.org

2210, 3250, 24650, 94250, 117650, 199810, 214370, 310250, 351650, 485810, 499850, 700570, 727610, 744770, 915850, 986050, 1306450, 1545050, 1814410, 1841450, 1885130, 2582450, 2699450, 3052010, 3583450, 4028050, 4678570, 5094050, 5257850, 6466850, 7059650, 7144930

Offset: 1

John L. Drost, Dec 22 2010

nonn

			2210 = 2*5*13*17, sigma(2210) = 4536 > 2*2210 and 47^2 + 1 = 2210.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A002522, A005101.

Select[Range[3000]^2 + 1, DivisorSigma[1, #] > 2# &]

A178050 n=x^2+17, n and n+2 are prime.

Original entry on oeis.org

17, 26261, 90017, 138401, 176417, 562517, 788561, 2022101, 2683061, 4743701, 5336117, 9622421, 11614481, 13927841, 21344417, 21734261, 22184117, 38192417, 59629301, 64448801, 68558417, 79923617, 82301201, 89302517, 90098081, 91814741, 95648417

Offset: 1

John L. Drost, Dec 16 2010

nonn

			17=0^2+17, and 19 are prime.
26261=162^2+17 and 26263 are prime.

Subsequence of A228244.

isok(n) = isprime(n) && isprime(n+2) && issquare(n-17); \\ Michel Marcus, Aug 18 2013

a(26)-a(27) from Michel Marcus, Aug 18 2013

A178066 Primes p of the form x^2+59, such that p+2 is also prime.

Original entry on oeis.org

59, 108959, 176459, 4040159, 5904959, 10497659, 25401659, 26625659, 38192459, 89302559, 105884159, 117288959, 155750459, 156500159, 228614459, 251856959, 306950459, 432224159, 491508959, 508953659, 624500159, 682776959, 934524959, 1092963659, 1106892959

Offset: 1

John L. Drost, Dec 16 2010

All sequence members end in '59' since x must be divisible by 10 (30 actually).

			59 = 0^2+59, 59 and 61 are prime.
108959 = 330^2+59, 108959 and 108961 are prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Select[(30*Range[0,2000])^2+59,PrimeQ[#]&&PrimeQ[#+2]&] (* Harvey P. Dale, Apr 14 2012 *)

More terms from Harvey P. Dale, Apr 14 2012

New name from Jason Yuen, Sep 01 2025

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User: John L. Drost

A329963 Numbers k such that sigma(k) is not divisible by 3.

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A287298 a(n) is the largest square with distinct digits in base n.

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A225888 Primes p such that neither 2 nor 3 are primitive roots, but together 2 and 3 generate the nonzero residues mod p.

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A212605 a(n) is the smallest prime such that it and the previous four primes are all of the form x^2 + n * y^2.

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A212604 a(n) is the smallest prime such that it and the previous three primes are all of the form x^2 + n * y^2.

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A212603 a(n) is the smallest prime such that it and the previous two primes are all of the form x^2 + n * y^2.

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A212602 a(n) is the smallest prime such that it and the previous prime are both of the form x^2 + n * y^2.

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Mathematica

A178458 Abundant numbers of the form k^2 + 1.

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A178050 n=x^2+17, n and n+2 are prime.

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