A102216 -id:A102216 - cp's OEIS frontend

A102615 Nonprime numbers of order 2.

1, 8, 10, 14, 15, 16, 20, 22, 24, 25, 27, 30, 32, 33, 35, 36, 38, 39, 40, 44, 46, 48, 49, 50, 51, 54, 55, 56, 58, 62, 63, 64, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 82, 85, 86, 87, 88, 90, 92, 93, 94, 96, 99, 100, 102, 104, 105, 108, 110, 111, 114, 115, 116, 117, 118, 120

Offset: 1

Cino Hilliard, Jan 31 2005

nonn

nps(n,0) -> list nonprime(n) or the sequence of nonprime numbers. nps(n,1) -> list nonprime(nonprime(n)) or nps of order 1 nps(n,2) -> list nonprime(nonprime(nonprime(n))) or nps of order 2 ..... The order is the number of nestings - 1. We avoid the nestings in the script with a loop.

Nonprimes (A018252) with nonprime (A018252) subscripts. a(n) U A078782(n) = A018252(n), a(n+1) U A175250(n) = A018252(n) for n >= 1. a(n) = nonprime(nonprime(n)) = A018252(A018252(n)). a(4) = 14 because a(4) = b(b(4)) = b(8) = 14, b = nonprime. a(1) = 1, a(n) = nonprimes (A018252) with composite (A002808) subscripts for n >=2. [Jaroslav Krizek, Mar 13 2010]

			Nonprime(2) = 4.
Nonprime(4) = 8 the second entry.

Cf. A018252.

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270796, A102216.

# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622.  - N. J. A. Sloane, Mar 30 2016

nonPrime[n_] := FixedPoint[n + PrimePi[ # ] &, n]; Nest[nonPrime, Range[66], 2] (* Robert G. Wilson v, Feb 04 2005 *)

\We perform nesting(s) with a loop. cics(n,m) = { local(x,y,z); for(x=1,n, z=x; for(y=1,m+1, z=composite(z); ); print1(z",") ) } composite(n) = \ The n-th composite number. 1 is defined as a composite number. { local(c,x); c=1; x=0; while(c <= n, x++; if(!isprime(x),c++); ); return(x) }

Edited by Robert G. Wilson v, Feb 04 2005

A078782 Nonprimes (A018252) with prime (A000040) subscripts.

Original entry on oeis.org

4, 6, 9, 12, 18, 21, 26, 28, 34, 42, 45, 52, 57, 60, 65, 74, 81, 84, 91, 95, 98, 106, 112, 119, 128, 133, 135, 141, 143, 147, 165, 170, 177, 180, 192, 195, 203, 209, 214, 220, 228, 231, 244, 246, 250, 253, 267, 284, 288, 290, 295, 301, 303, 316, 323, 329, 336

Offset: 1

Joseph L. Pe, Jan 09 2003

a(n) = A018252(A000040(n)). Subsequence of A175250 (nonprimes (A018252) with noncomposite (A008578) subscripts), a(n) = A175250(n+1). a(n) U A102615(n) = A018252(n). [From Jaroslav Krizek, Mar 13 2010]

			a(4) = nonprime(prime(4)) = nonprime(7) = 12.

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270796, A102216.

from sympy import prime, composite
def A078782(n): return composite(prime(n)-1) # Chai Wah Wu, Nov 13 2024

Corrected by Jaroslav Krizek, Mar 13 2010

A102617 Primes p(n) such that n is a second-order nonprime number.

Original entry on oeis.org

2, 19, 29, 43, 47, 53, 71, 79, 89, 97, 103, 113, 131, 137, 149, 151, 163, 167, 173, 193, 199, 223, 227, 229, 233, 251, 257, 263, 271, 293, 307, 311, 317, 337, 347, 349, 359, 379, 383, 389, 397, 409, 421, 439, 443, 449, 457, 463, 479, 487, 491, 503, 523, 541

Offset: 1

Cino Hilliard, Jan 31 2005

nonn

The prime/nonprime compound sequence ABB. - N. J. A. Sloane, Apr 06 2016

			Nonprime(4) = 8.
The 8th prime is 19, the second entry.

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270796, A102216.

For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622. - N. J. A. Sloane, Mar 30 2016

nonPrime[n_Integer] := FixedPoint[n + PrimePi[ # ] &, n]; Prime /@ nonPrime /@ nonPrime /@ Range[54] (* Robert G. Wilson v, Feb 04 2005 *)

\We perform nesting(s) with a loop. cips(n,m) = { local(x,y,z); for(x=1,n, z=x; for(y=1,m+1, z=composite(z); ); print1(prime(z)",") ) } composite(n) = \ The n-th composite number. 1 is defined as a composite number. { local(c,x); c=1; x=0; while(c <= n, x++; if(!isprime(x),c++); ); return(x) }

Edited by Robert G. Wilson v, Feb 04 2005

A270792 The prime/nonprime compound sequence ABA.

Original entry on oeis.org

7, 13, 23, 37, 61, 73, 101, 107, 139, 181, 197, 239, 269, 281, 313, 373, 419, 433, 467, 499, 521, 577, 613, 653, 719, 751, 761, 811, 823, 853, 977, 1013, 1051, 1069, 1163, 1187, 1237, 1289, 1307, 1373, 1439, 1453, 1549, 1559, 1583

Offset: 1

N. J. A. Sloane, Mar 30 2016

nonn

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270796, A102216.

# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622.  - N. J. A. Sloane, Mar 30 2016

A270794 The prime/nonprime compound sequence BAA.

Original entry on oeis.org

6, 9, 18, 26, 45, 57, 81, 91, 112, 143, 165, 203, 228, 244, 267, 303, 345, 354, 411, 437, 454, 495, 530, 564, 623, 668, 687, 714, 728, 749, 856, 893, 931, 959, 1032, 1054, 1104, 1158, 1185, 1233, 1268, 1298, 1372, 1392, 1425, 1445, 1539, 1672, 1698, 1714, 1742, 1773, 1802, 1886, 1914, 1966, 2031, 2050, 2104

Offset: 1

N. J. A. Sloane, Mar 30 2016

nonn

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270796, A102216.

# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622.  - N. J. A. Sloane, Mar 30 2016

A270796 The prime/nonprime compound sequence BBA.

Original entry on oeis.org

8, 10, 15, 20, 27, 32, 38, 40, 49, 58, 63, 72, 78, 82, 88, 99, 110, 114, 121, 125, 129, 140, 146, 155, 166, 172, 175, 183, 185, 189, 212, 217, 225, 230, 245, 248, 258, 265, 272, 279, 289, 292, 306, 309, 315, 319, 334, 355, 360, 362, 368, 375, 377, 393, 402, 408, 416, 420, 427, 435, 438, 452, 473, 478, 482, 486, 507

Offset: 1

N. J. A. Sloane, Mar 30 2016

nonn

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270796, A102216.

# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622.  - N. J. A. Sloane, Mar 30 2016

A102217 3-Suzanne numbers; composite multiples of 3 whose sum of prime factors with multiplicity is a multiple of 3.

Original entry on oeis.org

9, 24, 27, 42, 60, 72, 78, 81, 105, 114, 126, 132, 150, 180, 186, 192, 195, 204, 216, 222, 231, 234, 243, 258, 276, 285, 315, 330, 336, 342, 348, 357, 366, 375, 378, 396, 402, 429, 438, 450, 465, 474, 480, 483, 492, 510, 540, 555, 558, 564, 576, 582, 585

Offset: 1

Eric W. Weisstein, Dec 30 2004

nonn

Composite numbers k such that the sum of digits of k (A007953) and the sum of sums of digits of the prime factors of k (taken with multiplicity, A118503) are both divisible by 3. - Amiram Eldar, Apr 23 2021

The new secondary definition is equal to the original because taking the decimal digit sum preserves congruence modulo 3. This is a multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Jun 08 2024

			From _Antti Karttunen_, Jun 08 2024: (Start)
42 = 2*3*7 is a term as it is a multiple of 3, and also 2+3+7 = 12 is a multiple of 3.
60 = 2*2*3*5 is a term is it is a multiple of 3, and also 2+2+3+5 = 12 is a multiple of 3.
(End)

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384.
James J. Tattersall, Elementary Number Theory in Nine Chapters, 2nd ed., Cambridge University Press, 2005, p. 93.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Michael Smith, Cousins of Smith Numbers: Monica and Suzanne Sets, Fibonacci Quarterly, Vol. 34, No. 2 (1996), pp. 102-104.
Eric Weisstein's World of Mathematics, Suzanne Set.

Subsequence of A177927.
Intersection of A008585 and A289142 without the initial 3.
Positions of multiples of 3 in A082299, after A082299(3).
Cf. A001414, A007953, A018252, A102216, A118503, A373371.

s[n_] := Plus @@ IntegerDigits[n]; f[p_, e_] := e*s[p]; sp[n_] := Plus @@ f @@@ FactorInteger[n]; suz3Q[n_] := CompositeQ[n] && And @@ Divisible[{s[n], sp[n]}, 3]; Select[Range[600], suz3Q] (* Amiram Eldar, Apr 23 2021 *)

isA102217(n) = if(n<=3 || (n%3), 0, my(f=factor(n)); 0==(sum(i=1, #f~, f[i, 2]*sumdigits(f[i, 1]))%3)); \\ Antti Karttunen, Jun 08 2024

isA102217(n) = (n>3 && !(n%3) && A373371(n)); \\ Antti Karttunen, Jun 08 2024

a(n) = 3*A289142(1+n). - Antti Karttunen, Jun 08 2024

Alternative definition added and keyword:base removed by Antti Karttunen, Jun 08 2024

A102616 Nonprime numbers of order 3.

Original entry on oeis.org

1, 14, 16, 22, 24, 25, 30, 33, 35, 36, 39, 44, 46, 48, 50, 51, 54, 55, 56, 62, 64, 66, 68, 69, 70, 75, 76, 77, 80, 85, 86, 87, 90, 92, 93, 94, 96, 100, 102, 104, 105, 108, 111, 115, 116, 117, 118, 120, 122, 123, 124, 126, 130, 132, 134, 136, 138, 142, 144, 145, 148, 150

Offset: 1

Cino Hilliard, Jan 31 2005

nonn

nps(n,1) -> list nonprime(n) or the sequence of nonprime numbers. nps(n,2) -> list nonprime(nonprime(n)) or nps of order 2. nps(n,3) -> list nonprime(nonprime(nonprime(n))) or npcs of order 3 ..... The order is the number of nestings - 1.

			Nonprime(2) = 4.
Nonprime(4) = 8.
Nonprime(8) = 14, the 2nd entry.

Cf. A018252, A102615.

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270796, A102216.

# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622. - N. J. A. Sloane, Mar 30 2016

nonPrime[n_] := FixedPoint[n + PrimePi[ # ] &, n]; Nest[ nonPrime, Range[62], 3] (* Robert G. Wilson v, Feb 04 2005 *)

\\ We perform nesting(s) with a loop.
cics(n,m) = { local(x,y,z); for(x=1,n, z=x; for(y=1,m+1, z=nonprime(z); ); print1(z",") ) }
nonprime(n) = { local(c,x); c=1; x=0; while(c <= n, x++; if(!isprime(x),c++); ); return(x) }

Edited by Robert G. Wilson v, Feb 04 2005

A270795 The prime/nonprime compound sequence BAB.

Original entry on oeis.org

4, 12, 21, 28, 34, 42, 52, 60, 65, 74, 84, 95, 98, 106, 119, 128, 133, 135, 141, 147, 170, 177, 180, 192, 195, 209, 214, 220, 231, 246, 250, 253, 284, 288, 290, 295, 301, 316, 323, 329, 336, 339, 351, 365, 382, 387, 390, 394, 417, 429, 432, 445, 462, 470, 474, 481, 490, 505, 516, 518, 532, 538, 543, 550, 559, 566

Offset: 1

N. J. A. Sloane, Mar 30 2016

nonn

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270796, A102216.

# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622.  - N. J. A. Sloane, Mar 30 2016

A102218 2-Monica numbers.

Original entry on oeis.org

4, 8, 10, 12, 14, 15, 22, 26, 27, 35, 42, 44, 45, 54, 56, 58, 60, 62, 63, 64, 65, 68, 78, 84, 85, 88, 90, 92, 94, 96, 99, 102, 108, 111, 118, 119, 121, 122, 123, 126, 129, 133, 136, 138, 141, 143, 145, 152, 155, 158, 159, 160, 161, 164, 165, 166, 169, 174, 175

Offset: 1

Eric W. Weisstein, Dec 30 2004

Composite numbers k such that the difference between the sum of digits of k (A007953) and the sum of sums of digits of the prime factors of k (taken with multiplicity, A118503) is even. - Amiram Eldar, Apr 23 2021

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384.
James J. Tattersall, Elementary Number Theory in Nine Chapters, 2nd ed., Cambridge University Press, 2005, p. 93.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Michael Smith, Cousins of Smith Numbers: Monica and Suzanne Sets, Fibonacci Quarterly, Vol. 34, No. 2 (1996), pp. 102-104.
Eric Weisstein's World of Mathematics, Monica Set.

Cf. A007953, A018252, A102216, A102219, A118503, A177927.

s[n_] := Plus @@ IntegerDigits[n]; f[p_, e_] := e*s[p]; sp[n_] := Plus @@ f @@@ FactorInteger[n]; mon2Q[n_] := CompositeQ[n] && EvenQ[s[n] - sp[n]]; Select[Range[200], mon2Q] (* Amiram Eldar, Apr 23 2021 *)

cp's OEIS Frontend

A102615 Nonprime numbers of order 2.

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A078782 Nonprimes (A018252) with prime (A000040) subscripts.

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A102617 Primes p(n) such that n is a second-order nonprime number.

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A270792 The prime/nonprime compound sequence ABA.

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A270794 The prime/nonprime compound sequence BAA.

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A270796 The prime/nonprime compound sequence BBA.

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A102217 3-Suzanne numbers; composite multiples of 3 whose sum of prime factors with multiplicity is a multiple of 3.

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A102616 Nonprime numbers of order 3.

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